Real Computability and Hypercomputation

Real Computability and Hypercomputation Martin Ziegler December 20, 2007

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Summary: Turing machines are generally agreed to be the appropriate mathematical idealization describing the capabilities of actual digital computers. With regard to computations on real numbers (being continuous rather than digital objects), two major extensions have become well established: • Recursive Analysis considers computation on reals in terms of infinite approximations by fractions of integers. • BCSS machines (‘algebraic model’) perform finitely many arithmetic operations on real numbers exactly. However, doubts about the validity of the so-called Church-Turing Hypothesis have led to the study of • notions of (discrete) computability that go beyond the Turing machine, so-called hypercomputers. The present work surveys the three extensions, investigating and comparing the computational power of their various combinations. I report on external as well as on own work. Concerning the latter, Section 2.3.4 is roughly based on [26, §3.1], Section 2.4.5 on [27, Section 5], Section 3.3.3 on [24], Section 5.3 on [22], Section 5.5 on [30], Section 4.3 on [28], Section 4.4 on [29], and Section 4.4.4 on [33, Section 3.2].

Acknowledgements: Most of this research has been performed as Project Zi 1009/1-1 supported by the German Research Foundation (DFG); I am grateful to the department of Electrical Engineering, Computer Science, and Mathematics of the University of ¨rgisser and Meyer Paderborn and particularly to Professors Bu auf der Heide for they kind hospitality, advice, and latitude. First contributions had been obtained while working at the department of Mathematics and Computer Science (IMADA) under host Klaus Meer with project 21-04-0303 of the Statens Naturvidenskabelige Forskningsr˚ ad (SNF). Another major part, generously funded by JSPS scholarship PE 05501, was developed during an inspiring stay at the Japan Advanced Institute of Science and Technology (JAIST) under host Ishihara Hajime.

Contents 1 Introduction 1.1 Computability . . . . . . . . . . . . . . . . . . . . 1.1.1 Basics . . . . . . . . . . . . . . . . . . . . . 1.1.2 Applications . . . . . . . . . . . . . . . . . 1.2 Models of Computation . . . . . . . . . . . . . . . 1.3 Computational Complexity . . . . . . . . . . . . . 1.4 Church–Turing Hypothesis . . . . . . . . . . . . . . 1.5 Physics of Computation . . . . . . . . . . . . . . . 1.5.1 Theoretical Physics and Computer Science 1.5.2 Meta-Theory of Physics . . . . . . . . . . . 1.5.3 Constructivism in Physics . . . . . . . . . . 1.6 General Outline . . . . . . . . . . . . . . . . . . . .

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9 9 9 10 11 11 12 12 13 13 13 14

2 Real Computation 2.1 Meta-Mathematical Remarks . . . . . . . . . . . . 2.2 Real Numbers . . . . . . . . . . . . . . . . . . . . . 2.2.1 Algebra . . . . . . . . . . . . . . . . . . . . 2.2.2 Topology . . . . . . . . . . . . . . . . . . . 2.2.3 Continuity and Calculus . . . . . . . . . . . 2.2.4 Semi-Algebraic Geometry . . . . . . . . . . 2.3 BCSS Machines . . . . . . . . . . . . . . . . . . . . 2.3.1 BCSS Recursion Theory . . . . . . . . . . . 2.3.2 Uncomputability . . . . . . . . . . . . . . . 2.3.3 Computability . . . . . . . . . . . . . . . . 2.3.4 Use of Real Constants . . . . . . . . . . . . 2.3.5 Quantifier Elimination . . . . . . . . . . . . 2.4 Recursive Analysis . . . . . . . . . . . . . . . . . . 2.4.1 Binary Computability . . . . . . . . . . . . 2.4.2 Recursive Real Functions . . . . . . . . . . 2.4.3 Semi–Computability and Semi–Decidability 2.4.4 Enumerable Real Open Sets . . . . . . . . . 2.4.5 Gain and Vain of Non-Uniformity . . . . . 2.4.6 Type-2 Theory of Effectivity . . . . . . . . 2.5 BCSS versus/with Recursive Analysis . . . . . . . 2.5.1 Variations of BCSS Machines . . . . . . . . 2.5.2 Aspects of Classification . . . . . . . . . . . 2.5.3 Analytic Machines . . . . . . . . . . . . . . 2.6 Further Models . . . . . . . . . . . . . . . . . . . . 2.6.1 Domain Theory . . . . . . . . . . . . . . . . 2.6.2 Constructive Mathematics . . . . . . . . . . 2.6.3 Markov Computability . . . . . . . . . . . . 2.6.4 Analog Computation . . . . . . . . . . . . .

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15 15 16 16 18 20 20 21 22 23 26 27 28 30 30 31 33 34 35 36 39 39 40 40 41 42 42 43 44

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CONTENTS

2.7 2.8

2.6.5 Exact Computation Paradigm . . . . . . . . . . . . . . . . . . . . . . . . . . Practice and Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . My Manifesto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Hypercomputation 3.1 Oracle Turing Machines . . . . . . . . . . . . . . . 3.1.1 Power of the Halting Oracle . . . . . . . . . 3.1.2 Arithmetical Hierarchy . . . . . . . . . . . . 3.1.3 Hyperarithmetical and Analytical Hierarchy 3.1.4 Post’s Problem . . . . . . . . . . . . . . . . 3.2 Physical Hypercomputers . . . . . . . . . . . . . . 3.2.1 Relativistic Hypercomputation . . . . . . . 3.2.2 Quantum Mechanical Hypercomputation . 3.2.3 Hypercomputers in Classical Physics . . . . 3.2.4 N-Body Problem . . . . . . . . . . . . . . . 3.3 Further Discrete Hypercomputers . . . . . . . . . . 3.3.1 Infinite Time . . . . . . . . . . . . . . . . . 3.3.2 Fair Nondeterminism . . . . . . . . . . . . . 3.3.3 Infinite Parallelism . . . . . . . . . . . . . . 3.4 Critique of Hypercomputing . . . . . . . . . . . . . 3.4.1 Harnessability . . . . . . . . . . . . . . . . . 3.4.2 Practical Feasibility . . . . . . . . . . . . . 3.4.3 Relevance . . . . . . . . . . . . . . . . . . . 3.5 Real Hypercomputation . . . . . . . . . . . . . . .

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Real Hypercomputation∗

II

4 Arithmetical Hierarchies in Recursive Analysis 4.1 Effective Borel Hierarchy . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Transfinite Levels . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Representing the Borel Hierarchy . . . . . . . . . . . . . . . 4.2 Arithmetical Hierarchy of Real Numbers . . . . . . . . . . . . . . . 4.2.1 Some Relations to the Effective Borel Hierarchy . . . . . . 4.2.2 Transfinite Levels . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Hypercomputing Real Functions . . . . . . . . . . . . . . . . . . . 4.3.1 Weak Function Evaluation and Continuity . . . . . . . . . . 4.3.2 Hierarchy of Weakly Evaluable Functions . . . . . . . . . . 4.3.3 Relative Weierstraß Computability versus Weak Evaluation 4.3.4 Weak Function Evaluation and Borel’s Hierarchy . . . . . . 4.3.5 Are Discontinuous Real Functions Hyper computable? . . . 4.3.6 Markov with Oracles . . . . . . . . . . . . . . . . . . . . . . 4.4 Revising Type-2 Computation . . . . . . . . . . . . . . . . . . . . . 4.4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 The Jump of a Representation . . . . . . . . . . . . . . . . 4.4.3 Examples of Known Jumps . . . . . . . . . . . . . . . . . . 4.4.4 Jump of Set Representations θ< and ψ> . . . . . . . . . . . 4.4.5 Iterated Jumps and Applications . . . . . . . . . . . . . . . 4.5 Type-2 Nondeterminism . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Computational Power on Finite Inputs . . . . . . . . . . . . 4.5.2 Computational Power on Infinite Inputs . . . . . . . . . . . ∗A

very brief summary of this part (extended abstract) has appeared as [32]

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CONTENTS

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5 BCSS Hypercomputation 5.1 Machines with Additional Operations . . . . . . . . . . . . . . . . . . . . . 5.1.1 . . . and Nondeterminism . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Machines with Oracles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Relativized BCSS Recursion Theory . . . . . . . . . . . . . . . . . . 5.2.2 Relative Function Computability . . . . . . . . . . . . . . . . . . . . 5.3 Post’s Problem over the Reals . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Embedding any Countable Lattice as BCSS Degrees . . . . . . . . . 5.3.2 The Case of Linear BCSS Machines . . . . . . . . . . . . . . . . . . 5.4 Arithmetical Hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Completeness and Other Applications of the Normal Form Theorem 5.4.2 Nondeterministic and Analytic Hierarchy . . . . . . . . . . . . . . . 5.5 Real Computational Universality . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Word Problem for Groups . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Classical Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Presenting Real Groups . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.4 Examples of Presented Real Groups . . . . . . . . . . . . . . . . . . 5.5.5 Reducibility to the Real Halting Problem . . . . . . . . . . . . . . . 5.5.6 Basics from Group Theory and Their Presentations . . . . . . . . . 5.5.7 First Effectivity Considerations . . . . . . . . . . . . . . . . . . . . . 5.5.8 Benign Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.9 Reduction from the Real Halting Problem . . . . . . . . . . . . . . .

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105 105 106 106 107 108 109 111 112 114 116 118 118 119 119 120 121 123 124 125 127 129

6 Concluding Comparison 6.1 Discrete versus Real Hypercomputation . . . . . . . . . . 6.2 Superrecursive Analysis versus BCSS Hypercomputers . . 6.2.1 Comparing their Arithmetical Hierarchies . . . . . 6.2.2 Effect of Non-/Uniformity . . . . . . . . . . . . . . 6.3 BCSS to Hyperrecursive Analysis: Equality is a Jump . . 6.3.1 Open BCSS Semi-Decidable Subsets . . . . . . . . 6.3.2 Continuous BCSS-Computable Functions . . . . . 6.4 Robust Quasi-Strongly δ–Q–Analytic Functions . . . . . . 6.4.1 . . . and BCSS–Computation or Recursive Analysis 6.4.2 . . . and Type-2 Nondeterminism . . . . . . . . . . . 6.4.3 Finitely Revising Type-2 Computation . . . . . . . 6.4.4 Finitely Revising Real Representation . . . . . . . 6.5 Omissions . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Bibliography 140 Own Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Index

155

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CONTENTS

List of Figures 2.1 2.2 2.3

Simple BCSS Algorithm unrolled into a directed acyclic graph and the set it decides 24 Heaviside’s function h and its flipped variant h = 1 − h . . . . . . . . . . . . . . . 32 Relations between some Analytic Machines . . . . . . . . . . . . . . . . . . . . . . 41

3.1 3.2

Infinite comb with a wedge to probe . . . . . . . . . . . . . . . . . . . . . . . . . . Sample Initial and Two Successor Configurations of Life. . . . . . . . . . . . . . . .

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11

Classical (bold-face) and effective (light-face) Borel Hierarchy . . . . . . . . . . . . 66 The sets Uz defined in the proof of Proposition 4.7 . . . . . . . . . . . . . . . . . . 68 Illustration of the iterative construction employed in the proof of Claim 4.27 . . . . 75 Illustration of the iterative construction employed in the proof of Claim 4.28 . . . . 76 The first infinitely long iterative construction employed in the proof of Claim 4.29 77 Second infinitely long iterative construction employed in the proof of Claim 4.29 . 77 Further infinitely long iterative constructions employed in the proof of Claim 4.29 . 78 Nesting of some rational intervals of dyadic length contained in f −1 (y − , ∞) . . 81 Piecewise linear and smooth unit pulse, and a non-overlapping superposition . . . 84 Converting a sequence of stabilizing sequences into a sequence with revocations. . . 94 Models of Computation in Chomsky’s Hierarchies over finite/infinite strings . . . . 100

5.1

Correspondence between Classical Discrete and New Real Group Presentations . . 121

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LIST OF FIGURES

Chapter 1

Introduction Complexity is fundamental to theoretical computer science. For a problem P , it deals with the question: How expensive (in terms of resources like memory or, primarily, time) is a computational solution to P ? However, logically, the question of computability comes before that of complexity: Does P admit of a computational solution at all? Many problems are clearly computable (e.g. by means of exhaustive search); but for others, computability is not obvious or even fails.

1.1

Computability

Historically, computability was not considered an issue at all for a long time. More precisely, in 1900 David Hilbert, as number 10 of his famous list of mathematical problems, asked for a ‘procedure’ (which would nowadays be termed an algorithm) deciding the validity of number theoretical propositions, implying that such an algorithm should ‘obviously’ exist. This common belief, and in fact the entire implicit belief in the foundations of mathematics at that time expressed ¨ del’s Incompleteness Theorem, in Hilbert’s Program, was blown to pieces in 1931 with Kurt Go revealing that even the theory of integers allows propositions which can be neither proven nor refuted, not to mention decided in an automated way. For a problem which is either true or false mathematically but undecidable to a computer, in 1936 Alan Turing introduced what has become known as the Turing Machine1 (TM) and proved it incapable of deciding the termination of another given TM—the famous Halting Problem H. Although this result is considered the dawn of computability theory, one should not forget that at that time ‘computer’ referred to a professional person, the first computing machines in the present sense having been built only in the 1940s. Still, the Turing Machine is nowadays employed as a model less of a human mathematician than of automated digital computing devices, see Section 1.2.

1.1.1

Basics

A language (also called a problem) is a subset L of either the integers N or the set {0, 1}∗ of finite binary strings. It is decidable (or recursive) if a Turing machine can, upon input of x, within finite time (terminate and) determine whether x belongs to L or not. L is semi–decidable if a Turing machine can terminate on inputs x ∈ L and diverge for x 6∈ L. L is recursively enumerable (commonly abbreviated as “r.e.”) if a Turing machine is able to output in arbitrary order all elements of L. If the complement of L is r.e., L is called co–r.e. 1 Regarding

a formal definition, refer to any introductory text.

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CHAPTER 1. INTRODUCTION

With the important technique of dove-tailing (basically the interleaved simulation of countably many computations), semi–decidability can be shown to be equivalent to recursive enumerability; and recursiveness to both r.e. and co-r.e. Another crucial ingredient is an effective enumeration of all TMs: there exists a universal Turing machine U which, given an encoding of a TM M and input x, can simulate M on x. The code hM i of M is called its G¨ odel index , the mapping M 7→ hM i a G¨ odelization. Several codes and inputs may be combined into, and extracted back from, a single integer by virtue of a pairing function like for instance N × N 3 (x, y) 7→ hx, yi := 2x · (2y + 1) .

(1.1)

A given machine M and input x can be recoded into some M 0 which already contains x and, more precisely, ignores its own input x0 but always behaves like M on x. Slightly more generally, the SMN Theorem asserts that any (code of a) TM realizing a partial function (x, y) 7→ f (x, y) and a given x0 may be converted into (the code of) a machine realizing y 7→ f (x0 , y). The reverse conversion is known as the UTM Theorem, namely from a mapping x 7→ hMx i to the mapping hx, y 7→ fx (y), where fx denotes the partial function realized by TM Mx . SMN and UTM together permit effective adaptation of the arity of functions under consideration and are sometimes referred to as type conversion. They also imply that a Gödelization must enumerate every computable function infinitely often: fx1 = fx2 = . . . for some increasing sequence x1 , x2 , . . .; cf. e.g. [Soar87, Padding Lemma I.3.3]. Now suppose that the Halting problem H = hM, xi : M terminates on input x

(1.2)

were decidable. Its complement could then be semi-decided. This in turn would imply that the diagonal problem D = hM i : M does not terminate on hM i (1.3) could be accepted by some machine M0 . Now either hM0 i ∈ D or hM0 h6∈ D holds. In the first case, by the definition of D, M0 does not terminate on input hM0 i—contradicting that M0 accepts D, i.e. terminates exactly on inputs from D like hM0 i. Similarly, the second case hM0 i 6∈ D means divergence of M0 on input hM0 i and implies hM0 i ∈ D: again, this is a contradiction. The restricted (and thus seemingly easier) problem hM i : M terminates on empty input is sometimes also referred to as H. In fact each can be reduced to the other by virtue of the UTM theorem and are thus equally undecidable.

1.1.2

Applications

The easy part of the Halting Problem is of course verifying that a TM will halt; H is semidecidable. The difficulty lies in identifying that a machine will continue running forever—which one can never be sure of within finite time. Were that possible, many important open mathematical claims (Goldbach Conjecture, Riemann Hypothesis etc.) could be settled just by setting up a TM M to look for a counter example and checking whether this search eventually succeeds, that is, whether M terminates. One generally expects that these open questions may ultimately be solved without having to resort to the Halting Problem but by mathematical reasonings (and the insights they provide). This has been achieved, e.g. for Fermat’s Last Theorem [Wile95]. However, according to Rice’s Theorem, many other problems can provably be solved only in connection with H. Automated software verification, for instance, as desirable as it may be, is principally infeasible in that even the simplest requirement of correct software—termination upon empty input—cannot be decided algorithmically. Since Turing’s start, a rich variety of further problems has been proven undecidable; to mention two rather famous ones:

1.2. MODELS OF COMPUTATION

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• the Word Problem for finitely presented groups [Novi59, Boon58]; • the integral solvability of Diophantine Equations (also known as Hilbert’s Tenth Problem) [Mati70]. These exhibit serious principal limitations of computational group theory and number theory. Their real counterparts will be among the topics of the present work in Sections 5.3 and 2.3.5, respectively. A positive result in computability theory has recently been awarded the Gödel Prize 2002: • The equivalence of linear pushdown automata can be decided by a Turing machine [Seni01].

1.2

Models of Computation

A formal abstraction of a computer M is called a model of computation. It describes in sound mathematical terms which operations M is capable of (and what cost they involve in terms of resources). Elevator controls or digital watches, for instance, are conveniently modeled by finite automata. Algorithms often refer to an underlying model only implicitly. With regard to operations on bits, for example, the TM is generally accepted as the appropriate model. Euclid’s algorithm for calculating greatest common divisors, on the other hand, operates natively by arithmetic on integers, which is more naturally reflected by the Random Access Machine (RAM) model of computation or a von Neumann Machine. Regarding computations involving real numbers, Gaussian Elimination is an example of an algorithm pertaining to the BCSS model, while the Ellipsoid Method exemplifies a model commonly employed in Computable Analysis; see Chapter 2. A problem’s complexity, as well as its computability, in general largely depend on the model of computation under consideration. The latter should thus be chosen carefully so as to acutely reflect actual computers’ capabilities. Too accurate a model, on the other hand, may be mathematically awkward and intractable to analysis. In any case, every model of computation is an idealization reflecting only some aspects of reality while neglecting others: The Halting Problem for a bounded memory computer (e.g. a nostalgic Commodore C64 or even a state-of-the-art 4GB iMac) is that of a finite automaton and thus decidable by a Turing machine, albeit at a ridiculously high cost in terms of running time.

1.3

Computational Complexity

The complexity of a calculation or decision problem P , subject to a model of computation, is the least cost incurred by any algorithmic solution. It thus describes the inherent difficulty of P and allows the performance of a specific algorithm solving P to be assessed. Since there may be an infinite number of solution procedures, the word ‘least’ mathematically corresponds to an infimum, meaning that optimal cost need not be attainable. This holds even when constant factors are neglected as in asymptotic big-Oh notation. Constants can also be quite sensitive to minor modifications of the model of computation under consideration (For example, on a Turing machine, does a combined bit read/write and head move take one, two, or three steps?) and are therefore best ignored anyway. Determining the complexity of P amounts to a) devising algorithms for it and analyzing their respective costs: these constitute upper complexity bounds; b) complementing the latter by (ultimately matching) lower bounds, that is, formal proofs establishing that, within the given model, every algorithmic solution of P will incur a cost (asymptotically) at least as high as in a).

12


The uncomputability of P can thus be regarded as an ultimate form of lower bound: the problem is intractable to algorithmic solution regardless of the allowed cost. Computability on the other hand, that is the construction (or merely the proof of existence) of an algorithm to its solution, precedes and opens the door for further complexity considerations.

1.4

Church–Turing Hypothesis

The Turing machine is nowadays generally employed as the model describing the principal capabilities of digital computers. More precisely, it is widely believed that “every function which would naturally be regarded as computable can be computed by a Turing Machine.” This has become known as the Church–Turing Hypothesis and is supported by a) the continuous failure to come up with a device capable of solving, say, the Halting Problem; (In fact, all computers realized so far, from Zuse’s Z3 to a MacBook pro , are basically (extremely fast but still) merely Turing Machines.) b) the successful simulation of a vast variety of physical (and in particular of computing) systems on a Turing machine; c) formal proofs that several other reasonable yet seemingly unrelated models of computation (e.g. µ–recursion, WHILE-programs, λ–calculus) are in fact as powerful as the Turing machine. Nevertheless, it should be kept in mind that the Church–Turing Hypothesis itself has not been and cannot be proven, simply because it is far too informal, with ambiguous notions like ‘naturally’. As a matter of fact, neither Alonzo Church nor Alan Turing actually put forward a claim as bold as the one above, and current science more carefully distinguishes various variants of this hypothesis; see e.g. [Ord02, Section 2.2] or [Cope02]. For example, the Physical Church–Turing Hypothesis claims that b) above generalizes: “Every physical process can be simulated by a Turing machine.” The Strong or Extended Church–Turing Hypothesis also includes complexity considerations in addition to computability, claiming that “any ‘reasonable’ model of computation can be efficiently simulated on a probabilistic Turing machine.”

1.5

Physics of Computation

Both ambiguities, the notion of ‘the’ Church–Turing Hypothesis (while there are several of them) and the use of terms like ‘natural’ and ‘reasonable’, contribute to intense (and heated) disputes about their validity. Moreover, the involvement of both computer science and physics in the hypothesis increases the sources of possible misunderstandings. Still, there seems to be a (slowly) growing consensus that the Physical Church–Turing Hypothesis is too bold and to be considered as wrong. For instance it has been pointed out [Gero86, p.546] that the theory of Quantum Gravitation (compare also Section 3.2.2) contains a sum ranging over non-isomorphic simplicial complexes, yet the isomorphism of 4-dimensional simplicial complexes is known to be undecidable by reduction to the Word Problem for groups (Markov 1958); see also Section 5.5.

1.5. PHYSICS OF COMPUTATION

1.5.1

13

Theoretical Physics and Computer Science

A common restricted version of the physical Church–Turing Hypothesis considers not arbitrary physical processes but only those harnessable for universal computation: Hypothesis 1.1. There exists no physical computing device capable of solving the Halting Problem. It has only recently been noticed and stated explicitly [Smi06a, BeTu06] that the existence and non-existence of a physical object must be seen as subject to an underlying physical theory, that is as relative to a sound mathematical abstraction of (parts of) nature. This is quite similar to un/computability claims also being subject to a model of computation as a mathematical abstraction of a computing system. Newtonian Mechanics, for instance, does not allow Black Holes but does allow faster-than-light travel; whereas in General Relativity, it is the other way round. A further example, the Second Law of Thermodynamics (non-decreasing entropy), yields a rather interesting ultimate (though very small) lower bound on the speed of irreversible computation (i.e., regarding complexity)—yet seemingly not on computability, i.e., regarding the principal computing power of physical systems [BeLa85]. And it turns out (see Chapter 3) that some physical theories do permit the Halting Problem to be solved!

1.5.2

Meta-Theory of Physics

As we have just seen, there is not just one Theory of Physics but a great variety of such theories; and the status of Hypothesis 1.1 may depend on the term ‘physical’ that it contains being understood according to one theory or to the other. Thus, a thorough investigation of the Church– Turing Hypothesis should consider several theories of physics and their relations to each other (e.g. sub-theories), which brings us to the field of meta-theory. As promising as such an approach to a sound physical theory of computation may be, it has to cope with a major obstacle: As opposed to a model of computation (which usually somebody suggests and describes formally and which the scientific community then either rejects or accepts and uses), a physical theory is far more provisional in character: once proposed, it usually remains ‘under construction’ with new terms being added or others discarded (see below); and once the theory eventually ‘converges’ towards a stable formulation, it is already on the verge of being superseded by a superior one [Kuhn62]. Newtonian Mechanics for example, though successful for centuries in describing both small falling apples and large planets circling the sun, was eventually replaced by Quantum Mechanics describing the very, very small (e.g. elementary particles), and by Relativity Theory describing the very, very large (e.g. galaxies). General Relativity in turn used to contain a parameter, the cosmological constant Λ, whose introduction Einstein himself later called the “biggest blunder” of his life and reverted, declaring Λ = 0; whereas nowadays observations of the relationship between distance and redshift indicate that Λ might in fact have a very small yet non-zero value. Moreover, reconciling Relativity with Quantum Mechanics has led to Relativistic Quantum Field Theories, though these still lack mathematical well-foundation. Consequently, so very few physical theories have actually been formalized far enough (in the sense of successfully inhibiting misinterpretations by other physicists) that even the question “What is a physical theory anyway? ” splits the community [Weiz71, Ludw78, Schr96].

1.5.3

Constructivism in Physics

Apart from such issues, the term “exists” in Hypothesis 1.1 involves further ambiguity of interpretation. As has been pointed out in [24, Remark 1.4], this is quite similar to the case of ‘idealistic’ versus constructive mathematics: In the former, the following proposition holds true due to the principle of excluded middle (tertium non datur): Principle 1.2. A sequence (xn )n∈N is either identically zero; or there exists some index n ∈ N with xn 6= 0.

14


In constructive mathematics, however, this Limited Principle of Omniscience (LPO), is deprecated because the existence proof in the second case does not actually construct an n with an 6= 0. (Section 2.6.2 below expands on the close relations between constructive analysis and real number computability.) This is to be distinguished from LPO being non-contradictory, that is, from its double negation [Brid04]. Even among mathematicians (actually the majority) who do accept the LPO, attitudes differ concerning what can be regarded as its uncountable version: Q Principle 1.3 (Axiom of Choice). Let I and Xi , i ∈ I denote arbitrary sets. If i∈I Xi = ∅, then there exists some i ∈ I where Xi = ∅. In classical mathematics, i.e. with the permission to use proof by contradiction, this principle indeed also seems ‘obvious’; and is furthermore equivalent to several other famous and natural mathematical propositions: e.g. that every vector space admits a basis [Blas84]. Nevertheless, Principle 1.3 neither follows from nor contradicts conservative set theory [Goed40]; this justifies calling it an axiom and leaves it a matter of taste whether to include it in or to exclude it from a formal system. These examples illustrate that some mathematical object to exist can be interpreted as A) to insist that one actually constructs this object (and thus to reject LPO); B) its non-existence leading to a contradiction (and thus to accept LPO but not necessarily the Axiom of Choice); C) the object’s existence to not lead to a contradiction (and to accept e.g. the Axiom of Choice). Similarly, the existence of a physical object as in Hypothesis 1.1 may be understood in the strong sense (A) of actually constructing it. Yet in most cases, showing it (C) to be consistent with physical laws is accepted as well; observe that this is how both positrons and Black Holes came into ‘existence’: as putative solutions of (and thus consistent with) Dirac’s Equation and Einstein’s General Relativity, respectively. It was only later that new experimental observations upgraded our conception of their existence to type (B).

1.6

General Outline

This concludes our brief introduction and background information to computability in the classical setting, dealing with well-established models of computation (primarily the Turing machine) for bits, words, integers, and rational numbers. We will now proceed to review more recent research, dropping the restriction to discrete problems (Chapter 2) and considering super-Turing models of computation (Chapter 3). Their synthesis then leads to the most current area of real hypercomputation (Part II), which constitutes the core, and justifies the title, of the present work.

Chapter 2

Real Computation ‘Standard’ models of computation—Turing machines, Random Access Machines (RAMs), and even parallel variants like the PRAM –all operate on discrete objects: bits, Booleans, integers; at most, rational numbers (fractions) are considered in the form of numerator/denominator pairs of integers. These are the entities to be read, stored, processed, and output. A large amount of scientific computation, however, evolves around continuum rather than discrete problems. Fields like fluid dynamics, computational material science, and space mission design are just a few of them. In fact, most of applied mathematics amounts to solving various classes of ordinary or partial differential equations over the reals. This raises the need for a formal model (see Section 1.2) to describe and analyze the prospects and limits of computations over R. Introducing the two major ones will be the purpose of Sections 2.3 and 2.4.

2.1

Meta-Mathematical Remarks

Some people question the need for a model of real (not to mention complex) computation. They argue, convincingly indeed, that scientific computation deals with rationals only, if not with floating point numbers. This is of course mostly true (except for the algebraic numbers handled in LEDA [BKM*95], for example, and even transcendentals as in computer algebra systems): as true as the observation that all quantities obtainable from physical √ experiments—and thus, by extensionality, entire nature itself—are but rationals; objects like 2 and π being made up ‘artificially’. In the last consequence, such considerations lead to an ultrafinitistic point of view (quite consistent with floating point numbers). While I secretly agree with these opinions, the majority of mathematicians nevertheless consider real numbers and computations on them for a simple x2n +2 reason: assigning to certain sequences of rational numbers (like xn+1 = 2x ) a short name (like n √ 2) to talk about, and granting this abbreviation a mathematical identity to calculate with (like √ √ 2· 2 = 2), yields such enormous succinctness, elegance, and ease of comprehension (for instance, what is the uniform probability distribution on [0, 1] considered as the unit interval of rationals?) as to make this idealization well worth the ontological overhead. Moreover, insights obtained in an idealized realm have often led to interesting implications back in a more applied world; e.g. complex to real analysis. We are thus facing two sides of Occam’s Razor: extending (finitely constructable, countable cardinality) rationals Q to (not finitely constructable, continuum cardinality) R yields a tremendous increase in conceptional complexity, which it is fair to call unnecessary; on the other hand the transition from the minimal-ontological, ‘practical’ world Q to the Platonic realm of idealizations R does benefit from an elegance and ease of description within R which brings with it a decrease in conceptional complexity: compare, for example, the simplicity of Newton’s Law in its differential form (i.e. referring to R) to one based on Q only. Equipping R with a notion of computability (i.e. a certain sense of infinite constructability by finitely describable machines) can be regarded as a compromise between the two extremes. 15

16

CHAPTER 2. REAL COMPUTATION

To sum up, pure mathematics and theoretical computer science have both contributed and still contribute essentially to the undeniable success of scientific computing—and vice versa [DaRe06, HHLR06]. Having said this, the present section proceeds to review common models of real number computation: the two most common being the algebraic BCSS machine and the Type-2 machine (approximation paradigm) in Sections 2.3 and 2.4, respectively. (Also prepare to encounter another personal comment in Section 2.8. . . )

2.2

Real Numbers

Before addressing computation on reals, √ let us look at R itself. For many purposes, common high school intuition is fairly sufficient: 31 , 2, and π are real numbers; there are arithmetic operations +, −, ×, ÷ and a linear order Card(Q), there exists no surjective mapping from Q onto R. The question of whether there exist sets of cardinality strictly between Q and R is known as the Continuum Hypothesis and has been proven logically ¨ del and Cohen. independent of ZFC (Zermelo-Fraenkel set theory with axiom of choice) by Go We briefly recall two classes of properties of R which are trivial on Q: algebraic and topological ones. Each turns out to correspond to one major model of real number computation.

2.2.1

Algebra

Like Q, R is an (ordered) field to which the general concepts of abstract algebra apply: Definition 2.1. Fix a subfield E of the field F . a) For A ⊆ E, let F (A) denote the smallest subfield of E containing F and A. b) The degree [E : F ] is the dimension of E considered as an F –vector space. c) For e ∈ E, call degF (e) := [F (e) : F ] the degree of e. d) If degF (e) < ∞, e is algebraic over F otherwise transcendental over F . e) Field extension E is algebraic over F if every element e ∈ E is algebraic over F . f ) For an integral domain R, a non-zero polynomial p ∈ R[X] is called irreducible (over R[X]) if it is not constant and if every factorization p = q1 · q2 with q1 , q2 ∈ R[X] requires q1 or q2 to be constant. When dealing with E = R, we occasionally omit F in the case that F = Q and denote by A the set of algebraic reals. Fact 2.2. a) The degree of iterated field extensions is multiplicative: For fields F ⊆ E ⊆ D, [D : F ] = [D : E] · [E : F ]. b) For fields F ⊆ E 3 e, e is algebraic over F iff there exists a non-zero polynomial p with coefficients in F such that p(e) = 0. c) More precisely, let e be algebraic over F . Then F (a) = F [a] and degF (e) < ∞ coincides with the degree of every irreducible p ∈ F [X] \ {0} satisfying p(e) = 0. d) If R is a unique factorization domain and F its field of fractions, then a polynomial p ∈ R[X] irreducible over R[X] is also irreducible over F [X].

2.2. REAL NUMBERS

17

e) To an algebraic field extension E over F ⊇ Q of finite degree, there exists some e ∈ E (a so-called primitive element) such that E = F [e]. Proof. Cf. e.g. [Lang93, Proposition VII.§1.4]. Claim d) is known as Gauss’ Lemma [Lang93, Section V.§6]. Regarding e), E has characteristic 0 and is therefore a separable extension of F [Cohn91, Corollary 3.4.3]; now apply [Cohn91, Theorem 3.9.2]. Roughly 2,500 years ago, Hippasus √ of Metapontum proved (and was drowned for doing so) √ that 2 6∈ Q; equivalently: that [Q( 2) : Q] = 2. This proof immediately extends to squarefree numbers t and single higher roots: [Q(t1/n ) : Q] = n for n ∈ N. But how about field extensions by several roots? Here are three answers: Lemma 2.3. a) √Let Q ⊆√ F⊆ t ∈ N be squarefree, and n1 , . . . , nk ∈ √ fields, R denote a tower of N. Then F n1 t, . . . , nk t coincides with F N t , where N := lcm(n1 , . . . , nk ) denotes the √ √ least common multiple. In particular [F n1 t, . . . , nk t : F] = N . √ √ b) Let t1 , . . . , tk ∈ N be distinct and squarefree. Then [Q( t1 , . . . , tk ) : Q] = 2k . c) For d ∈ N, let p1 , . . . , pk denote distinct prime numbers and n1 , . . . , nk ∈ N. Then √ √ √ Q n1 p1 , n2 p2 , . . . , nk pk : Q = n 1 · n2 · · · n k . The first two claims admit elementary proofs; that of b) is quoted from [Rick00]. Proof.

a) Recall that the following properties of lcm:

and

ni | lcm(n1 , . . . , nk ) = lcm lcm(n1 , . . . , nk−1 ), nk lcm(a, b) = ab/ gcd(a, b) = ab/(ra + st) with r, s ∈ Z .

√ conversely, t1/ lcm(a,b) = Therefore, each t1/ni √is a√power of t1/N and thus in F( N t); √ while, n√ a b n1 1/a s 1/b r 1/N (t ) · (t ) ∈ F( t, t) yields t to belong to F( t, . . . , k t) by induction on k. Hence we have indeed established t1/N as a primitive element. b) Define Sn (n ≥ 0) to be the set of square roots of √ products of subsets √ √ of√some n distinct primes. (So Sn has 2n elements, S0 = {1}, S1 = {1, 2}, S2 = {1, 2, 3, 6}, . . . ) Define Fn to be the subset of R generated as a vector space over Q by the elements of Sn . Then: i) The elements of Sn are linearly independent over Q.

(This yields the claim!)

ii) Every element of Fn whose square is rational is a rational multiple of an element of Sn . iii) Fn is a field. I prove this proposition by induction over n. It is clearly true for n = 0 (S0 = {1}, F0 = Q). √ Suppose it is true for n = k. Let p be the (k + 1)-st prime, r = p. Then every element of Fk+1 is of the form a + br, where a and b are in Fk . If Sk+1 is not linearly independent, then 0 can be expressed in this form with a and b linear combinations of elements of Sk , not both a and b being the trivial linear combination: since Sk is linearly independent, this implies that a and b are not both zero. Since a + br = 0, it follows that neither a nor b is zero; we have r = −a/b (using the fact that Fk is a field), and p = (a/b)2 ; this contradicts (ii) for n = k, using unique factorization in the integers and counting powers of p. Hence we have (i) for n = k + 1. Now (a+br)2 = a2 +pb2 +2abr; if this is rational, then since S( k +1) is linearly independent, 2ab = 0, so either a = 0 or b = 0. If a = 0, then (a + br)2 = pb2 , so if this is rational then b2 is rational, so b is a rational multiple of an element of Sk , so a + br = br is a rational multiple of an element of S( k + 1); similarly if b = 0 then a + br is a rational multiple of an element of Sk+1 (indeed, of an element of Sk ). This shows (ii) for n = k + 1.

18

CHAPTER 2. REAL COMPUTATION Fk+1 is clearly a ring; to show that it is a field, it suffices to show that every non-zero element is invertible. This is true because (a + br)−1 = (a − br)/(a2 − pb2 ); a2 − pb2 is non-zero because it equals (a + br) · (a − br), so it is invertible because it is in Fk , which is a field. This shows (iii) for n = k + 1. c) Cf. [Besi40, Theorem 2]; see also [Albu03, bottom of p.2].

Transcendence Definition 2.4. For fields F ⊆ E, a set A ⊆ E is algebraically dependent over F if there exists n ∈ N, distinct a1 , . . . , an ∈ A, and a non-zero polynomial p ∈ F [X1 , . . . , Xn ] such that p(a1 , . . . , an ) = 0. The transcendence degree tr. degF (E) of E over F is the cardinality of a maximal, algebraically independent subset of E. Fact 2.5.

a) e and π are transcendental.

b) The transcendence degree of R over Q is of continuum cardinality [Cohn91, Exercise 5.1(4)]. c) Let a1 , . . . , an be algebraic yet linearly independent over Q. Then ea1 , . . . , ean are algebraically independent over Q [Bake75, Theorem 1.4]. d) For fields F ⊆ E ⊆ L, tr. degF (L) = tr. degF (E) + tr. degE (L). Claim a) follows from c) which in turn is known as the Lindemann-Weierstraß Theorem. Claim d) can be regarded as a transcendent and additive variant of the multiplicativity of the algebraic degree (Fact 2.2a); cf. e.g. [Lang93, Exercise X.5]. Regarding the following claim, refer for instance to [Lang93, Exercise X.6]: Fact 2.6. Let E denote a field extension of F which is finitely generated in the sense that E = F (x1 , . . . , xd ) for finitely many x1 , . . . , xd ∈ E. Then any intermediate field between E and F is also finitely generated.

2.2.2

Topology

R is by construction the metric completion of Q: the set of limits of rational Cauchy sequences: (xn )n ⊆ Q,

|xn − xm | ≤ 2−n ∀m ≥ n

∃x ∈ R : |x − xn | ≤ 2−n+1 .

⇒

The same applies to the finite Cartesian product Rk , k ∈ N. Equipped with the Euclidean norm, Rk is a locally compact, separable, connected space with a countable base of open rational balls B(~q, r) = {~x : |~x − ~q| < r}, ~q ∈ Qk , 0 < r ∈ Q. The class of open subsets of Rk is denoted by Ok ; we write Ak for the class of closed subsets; S and S ◦ for the closure and interior, respectively, of a set S. Fact 2.7. Let X denote an arbitrary topological space, I an arbitrary index set, and G, H, Mi ⊆ X, i ∈ I. Then \ ◦ [ ◦ [ ◦ \ ◦ 1. Mi ⊆ Mi 5. Mi ⊇ Mi i∈I

2.

\

i∈I

Mi ⊇

\

i∈I

Mi

6.

3. G◦ ∩ H ◦ = (G ∩ H)◦

[

i∈I

Mi ⊆

[

Mi

7. G ∪ H = G ∪ H

◦

◦

4. (X \ G) = X \ G

8. X \ H = X \ H

Let U denote an open and A a closed subset of X. ◦

◦

9. U ⊆ U

11. U = A◦

⇒

U =U

10. A ⊇ A◦

12. A = U

⇒

A◦ = A.

2.2. REAL NUMBERS

19

In Euclidean space, every non-empty open set U is of second category (Baire’s Category Theorem), S ◦ that is U = n An implies ∃n : A 6= ∅. Cantor Space arises by taking the binary expansions of real numbers as the set {0, 1}ω of infinite 0/1–sequences equipped with metric d(¯ σ , τ¯) = max{2−n : n ∈ N, σn 6= τn } . It is compact, separable, and totally disconnected. Borel Hierarchy Fix a topological space X. We write (bold-face) Σ1 for the class of open subsets of X, Π1 for its complements, i.e. the closed subsets of X, and ∆1 for those which are both open and closed (clopen): ∆1 ⊇ {∅, X} with equality iff X is connected. Countable unions of closed sets, known as Fσ , form the class Σ2 ; Π2 consists of Gδ sets, that is countable intersections of open ones; and ∆2 = Σ2 ∩ Π2 . Example 2.8. For X = R, the set Q of rational numbers, being a countable union of (closed) singletons, belongs to Σ2 ; but not to Π2 [Boto79, A13.5]. The Cantor set, consisting of all real numbers in [0, 1] having a ternary expansion involving only digits 0 and 2 (but not 1), is closed (i.e. in Π1 ) and uncountable yet nowhere dense [GeOl03, §8.1]. Σ3 contains all countable unions of members from Π2 ; Π3 its complements and so on for Σd and Πd , d ∈ N — the (bold-face) Borel Hierarchy of subsets of X. For X = Rk and X = {0, 1}ω one can show it to be strict [Kech95]: ∆1 ( Σ1 ( ∆2 ( Σ2 ( . . . ( ∆d ( Σd ( . . .

(2.1)

Transfinite Levels S To the finite Borel Hierarchy Σ 0 given by the rational sequence (˜ qn ) := (q0 , q1 , . . . , qN , qN , . . .) coinciding with (pn ) for n ≤ N . Being a deterministic machine, M will then proceed exactly as before for its first N steps; in particular the output (˜ pm ) agrees with (pm ) up to m = 2. Hence |˜ p2 − y˜| > 2−2 contradicting that M is supposed to output fast converging approximations to y˜ = h(˜ x). For the case of a general function f : R → R with discontinuity at some x ∈ R, let y = f (x) 6= limk f (xk ) = y˜ with a real sequence xk converging to x. There exists M ∈ N with |y − y˜| > 2−M +2 ; by proceeding to an appropriate subsequence of (xk ), we may suppose w.l.o.g. that |x − xk | ≤ 2−k−2 and |f (xk ) − y˜| ≤ 2−M . Then there is a rational double sequence (qk,n ) such that |xk − qk,n | ≤ 2−n−1 ; thus |x − qn,n | ≤ 2−n . One can therefore feed rational approximations

2.4. RECURSIVE ANALYSIS

33

(qn,n ) in order to evaluate f at x and obtain in turn approximations (pm ) ⊆ Q to y. As before, pM is output after only some finite initial part (qn,n )n≤N of the input has been read. Then |qn,n − xN | ≤ |qn,n − xn | + |xn − x| + |x − xN | ≤ 2−n−1 + 2−n−2 + 2−N −2 ≤ 2−n for n ≤ N reveals this very initial part to also be the start of a sequence of fast converging rational approximations to x ˜ := xN , whereas 2−M +2 < |y − y˜| ≤ |y − pM | + |pM − f (˜ x)| + |f (˜ x) − y˜| ≤ 2−M + |pM − f (˜ x)| + 2−M shows that (pm )m≤M do not constitute approximations to f (˜ x): contradiction. Recall that a function f : X → Y is continuous iff the pre-image f −1 [V ] ⊆ X is open for every open V ⊆ Y ; and a set U ⊆ R is open iff it is a countable union of open intervals with rational centers and radii [ U = B(qn , rn ), qn ∈ Q, 0 ≤ rn ∈ Q, B(q, r) = {x : |x − q| < r} . (2.8) n∈N

This suggests a further notion of computability for continuous real functions based on the requirement that the mapping V 7→ f −1 [V ] from open sets to open sets be effective: Definition 2.42 (cont.) Call f : [0, 1] → R c) effectively continuous if some Turing S machine can, upon input of open V by means of a rational sequence (~ q , r ) with V = qn , rn ), output a rational sequence (~ pm , sm ) with n n n B(~ S f −1 [V ] = m B(~ pm , sm ). Now the good news is that the above three notions of computable real functions have turned out to be equivalent: Fact 2.45. Function f : [0, 1] → R is effectively evaluable iff it is Weierstraß–computable and iff it is effectively continuous. (The equivalence holds even uniformly in the sense that a Turing machine can convert back and forth between the three above ways of representing f ; cf. Section 2.4.6.) This follows from the works of Grzegorczyk [Grze57], Pour-El, Caldwell, Hauck [PECa75, Hauc76], and Lacombe [Laco57, Laco58]. Equivalence to Definition 2.42b) is known as Effective Weierstraß Theorem; see e.g. the textbook [PERi89, Section 0.7]. Example 2.46. computable.

a) A real c is (Cauchy–) computable iff the constant function f (x) ≡ c is

b) The characteristic function of {0} is not computable (i.e., the test “x = 0?” is undecidable in Recursive Analysis). Indeed, χ{0} : R → {0, 1} has a discontinuity and thus fails to be computable by virtue of the ‘Main Theorem’. Recursive Analysis is often criticized for this negative example; particularly by the BCSS community, but also by numerical analysts: to whom on the other hand it is gospel that numerical programs must not use tests for equality. Section 4.3 of the present work addresses Question 4.22, whether hypercomputers may allow discontinuous real functions to be evaluated.

2.4.3

Semi–Computability and Semi–Decidability

P −n Since H is undecidable, the real number xH = lacks Cauchy–computability Examn∈H 2 ple 2.37c); on the other hand, the semi–decidability of H allows to computably approximate xH from below; in fact to compute all rational lower approximations:

34


Definition 2.47. Call x ∈ R lower computable if the Dedekind cut Qx = {q ∈ Q : q > x} means right computability of x. In the same way as recursivity amounts to both r.e. and co-r.e. (Section 1.1.1), a real number is Cauchy–computable iff it is both left and right computable. This equivalence in fact holds uniformly (Fact 2.61); here, the strictness of the Dedekind cuts Qx in Definition 2.47— i.e., is both are not complementary (in Q) to each other for x ∈ Q—enters crucially [Weih00, Example 4.1.14.2]. A different notion of real semi-computability, related to classical semi-decidability, is exemplified as follows: Although the equality of real numbers is ρ–undecidable due to the Main Theorem, inequality is at least semi-decidable. Observation 2.48. There exists a Turing machine which, given a rational sequence (qn )n with |x − qn | ≤ 2−n , terminates iff x = 6 0: simply search for some n ∈ N with |qn | > 2−n .

2.4.4

Enumerable Real Open Sets

As pointed out at the end of Section 2.4.2, the concept of decidability of real subsets makes no sense in Recursive Analysis. That of semi-decidability, however, does lead to a natural notion, nicely related to Definitions 2.47 and 2.42a+c): Fact 2.49. Call U ⊆ R semi-decidable if some Turing machine can, upon input of x ∈ R by means of (qn ) ⊆ Q with |x − qn | ≤ 2−n , terminate in the case of x ∈ U and diverge in the case of x 6∈ U . Then every semi-decidable set is necessarily open. More precisely, for U ⊆ R the following are equivalent: i) U is semi-decidable. ii) Some Turing machine can enumerate rational centers qn and radii rn ≥ 0 of open intervals exhausting U according to Equation (2.8). iii) The characteristic function χU : R → {0, 1} of U is effectively left-evaluable (also called (ρ → ρ< )–computable) in the sense that a Turing machine can, upon input of (qn ) for x ∈ R as above, output the Dedekind cut Q 0”. The third case “p(~x) = 0” is decidable by a). c) S is the set of solutions ~x to some finite system of polynomial (in)equalities pi (~x) = 0 and qj (~x) > 0 with pi , qj ∈ Rc [X1 , . . . , Xn ]. For ~x ∈ Qn , these are decidable according to b). The non-uniformity implicit in this result consists in the arbitrary but fixed polynomial (system) and, more specifically, in finding, for its coefficients, a Q–basis and decomposition therein. Not only Turing machines fail at this task: a Q–basis for the seemingly simple (since at most 3-dimensional) vector space spanned by (1, e · π, e + π) is unknown to date [EHH*91, p.153].

2.4.6

Type-2 Theory of Effectivity

Recursive Analysis used to suffer from a vast variety of notions, some subtly distinct, some different but equivalent. For instance consider the Definitions of a computable real number 2.36 (with its two variants: n arbitrary or n = 2−n ), 2.38, and 2.47 (with alternative Dedekind cuts Q≤x , Qx ): Which of them are equivalent? uniformly or non–uniformly? It was the achievement of Weihrauch’s Type-2 Theory of Effectivity (TTE ) to have formalized these notions by means of so-called representations; see [Weih95, Weih00]. Within this framework, computability notions like the above ones have been systematically compared and the results formulated concisely both for the uniform and the non–uniform case [Weih00, Section 4.1]. In order to prevent misconceptions, I should emphasize that TTE does not constitute a new model of computation but a framework for and (if you like, meta-) theory of (both previous and new) notions of computability. It extends, from countable universes to those of continuum cardinality, the (classically usually implicit) requirement that objects (such as graphs or integers) need to be encoded into strings over a finite alphabet (e.g. {0, 1}). Definition 2.56. Let A be an arbitrary set. i) A representation α of A is a surjective partial mapping α :⊆ {0, 1}ω → A from the Cantor space of infinite strings onto A. ii) An infinite string σ ¯ with α(¯ σ ) = a is called an α–name of a. We say that a is α–computable if it has a computable α–name. iii) For an additional set B with representation β, a partial function f :⊆ A → B is called (α → β)–computable if a Turing machine can convert every α–name of every element a ∈ A to some β–name of b = f (a). f is (α → β)–continuous if there exists a (Cantor-) continuous function F :⊆ {0, 1}ω → {0, 1} such that β F (¯ σ ) = f α(¯ σ ) for all σ ¯ ∈ dom(f ◦ α). Such F constitutes an (α → β)–realization of f . iv) For two representations α and β of the same set X, α is reducible to β, written as “α β”, if the identity function on X is (α → β)–computable. Equivalence “α ≡ β” of α and β means that in addition β is conversely reducible to α. We write “α t β” if the identity on X is (α → β)–continuous.

2.4. RECURSIVE ANALYSIS

37

Whereas classical (i.e. Type-1) computability deals with objects from a countable universe (e.g. {0, 1}∗ , N, or Q) consisting of (or encoded in) finite strings, the Type-2 computability of objects a in some uncountable universe A is thus defined in terms of infinite strings as names of a; and computability of a function amounts to effective conversion of names of arguments to names of values, that is to the commutativity of the following diagram with some realization F :⊆ {0, 1}ω → {0, 1}ω as a computable function on Cantor space: σ ¯ ∈ {0, 1}ω −−−−→τ¯ ∈ {0, 1}ω F     βy yα a∈A

f

−−−−→

(2.9)

b∈B

Note that the Turing machine in Definition 2.56iii) may behave arbitrarily both on inputs σ ¯ 6∈ dom(α) and for α(¯ σ ) 6∈ dom(f ). The Main Theorem now takes the form of Fact 2.57 (Main Theorem for functions on Cantor Space). Let F :⊆ {0, 1}ω → {0, 1}ω be computable. Then F is continuous (w.r.t. Cantor topology). A simple proof can be found in [Weih00, Theorem 2.2.3]. In particular, (α → β)–computability requires (α → β)–continuity! Items b), c), and d) below are TTE’s rephrasing of Definitions 2.36, 2.47, 2.35, respectively. Example 2.58. a) The standard representation ν :⊆ {0, 1}ω → Q has domain dom(ν) = {0, 1}◦ p q {1 0 1 0 : p, q ∈ N} ◦ {0, 1}ω ; it assigns to σ ¯ = 0 1p 0 1q 0 · · · the number ν(¯ σ ) = +p/q, to p q σ ¯ = 1 1 0 1 0 · · · the number ν(¯ σ ) = −p/q. b) The Cauchy representation ρ :⊆ {0, 1}ω → R is the mapping, from sequences (qn ) of rationals (encoded and delimited as in a) with |qn − x| ≤ 2−n for some x ∈ R, to this very x. c) The left representation ρ< :⊆ {0, 1}ω → R is the mapping from (appropriately encoded) enumerations of the Dedekind cuts Q . d) The naive representation ρCn assigns to (an encoding of ) a convergent rational sequence (qn ) its limit x = limn qn . e) The Weierstraß representation [ρ → ρ] of C(R) assigns, to (an encoding of the degrees and coefficients of ) a sequence (Qn ) of polynomials over Q with kQn − f k[−n,+n] ≤ 2−n for some f ∈ C(R), this very f . f ) Fix d ∈ N. Let θ< denote the following (inner) representation of the class Od of open subsets of Rd : a name of U ∈ Od is (an S encoding of ) a list of rational centers ~qn and radii rn of open Euclidean balls with U = n B(~qn , rn ). The representation in Item e) corresponds of course to Definition 2.42b); its surjectivity relies on the Weierstraß Approximation Theorem, recall Fact 2.9. Item f) corresponds to Fact 2.49ii) and permits Definition 2.42c) to be rephrased as (θ< → θ< )–computability of the pre-image mapping V 7→ f −1 [V ] on open subsets of R. Remark 2.59. It is central to the concept of TTE (Section 2.4.6) that a representation α :⊆ {0, 1}ω → A induces a notion of effectivity on all elements of the represented space A: both computable and uncomputable ones are assigned names and can be input to a Type-2 Machine computing some function f : A → B. It may at first seem unnatural to define effective function evaluation f : x 7→ f (x) even on uncomputable inputs x. However, this requirement avoids many pathologies (unlike, for instance, Markov-computability; see Section 2.6.3 below) and is crucial to the equivalence of effective evaluation to several other natural notions of function computability (recall Fact 2.45).

38


Constructing New Representations from Given Ones can often proceed canonically and categorically [Weih00, Section 3.3]. We will mention three important examples: Fact 2.60. Let index set I be either N or finite. a) If αi is a representation for Ai , i ∈ I, then there is a natural representation Q Cartesian product i∈I Ai .

Q

i∈I

αi for the

b) For each i ∈ I, let αi be a representation for (the same) A. There is a representation of A (which roughly incorporates the information provided by all αi ).

V

i∈I

αi

c) Let α be a representation for A and β one for B. There is a natural representation [α → β] for the class of (α → β)–continuous functions f : A → B (basically encoding an (α → β)realization F :⊆ {0, 1}ω → {0, 1}ω ); see [Weih00, Definition 3.3.13]. d) For the case A = B = R and α = β = ρ, the representation from c) is equivalent to the one in Example 2.58e) [Weih00, Section 6.1]. e) The representation from c) satisfies SMN and UTM–like properties, even uniformly [Weih00, Theorem 3.3.15]. A real sequence (xm )m for instance isQ computable in the sense of Definition 2.53 iff it is computable with respect to the representation m ρ according to Fact 2.60a); which is in turn equivalent to [ν|N → ρ] according to Fact 2.60c) where ν|N denotes the restriction (in image) to N of the representation ν :⊆ {0, 1}ω → Q from Example 2.58a) [Weih00, Lemma 3.3.16]. Fact 2.60d) is the uniform version of Fact 2.45. Analogously, the uniform equivalence of Cauchy-computability to simultaneous left and right computability mentioned in Section 2.4.3 can now be formulated (and easily proved) as Fact 2.61. It holds ρ ≡ ρ< ∧ ρ> . Every representation α :⊆ {0, 1}ω → A induces on A a topology inherited from Cantor space. If this final topology complies with the original one in the sense of [Weih00, Definition 3.2.7], α is called admissible. For instance, ρ is admissible with the usual Euclidean topology on R [Weih00, Lemma 4.1.4]. The following consequence of Fact 2.57 thus generalizes Proposition 2.44 and yields a further version of the Main Theorem: Fact 2.62 (Main Theorem of TTE). Let α and β be admissible representations of A and B, respectively. Then every (α → β)–computable function f :⊆ A → B is (α → β)–continuous and in particular continuous. Proof. See [Weih00, Theorem 3.2.11]. Multivalued Functions Many mathematical proofs involve ‘choosing’ an arbitrary member from a set asserted to be non-empty and continuing arguing about that element. The equivalence of –δ–continuity with sequential continuity for real functions, for instance, exploits the Archimedean Property of R: Example 2.63.

a) To any given real x, there exists an integer N with N > |x|.

b) Every mapping f : R → N with f (x) > |x| is discontinuous and thus uncomputable by the Main Theorem. c) Given (qn ) ⊆ Q with |x − qn | ≤ 2−n , N := d|q0 |e + 3 is easily computable and satisfies N > |x|.

2.5. BCSS VERSUS/WITH RECURSIVE ANALYSIS

39

Item c) exploits the arbitrariness of a choice N with N > |x|. This amounts to replacing (singlevalued) f by a multi-valued function f˜ : R ⇒ N with f˜(x) ⊆ {N ∈ N : |x| < N }, that is a relation Rf˜ = {(x, N ) : x ∈ R, N ∈ N, |x| < N }. More generally, relaxing (α → β)–computability from g :⊆ A → B to g˜ :⊆ A ⇒ B in the sense of [Weih00, Definition 3.1.3] means that not only the β–name output for b = g(a) but also the computed value b ∈ g˜(a) itself may depend on which α-name for a the Turing machine is given.

2.5

BCSS versus/with Recursive Analysis

Choosing an appropriate model of computation (recall Section 1.2) is generally a difficult and hotly disputed task. In the discrete realm (Type-1 setting: bits, integers, . . . ) the Turing machine has become widely accepted, a particular argument in its favor being its equivalence to several other seemingly unrelated reasonable notions of computability (Section 1.4, Item c). Over the reals, however, it holds Observation 2.64. BCSS machines and Recursive Analysis have noncomparable computational power: a) Heaviside’s function is computable by the first but, in spite of its simplicity, not to the latter (see Proposition 2.44). b) The equally simple exponential function on the other hand behaves in the opposite way, cf. Example 2.18. (Regarding the semi-decidability of open sets, however, a comparison has succeeded in terms of degree theory, see Section 6.3.) The question of which of the two models is ‘the’ appropriate one is a constant source of disputes and (meta-) arguments [Koep01, Brat00]. Other approaches try to combine both models and are described in the sequel. R-analytic machines (Section 2.5.3), for instance, can be regarded as a synthesis of Recursive Analysis with BCSS-computations. But let us first discuss

2.5.1

Variations of BCSS Machines

Every model of computation reflects some aspects of reality better and falls down on others due to the inherent idealizations. Some major criticisms of the BCSS machine concern a) their use of arbitrary (although only finitely many) real constants; n

b) exponentially long (in terms of bits) integers like 22 obtainable within linearly many steps O(n) by repeated squaring; c) the ability to perform all operations and in particular comparisons exactly; d) the uncomputability of elementary functions like square root, exponentiation, logarithm, sine, and cosine; compare Example 2.18d). We have investigated the impact of a) in Section 2.3.4. In order to counter Criticism d), we may enhance the BCSS machine’s set of operations to also include square root, exponentiation, or sine. This leads to hypercomputation and is discussed in Section 5.1. Conversely, Criticism b) can be circumvented by ex cluding multiplication and division as operational primitives. This leads to linear BCSS machines [Koir94, CuKo95, FoKo00], compare Section 5.3.2. Criticism c), too, can be avoided: e.g. by including numerical condition numbers in the analysis [CuSm99]— or by reconciling BCSS Theory with Recursive Analysis, for instance by disallowing tests for equality and by changing the semantics of inequality tests [BrHe98]. This naturally gives rise to a mathematically elegant synthesis of algebraic complexity theory with real computability theory [25]. Another approach to obtaining a more realistic algebraic complexity (not computability) theory charges, to a BCSS machine’s operations, costs depending on the size of their operands [Koi97a, MadHW97, Brav05].

40


2.5.2

Aspects of Classification

We may roughly classify a model of real computation according to whether • an admissible calculation takes finitely or infinitely many steps; • input is presented exactly or by means of approximations; • the entities under consideration (i.e., real or rational numbers) can be handled exactly or approximately • and subjected to which kinds of operations and constants; • the result is to be obtained exactly or approximately. Notions of Approximation In the case of approximate computations, one may furthermore distinguish • certified (i.e. with error bounds) versus ultimate (i.e. in the sense of eventually tending towards the limit) precision; • absolute (∆z = |z − z ∗ |) versus relative (δz =

|z−z ∗ | |z| )

errors,

• being forward (i.e. referring to error of y = f (x)) or backward bounded/stable (i.e. referring to deviating x). whereas computability in Recursive Analysis means infinitely long calculations on rational numbers approximating input and output with arbitrary classically-recursive operations and guaranteed absolute error bounds in the sense of forward stability. (This gives another way of looking at the ‘Main Theorem’: For a discontinuous function, the absolute forward error ∆y is unbounded in terms of the input error ∆x .) Uncertified approximation occurs in naive computations, recall Definition 2.35. Remark 2.65. Most models of approximate real computation refer to absolute error: for the simple reason that the difference x − y of two positive numbers x, y may have relative error unbounded in that of the input x, y. Determining the result up to arbitrarily prescribable relative error may thus be desirable in practice but is in general infeasible. Thus in order for this notion to make sense one has to resort to input given exactly, that is without any (relative or absolute) error.

2.5.3

Analytic Machines

Combining the items of the above classification results in a plethora of notions of real computation. Not all of them make sense (e.g. Remark 2.65) but most which do appear in the family of Analytic Machines proposed by Chadzelek and Hotz, introduced and compared in [HVS95, ChHo99]. In the sense of the above Classification 2.5.2, they name such a machine (or rather the function f : x 7→ y = f (x) it computes) according to whether • its primitive operations and comparisons apply to rational numbers (“Q”) or to real ones (“R”); • the result y appears after finitely many steps (“computable”) or is the limit of an infinite computation (“analytic”); • in the latter case: is input x ∈ R accessed directly and exactly or by means of a (specific) rounding function ρ˜ : R × N → Q, |x − ρ˜(x, n)| < 2−n as rational approximations of prescribable precision (“δ–Q” machine).

2.6. FURTHER MODELS

41

• If the result y of a computation x 7→ f (x) = y is independent of the rounding function employed, the adjective “robust” is affixed to it. • Are the (not necessarily rational) output approximations (pn ) to y accompanied by error bounds |y − pn | ≤ εn → 0? In this case call the machine “strongly analytic”. • Suppose a computation provides error bounds εn → 0 but violates |y −pn | ≤ εn an (arbitrary yet) finite number of times: this case is named “quasi-strongly analytic”. Figure 2.3, taken from [ChHo99], illustrates weaker/stronger relations among certain members of this family and indicates which ones satisfy closure under composition. Thus, exactly the BCSS–

Figure 2.3: Relations between some Analytic Machines

computable functions are “R–computable” and “Q–computability” is equivalent to classical (Type1) computability. The class of (ρ → ρ)–computable functions considered in Recursive Analysis coincides with the robust strongly δ–Q–analytic ones (which are in particular continuous); whereas (ρ → ρCn )–computability amounts to robust δ–Q–analyticity and includes some discontinuous functions but lacks closure under composition. Even stronger a model, an R–analytic machine, receives exact real number inputs yet has to only approximate the output. Apart from the practically unattainable power and structural lack of closure under composition, it does reflect how numerical analysis treat, for instance, integrals or nonlinear equations. It may therefore come as quite a surprise that even this machine provably cannot decide the stability of a dynamical system [ChHo99, Section 3.2]; more precisely it cannot decide, for a given initial value ~x(0), whether the solution ~x(t) of a fixed ordinary differential equation (ODE) in two dimensions converges as t → ∞. This is related to and generalizes previous results on discrete dynamical systems and Turing machines [Moor90, Delv06].

2.6

Further Models

The BCSS model (Section 2.3) and Recursive Analysis (Section 2.4) are the two major notions of computability for real number problems and the primary focus of the present work. Their

42


synthesis (exact real arithmetic/tests and approximate output computation, cf. the classification in Section 2.5.2) naturally leads to certain Analytic Machines dealt with in Section 2.5.3. However, for sake of completeness, we will now also briefly survey some other notions of real computation.

2.6.1

Domain Theory

dates back to Dana Scott [Scot70, Scot72]. It embeds real numbers R in the domain IR of closed real intervals; and continuous real (partial) functions f :⊆ R → R in order–continuous partial functions F :⊆ IR → IR. This leads to another formalization of interval computation very similar to Cauchy–computation, with two subtle differences: First, evaluating a real function f : R → R in the 2.36 and 2.42a) cor sense of Definitions responds to a computable transformation (qn ), (n ) 7→ (pm ), (δm ) on rational sequences. An output interval (pm − δm , pm + δm ) is thus allowed to depend on several input intervals qn , n ≤ N , where N may be computed adaptively, that is in turn depending on m and the values qn . However, F (restricted to the effective basis B of all rational intervals) has to assign a value to every single rational input interval. An involved compactness argument [Weih00, Theorem 9.5.2] shows that in principle this additional constraint can indeed always be satisfied. In practice, however, it renders implementing a Domain algorithm technically more involved than doing so within the setting of Recursive Analysis. Regarding partial real functions, the second difference makes Domain Computation truly more restrictive than Recursive Analysis: When a partial f is evaluated on an argument x 6∈ dom(f ), the latter allows the Turing machine to behave arbitrarily; whereas the former requires it to diverge (strong computation). For total functions, however, the two are equivalent [Weih00, Theorem 9.5.2].

2.6.2

Constructive Mathematics

Constructive mathematicians interpret quantifiers and connectives differently from classical mathematics, e.g. in logical formulas like ∀x ∀ > 0 ∃δ > 0 ∀y : |x − y| ≤ δ ⇒ |f (x) − f (y)| ≤ . Such use of the same words (‘continuity’) but with different semantics of course tends to cause misunderstandings; compare the titles of [BiBr85, Weih00] etc. (The term Church’s Thesis also has a different meaning.) Moreover, there is not just one constructive mathematics but an entire bunch of flavors [Brid04] adding to an almost Babylonian confusion of language: e.g. Intuitionistic [Heyt71], Bishop–style [BiBr85], Markov–style (see Section 2.6.3) and, more recently, Reverse Mathematics [Ishi05, Ishi06]. Their common underlying idea, however, is simple and indeed quite convincing: A claim “∃y : P (y)” that some object y with property P exists should be proven by constructing such an object, rather than indirectly by raising a contradiction from its non-existence. (The latter is denoted by “¬¬∃y : P (y)” and, lacking excluded middle, carefully distinguished from the former.) And “∀x∃y : P (x, y)” means that y is to be constructed from (and thus uniformly in) x; cf. Section 2.4.5. Constructive versus Recursive Analysis ‘Constructing y’ roughly amounts to specifying an algorithm ‘computing y’—thus the title of the text book [BiBr85]—and the relations between Constructive and Recursive Analysis are indeed quite close, with many proofs and theorems in the first having carried over to the second [Liet04]. It is the converse transfer direction which may cause problems: Weak K¨ onig’s Lemma, for instance, is considered non–constructive and easily seen to fail computably, yet has been proven equivalent to Weierstraß’ Approximation Theorem Fact 2.9 [Simp99, Theorem IV.2.5] which in turn does hold computationally [PERi89, Section 0.7]! Among the causes of such differences, we mention

2.6. FURTHER MODELS

43

• diverging purposes: A major goal of (in particular Reverse) Constructive Mathematics is the restriction and identification of minimal10 choices of logical axioms in order to prove certain claims. • the range of objects under consideration: In Recursive Analysis, a real function (and evaluation thereof; cf. Section 2.4.2) is also defined for noncomputable arguments x. For instance, constructive mathematicians say that a continuous function f on [0, 1] need not attain its supremum; whereas in Recursive Analysis, it is attained (even with a computable value y = sup f ) although on a possibly non-computable argument x. For a more thorough comparison between Recursive and (various flavors of) Constructive Mathematics, we recommend [Bees85, Chapter III]. As a consequence of the two items above, an algorithm in Recursive Analysis, although generally yielding a construction of some object x, is typically shown to be correct on the basis of classical logic. Such a proof may in particular refer to x (which constructively amounts to a circular argument) in order to assert its computability. Example 2.66. A function f : Rn → Rm is called open iff ∀¯ x ∀ > 0 ∃δ > 0 : f B(¯ x, ) ⊇ B f (¯ x), δ Computable openness of course means that from x and . The assertion that δ be computable the calculated δ indeed satisfies “f B(¯ x, ) ⊇ B f (¯ x), δ ” however may be shown classically [27, Example 34b)]; whereas constructively, the latter inclusion amounts to ∀¯ v ∈ B f (¯ x), δ ∃¯ u ∈ B(¯ x, ) : v¯ = f (¯ u) which requires that, in addition to δ, u ¯ also be ‘computed’. More generally, formula involving iterated alternating quantifiers “∀x∃y∀u∃v : P (x, y, u, v)”, interpreted constructively, require both y and v to be constructed; whereas in Recursive Analysis it may often suffice for the first existential claim to be asserted computably.

2.6.3

Markov Computability

Since most real numbers cannot be computed, why would one consider function computability (Definition 2.42a) in terms of effective evaluation x 7→ f (x) on all x ∈ R? This question has led Markov to study a notion where only computable reals x arise, input in the form of any Gödel index e (i.e., basically an encoding) of a Turing machine Me computing (rational approximations qn up to error 2−n to) x and output similarly as an index e0 for (rational approximations pm up to error 2−m to) f (x); compare [Weih00, Section 9.6]. On the basis of SMN and UTM Theorems, it is easy to see that every function computable in the sense of Definition 2.42 is also Markov computable. It turns out that some kind of ‘Main Theorem’ also holds for Markov Computation. Fact 2.67. Let f : (−1, 1) → R be total and Markov computable. Then f is continuous; even (ρ → ρ)–computable. The second claim is known as Tseitin’s Theorem [Weih00, Theorem 9.6.6]. We illustrate the first claim in a special case: Proof (Heaviside’s function is Markov-uncomputable). Assume the converse; then one can solve the Halting problem as follows: Given a(n encoding of a) Turing machine hM i, define qn,M := 2−n if M does not terminate within n steps;

qn,M := 2−n0 if M terminates at step no.n0 ≤ n .

10 Constructive Reverse Mathematics can thus be viewed as a kind of complexity theory with logical axioms as resources (recall Section 1.3) instead of time or space

44


Observe that, if hM i 6∈ H, then |qn,M − xM | ≤ 2−n for xM := 0; and |qn,M − xM | ≤ 2−n for xM := 2−n0 > 0 if hM i ∈ H with n0 denoting the number of steps M performs before termination; in particular it holds either h(xM ) = 0 or h(xM ) = 1, depending on whether hM i ∈ H or 6∈ H. Simulating M for n steps reveals (qn,M )n ⊆ Q as recursive uniformly in M ; and UTM and SMN Theorems ensure a G¨ odel index e for a Turing machine Me computing (qn,M )n (i.e. xM ) effectively obtainable from hM i. By hypothesis, one calculates from e in turn an index e0 of a Turing machine Me0 producing (pm )m ⊆ Q with |h(xM ) − pm | ≤ 2−m . In particular p2 ≤ 1/2 iff hM i ∈ H, i.e. by simulating the machine Me0 effectively obtained from hM i one can decide the termination of M —a contradiction. By taking only computable x into consideration, the notion of Markov computability may arguably be not fully a model of real number computation. Moreover, it leads to pathological counterexamples of discontinuous computable partial functions on a dense domain [Weih00, Example 9.6.5]. Also, Markov himself actually did not think in terms of Turing machines and Gödelizations but in terms of constructive mathematics—plus what has become known as Markov’s Principle: ∀x : N → {0, 1} : ¬∀n : x(n) = 0 ⇒ ∃n : x(n) = 1 . This proposition is classically true; it is well permitted in computability theory, corresponding to an unbounded search (i.e. µ–recursion as opposed to primitive recursion) with applications for instance in dove-tailing; but, in many flavors of constructive mathematics other than Markov–style, it is rather frowned upon.

2.6.4

Analog Computation

Analog machines like the Differential Analyzer [Bush31] preceded digital computers historically (and for decades flourished alongside them). An elegant theory of computability for such devices has been devised by C. Shannon [Shan41], resulting in a characterization in terms of algebraic differential functions, that is, (components of vector) functions ~y = ~y (t) of continuous time t ∈ R satisfying a differential equation P t, ~y (t), ~y 0 (t) = 0 (2.10) where P denotes an arbitrary multi-variate real polynomial. Since solutions to (2.10) may in general exist only locally, the domain of definition requires special attention lest the notion may become surprisingly ambiguous [Pour74, Kawa06]. Variations lead for example to a class of real recursive functions [Moor90, CMC02, GrCo03]. However, analog computation, in spite of its many relations to BCSS machines and Recursive Analysis [Pour74, Kawa05], exceeds the scope of the present survey. For further information, refer to [Orpo97, BoCa07].

2.6.5

Exact Computation Paradigm

The Exact Computation Paradigm is promoted, for example by Chee Yap [YaDu95], as a formalization of the requirements in Computational Geometry. Specifically, it considers inputs ~x given exactly in the form of a rational (or, slightly more general, algebraic) number/vector. Only the output real f (¯ x) is to be approximated by rationals with prescribable precision. A structural disadvantage is that the composition of two computable functions in general is not computable in this sense. Moreover, as with Markov (Section 2.6.3), restriction to rational or algebraic arguments makes Exact Computation not fully a model of real number computation. From a practical point of view, however, it may be closer to reality than any of the above. In Computational Geometry, specifically, most problems can be formalized as evaluating a function with both real (continuous) and discrete (combinatorial) values; see for instance Problem 2.68 below. Getting the discrete components calculated right (exactly or, equivalently, approximately up to error < 21 ) is crucial—lest the output be inconsistent [KMPSY04]—and can be achieved only on exact input.

2.7. PRACTICE AND EFFICIENCY

45

Problem 2.68 (Combinatorial Convex Hull). Given points ~x1 , . . . , ~xn in d-dimensional space, determine both the number and coordinates of those lying on the boundary of chull({~x1 , . . . , ~xn }), where chull(A) :=

\

C

=

A⊆C⊆Rd C convex

d nX

λi ~xi : ~x0 , . . . , ~xd ∈ A, λ0 , . . . , λd ≥ 0,

X

o λi = 1 . (2.11)

i=0

In this and many other cases, mere computability is easy11 to establish by means of exact rational/algebraic arithmetic because the input is given exactly. The challenge then consists in doing so efficiently, see Section 2.7.

2.7

Practice and Efficiency

All models are of course (and must be, recall Chapter 2) idealizations—in one direction or another—of actual numerical programming. Conversely, they have spurred (or are related to) attempts to actually realize and implement them. As already mentioned, computer algebra systems and libraries like LEDA [BKM*95, BFMS99] and Core [KLPY99] all come reasonably close to the capabilities (and semantics) of a BCSS machine; the latter is naturally biased towards the Exact Computation Paradigm (cf. Section 2.6.5). In view of its practical applications in Computational Geometry, LEDA is particularly suitable for low (algebraic) degree calculations and includes many sophisticated optimization heuristics such as filters [FoWy93] and constructive root bounds [BFMS00]; see also [LPY05]. Another implementation, based on the aforementioned publication [BrHe98] and thus realizing the computational model of Recursive Analysis, is the iRRAM [Muel01]. Its strengths consist in calculations involving high algebraic degree and transcendental numbers, functions, and operations. See for instance [Lamb05] for a competitor.

2.8

My Manifesto

As pointed out in the introduction, the purpose of classical Computability Theory is to identify, with regard to discrete problems, the limits of what computers can and what they cannot do even in principle. Correspondingly, Real Computability is concerned with problems involving real numbers. Blum, Cucker, Shub, and Smale write in [BCSS98, Section 1.4]: The developments described in the previous section (and the next) have given a firm foundation to computer science as a subject in its own right. Use of the Turing machines yields a unifying concept of the algorithm well formalized. [. . . ] The situation in numerical analysis is quite the opposite. Algorithms are primarily a means to solve practical problems. There is not even a formal definition of algorithm in the subject. [. . . ] Thus we view numerical analysis as an eclectic subject with weak foundations; this certainly in no way denies its great achievements through the centuries. Investigations on Real Computability thus have relevance to practical applications; the specific range of applicability, that is the question of which practical aspects such research reflects and which it does not, depend on the model of real computation under consideration; recall Section 1.2. Specifically, BCSS machines were introduced as an abstraction of floating point computations (one time step per operation) and neglect stability issues. In contrast, stability constitutes the major focus of Recursive Analysis as a formalization of arbitrary precision computations over R. Each model has its own counter-intuitive implications (e.g. Example 2.22 and Proposition 2.44) and rather sharply splits the community into unconditional supporters and fierce opponents. 11 For

an exception where this requires highly non-trivial arguments, see [CCK*04]

46


I myself prefer to avoid such dogmatism and choose from case to case either the one or the other model, depending on which one seems more appropriate and yields the more interesting and mathematically rich results. Moreover, combining the strengths of both models has proved to be a seminal synthesis, recall Sections 2.5.2 and 2.5.3 and see Section 6.4 below.

Chapter 3

Hypercomputation Following [CoPr99], we agree with Convention 3.1. Any (model of a) computer whose principal power strictly exceeds that of the Turing machine is called a hypercomputer. This notion is far from being a definition and may involve ambiguities. For instance, the BCSS machine’s inability to decide ‘its own’ termination H suggests it does not count as a hypercomputer; whereas the ability to solve the discrete Halting Problem H might also justify the contrary. For the purpose of the present work it is fortunately not necessary to choose sides on this issue; see Chapter 5. Also, the case of the model of Recursive Analysis is clear: being but an ordinary Turing machine, this definitely does not constitute a hypercomputer. Instead, we shall enhance it to produce a hypercomputer in various ways and investigate the resulting power in Chapter 4. But first a review of discrete hypercomputation: Section 3.1 recalls the basics of oracle computation. A device capable of deciding H (or even a higher undecidable one like ∅(d) for some d ≥ 1), plugged into a PC’s expansion slot, most naturally corresponds to the oracle Turing machine model. Approaches to physical solutions to H will be discussed in Section 3.2. Other theoretical models for them are deduced in Section 3.3.

3.1

Oracle Turing Machines

The study of hypercomputers (although they were not originally called that) actually also dates back tp Turing’s work: his PhD thesis [Turi39] deals with what are now known as Oracle Turing Machines. This formalization allows us to study, in a mathematically consistent way, a question like What could we compute—and what still not—if some (undecidable) problem O were decidable? It also yields a way to gauge (undecidable) problems according to their ‘difficulty’: Definition 3.2. Let M O denote an oracle Turing machine M ? equipped with oracle O ⊆ N. Problem A is many–one reducible to B, written as “A m B”, if there exists a computable total function f satisfying the following condition: x belongs to A if and only if f (x) belongs to B. A is (Turing–) reducible to B (“A T B”) if A can be decided by a Turing machine with oracle access to B. The Turing degree of A is the set of all problems B ≡ A equivalent to (i.e. reducible both to and from) A.

3.1.1

Power of the Halting Oracle

By querying H, an oracle machine can decide the termination of a given Turing Machine M (on a given specific input or, equivalently, on the empty input). 47

48

CHAPTER 3. HYPERCOMPUTATION

Fact 3.3. Every semi-decidable problem (or its complement) becomes decidable relative to H. On the other hand, the H–oracle is provably not sufficient (cf. Example 3.12) to decide the following problem even more relevant to automated software verification than H (cf. Section 1.1.2): Definition 3.4. Tot is the question of whether a given Turing machine M terminates on all possible inputs. Fin is the question of whether the language accepted by a given Turing machine M is finite (i.e. if M terminates on at most finitely many inputs). A characterization of computation with the support of a Halting oracle is given by Lemma 3.5 (Shoenfield’s Limit12 ). A function f :⊆ N → N is computable relative to H iff f (n) = limm g(n, m) is the pointwise limit of an ordinarily computable function g : dom(f ) × N → N. Since this claim can be concluded ‘bitwise’ from Theorem 3.9 below, we omit a separate proof but refer to, e.g., [Soar87, §III.3.3] and point out a putative pitfall. Suppose one naively simulates computation of n 7→ f (n) in the following way: For each query “x ∈ H?” answer no and, while continuing the simulation (and printing the result when it terminates), start simultaneously looking for x ∈ H (semi-decidable). If it turns out that x does belong to H, then restart the simulation with the corrected answer yes. Then the sequence of outputs g(n, 1), g(n, 2), . . . might never attain the correct value because the algorithm for f (n) can lead to infinitely many oracle queries when given incorrect answers. More specifically, consider the following computation: Upon input of n, query “x ∈ H?” for x = n, n + 1, . . . and for the first positive answer output x and terminate. The Padding Lemma (recall Section 1.1.1) ensures that this yields a total function, but (a careless distribution of priorities during dove-tailing in) the above naive simulation may never terminate! Closely related is the area of revising computation, also known as Limiting [Gold65], Trialand-Error [Putn65], Inductive [Burg04], or General [Schm02] Turing Machines. Indeed, even semidecidability can be regarded as a relaxation of decidability which permits finitely but unboundedly many one-sided errors in the form of false negatives: a machine may report “hM i 6∈ H” unless and until termination of M is detected. Let us generalize this to two-sided errors: Definition 3.6. A revising machine may make an infinite number of steps but has to write its result onto some tape in the form of a finite string. This output may be reverted an arbitrary yet finite number of times: A revising name for a (delimited) string ~σ ∈ {0, 1}∗ ◦ { }, ~σ = (σ1 , . . . , σN −1 , ), is a sequence (~τm )m of (delimited) strings ~τm ∈ {0, 1}∗ ◦ { } such that ~σ = limm ~τm , i.e. such that ∀n ≤ N ∃M ∀m ≥ M : τn,m = σn (pointwise convergence); equivalently: uniform convergence ∃M ∀m ≥ M : ~τm = ~σ . By Lemma 3.5, a function f :⊆ N → N is revisingly computable (i.e. computable by a machine with the above semantics) iff it is computable relative to H. Motivation for Revising Output Revising machines can be regarded as models that take into account that the display of, say, a video terminal may change; and, even if it eventually becomes fixed, we can never be sure when. Recall the two most basic ascii control characters, which even the earliest text display consoles understood: BS and CR . The first, called “backspace”, moves the cursor left by one position, allowing the last printed symbol to be overwritten; the second, “carriage return”, is the command to restart output from the beginning (of the present line). 12 The

name seems to be more historical folklore than to indicate the discoverer of the result.

3.1. ORACLE TURING MACHINES Example 3.7. The character sequence G o o d b y e CR H e l l o, will display as: Hello, world.

49

M r s BS BS BS w o r l d

Let us consider a non-terminating program (such as an operating system) generating an infinite sequence of characters including BS and CR ; how do they appear on an (unbounded, one-line) display? Let us make Requirement 3.8. Each character position has to settle down eventually, leading ultimately to the display of a finite string ~σ (without BS and CR ). This kind of output is obviously equivalent to the (classical) generation of a revising name for ~σ : every change from ~τm to ~τm+1 can be effectively expressed as a sequence of ‘editing operations’ involving BS ; and conversely, by considering, for m = 1, 2, . . ., the first m characters of ~σ and applying the edit instructions it contains, we obtain a string ~τm converging to ~σ due to Requirement 3.8.

3.1.2

Arithmetical Hierarchy

The question H =: ∅0 whether a given Turing machine terminates (on the empty input, say), is undecidable (relative to a decidable oracle like ∅) but, relative to ∅0 , it is decidable. The proof relativizes, that is, it carries over to arbitrary oracles: Termination of a given Turing machine with oracle H, that is the jump H 0 =: ∅00 of the ordinary Halting problem, is undecidable relative to ∅0 yet decidable relative to ∅00 . Iteration ? ∅(d−1) ∅(d) := hM , xi M terminates on input x (3.1) leads to Kleene’s Arithmetical Hierarchy ∅

¬m

∅0 = H

¬m

∅00 = H 0

¬m

...

¬m

∅(d)

¬m

... .

(3.2)

where “A ¬m B” means “A m B ∧ B 6 A”. A syntactical characterization of the levels of this hierarchy was obtained by E. Post: Theorem 3.9 (Normal Form). For L ⊆ X := N and d ∈ N, the following are equivalent: a) L is semi-decidable relative to ∅(d−1) b) There exists a decidable R ⊆ X such that L = x ∈ X ∃y1 ∈ N ∀y2 ∈ N ∃y3 ∈ N . . . θd yd ∈ N : hx; y1 , . . . , yd i ∈ R .

(3.3)

Here h · i : N∗ → N denotes a computable tupling function and θd either the universal or the existential quantifier, depending on d’s parity. c) L m ∅(d) . Obviously, L is decidable relative to an oracle O iff both L and its complement X \ L are semi-decidable relative to O. Definition 3.10. Write Σ0 = Π0 = ∆0 = ∆1 for the class of all decidable problems. With d ∈ N, Σd is the class of all problems L of the form (3.3); equivalently: of all problems semi-decidable relative to ∅(d−1) . Πd denotes the class of complements of languages in Σd ; and ∆d := Σd ∩ Πd the class of problems decidable relative to ∅(d−1) . Theorem 3.9 and Equation (3.2) translate to this picture: ∆ 1 ( Σ1 ( ∆ 2 ( Σ2 ( . . . ( ∆ d ( Σ d ( . . .

(3.4)

We shall encounter and study real variations of this hierarchy both for the BCSS model and in Recursive Analysis in Sections 5.4 and 4.1, respectively.

50

CHAPTER 3. HYPERCOMPUTATION

Proof (Theorem 3.9). c⇒a): ∅(d) is semi-decidable relative to ∅(d−1) . b⇒c) holds by induction on d, starting with d = 1 by definition. For d > 1, consider

K := x, y1 ∀y2 ∃y3 . . . θd yd : hx, y1 i; y2 , . . . , yd ∈ R . X \ K ∈ Σd−1 is, by induction hypothesis, many-one reducible to ∅(d−1) . Let f denote such a computable reduction function. Then some fixed x ∈ X belongs to L iff there exists some (d−1) y1 ∈ N such that f (hx, y1 i) ∈ ∅(d−1) ; that is, iff the oracle Turing machine Mx∅ searching (d−1) ∅ for such a y1 terminates. The computable mapping x 7→ hMx i therefore constitutes a reduction of L to ∅(d) . a⇒b) Let L be semi-decidable relative to K := ∅(d−1) by some oracle machine M . On input of x ∈ L, M makes finitely many oracle queries “z ∈ K?” where z ∈ O+ ∪ O− ; and O+ ⊆ K denote the set of those answered positively, O− ⊆ ¬K those answered negatively. In fact, x ∈ L iff there exist finite sets O+ ⊆ K and O− ⊆ ¬K such that M makes only queries ˜ that to O+ ∪ O− and terminates. Now it is easy to construct an oracle–free machine M semi-decides the language ˜ := L hx, O+ , O− i : M terminates on x, making oracle queries only to O+ ∪ O− . (Here we refer to a straightforward encoding of finite sets O ⊆ N into integers such that ˜ ∈ Σ1 ; and hOi ≥ max O.) Therefore, L ˜ ∧ O+ ⊆ K ∧ O− ⊆ ¬K . L = x ∃hO+ , O− i ∈ N : hx, O+ , O− i ∈ L Since K = ∅(d−1) is semi-decidable relative to ∅(d−2) , K ∈ Σd−1 by induction hypothesis. Condition “O+ ⊆ K” is equivalent to “∀o ≤ hO+ i : o 6∈ O+ ∨o ∈ K” (since max O ≤ hOi, see above): a formula whose first (i.e. the universal) quantifier has a bounded range. Regarding condition “O− ⊆ ¬K”, its negation is equivalent to “∃o : o ∈ O− ∧ o ∈ K”. Now apply Lemma 3.11 below. a) For K1 , K2 ∈ Σd , K1 ∪ K2 , K1 ∩ K2 ∈ Σd .

Lemma 3.11 (Closure Properties).

b) If K ∈ Σd , then {x ∈ X|∃y : hx, yi ∈ K} ∈ Σd . c) If K ∈ Σd−1 , then ¬K ∈ Πd−1 ⊆ Σd . d) If K ∈ Σd , then {x|∀y ≤ x : hx, yi ∈ K} ∈ Σd . Concerning the latter, observe that ∀y ≤ x∃z : R(x, y, z)

⇔

∃hz1 , . . . , zx i ∀y ≤ x : R(x, y, zy ), | {z } ˜ =:R(x,hz 1 ,...,zx i)

that is, quantifiers with bounded range may be swapped and partially eliminated. The following generalizes Fact 3.3: Example 3.12 (Complete Problems). Σd is many-one reducible to ∅(d) .

a) It holds ∅(d) ∈ Σd . Conversely, every problem in

b) The problem Fin belongs to Σ2 ; and every problem in Σ2 can be many-one reduced to Fin. c) The problem Tot belongs to Π2 ; and every problem in Π2 can be many-one reduced to Tot. See e.g. [Soar87, Theorem IV.3.2]. . . In particular, Tot is not semi-decidable relative to H!

3.1. ORACLE TURING MACHINES

3.1.3

51

Hyperarithmetical and Analytical Hierarchy

The many similarities of the Arithmetical Hierarchy (3.4) with the Borel one (2.1) have led to the effective Borel Hierarchy discussed in Section 4.1. It also resembles the Polynomial Hierarchy In complexity theory, extending the famous classes P and N P by iterated oracles gives rise to a hierarchy which can equivalently be described by alternating quantifiers [Papa94]: P = ∆P 1

⊆

N P = ΣP 1

⊆

P N P = ∆P 2

⊆

N P N P = ΣP 2

⊆

...

where ΣP d is the class of languages L of the form L

=

x ∈ {0, 1}∗ ∃y1 ∈ {0, 1}∗ ∀y2 ∈ {0, 1}∗ ∃y3 ∈ {0, 1}∗ . . . θd yd ∈ {0, 1}∗ : |y1 |, . . . , |yd | ≤ p(|x|) ∧ hx; y1 , . . . , yd i ∈ P

(3.5)

for some integer polynomial p and a language P ∈ P. (As opposed to strictness of the Arithmetical Hierarchy, that of the Polynomial one is a serious open question worth $1,000,000—recall Exercise 2.52.) All classes ΣP d , d ∈ N, are contained in PSPACE, the collection of problems decidable within polynomial space. A variant of Cook’s Theorem (PSPACE–hardness of TQBF) implies that PSPACE coincides with the class ΣP poly of problems of the form L

=

x ∈ {0, 1}∗ ∃d ∈ N ∃y1 ∈ {0, 1}∗ ∀y2 ∈ {0, 1}∗ ∃y3 ∈ {0, 1}∗ . . . θd yd ∈ {0, 1}∗ : d ≤ p(|x|) ∧ |y1 |, . . . , |yd | ≤ p(|x|) ∧ hx; y1 , . . . , yd i ∈ P (3.6)

for some integer polynomial p and P ∈ P. In particular, ΣP poly is closed under complement! Notice that, as opposed to Equation (3.5), the number d of quantifier alternations here varies polynomially in the input length |x|. Transfinite Arithmetical Hierarchy In a similar way to Equations (3.5) and (3.6), extend (3.3) and consider languages of the form LR = x ∈ X ∃d ∈ N ∃y1 ∈ N ∀y2 ∈ N ∃y3 ∈ N . . . θd yd ∈ N : hx; y1 , . . . , yd i ∈ R (3.7) for decidable R. S Equation (3.3) describes all problems in the finite Arithmetical Hierarchy Σ ) exchanged. . . Question 4.17. Nonuniformly, does it hold sup S ∈ Π2 (R) for every S ∈ Π2 (2R )? Or at least for every S ∈ Π2 ∩ Σ1 ? S Proof (Lemma 4.16). c) Let S = n An be given by ψ> –names of closed An ⊆ [0, 1]. For each n, compute by virtue of b) a rational sequence (qn,m )m with inf m qn,m = sup An . According to Claim 4.18 below, (qn,m ) is then a ρ0< –name for sup S = supn inf m qn,m . S Claim 4.18. For SiS ⊆ R, i ranging over an arbitrary index set I, it holds sup( i Si ) = supi sup Si and inf(T i Si ) = inf i inf Si . T Moreover, it is sup( i Si ) ≤ supi sup Si and inf( i Si ) ≥ inf i inf Si . It is easy to find (even topologically very simple) examples of strict inequalities in the case of intersection. S S Proof. sup( i Si ) for each i by monotonicity; and hence supi sup Si ≤ S Since Si ⊆ i Si , sup Si ≤ S sup(Si Si ). Conversely every x ∈ i Si has x ≤ supi sup Si (because x ∈ Si for some i); therefore sup( i Si ) ≤ supi sup Si . The next lemma can be regarded as a partial converse to Lemma 4.16. Again, Claims a) and b) are folklore. Lemma 4.19.

a) The mapping R2 3 (a, b) 7→ (a, b) ∈ O is (ρ> × ρ< → δΣ1)–computable.

b) The mapping R2 3 (a, b) 7→ [a, b] is (ρ< × ρ> → δΠ1)–computable. c) The mapping R2 3 (a, b) 7→ [a, b] is (ρ0< × ρ0> → δΠ2)–computable. In view of Fact 4.15, the mapping R2 3 (a, b) 7→ (a, b) ∈ O cannot be (ρ0> × ρ0< → δΣ2)–computable. Proof. b) For a = supn qn and b = inf m pm output open rational intervals (exhausting) (−∞, qn ) and (pm , +∞): these cover exactly (and hence constitute a δΣ1–name of) (−∞, a) ∪ (b, +∞); even in the degenerate cases a = b and a > b. c) For notational simplicity we consider only a = supn inf m qn,m as given and fix b = 0. We may w.l.o.g suppose that inf m qn,m < inf m qn+1,m < a holds: apply T Claim 4.24. According to Item a), δΣ1–compute Un := (inf m qn,m , 1/n) and observe that n Un = [a, 0] holds.

72

4.2.2

CHAPTER 4. ARITHMETICAL HIERARCHIES IN RECURSIVE ANALYSIS

Transfinite Levels

Similar to Section 4.1.1, one may extend the finite Arithmetical Hierarchy of real numbers (4.3) to transfinite levels. This is easy for the ambiguous classes by defining, in view of the first item in Theorem 4.9, X −n (4.4) ∆α (R) := 2 S ∈ ∆α (N) n∈S

with ∆α (N) according to Section 3.1.3. However an approach based on binary expansions does not work for classes Σα . A definition based on extending the last item of Theorem 4.10 to an infinite number of “sup / inf” alternations seems more natural; but a multi-sequence qhn1 ,...,nd i of course makes sense only for finite d. G. Barmpalias had the idea of letting d be finite but vary; he defined, analogously to Equation (3.7), Σω (R) := sup sup inf sup . . . lim qhd,n1 ,...,nd i (qn )n ⊆ Q computable . d

n1 n2 n3

nd

This approach generalizes to higher recursive ordinals and indeed yields a proper hierarchy [Barm03, Theorem 4]. For P the ambiguous classes it coincides with (4.4); and, just as for the finite levels α < ω, it holds n∈S 2−n ∈ Σα (R) if (but not only if) S ∈ Σα (N) [Barm03, Proposition 3].

4.3

Hypercomputing Real Functions

In addition to real numbers, [Ho99] has also studied real function computability (Definition 2.42a) relative to ∅0 and obtained, for instance, the following analogue of Fact 2.45 with non-effective uniform convergence: Proposition 4.20 ([Ho99, Corollary 17]). A function f : [0, 1] → R is computable relative to ∅0 iff there exists a Turing–computable sequence of (degrees and coefficients of ) rational polynomials (Pm ) ⊆ Q[X] such that kf − Pm k → 0. Notice the similarity to Proposition 4.8. In particular the Halting oracle, employed in this way, does not lift the continuity condition Proposition 2.44. More generally it immediately follows from the classical Fact 2.9 in combination with the relativization of Fact 2.45: Corollary 4.21. A function f : R → R is continuous iff there exists an oracle O ⊆ N such that f is computable relative to O. Compare [Mosc80, Exercise 3D.21]. . .

4.3.1

Weak Function Evaluation and Continuity

The Main Theorem of Recursive Analysis (Proposition 2.44) requires any computable real function to be continuous. This raises Question 4.22. Does real hypercomputation permit an effective evaluation of (certain) discontinuous functions? In view of Corollary 4.21, the answer seems negative. However, a characteristic of real hypercomputation is that it naturally includes non-oracle ways of transcending the Church-Turing Hypothesis. (d) Specifically, the present section investigates (ρ(k) → ρ(d) )–computable, (ρ(k) → ρ< )–computable, (k) (d) and (ρ< → ρ< )–computable real functions in the sense of Section 4.2. (Another interesting no(d) tion will be dealt with in Section 4.5.) Indeed, the representations ρ(d) for d ≥ 1 and ρ< for d ≥ 0 fail the admissibility hypothesis of Fact 2.62; and this kind of real hypercomputation does indeed open the door to ‘counterexamples’ to Proposition 2.44:

4.3. HYPERCOMPUTING REAL FUNCTIONS

73

Example 4.23. Heaviside’s function (cf. Equation (2.7) and Figure 2.2) is both (ρ< → ρ< )– computable and (ρ0< → ρ0< )–computable. Proof. Given (qn ) ⊆ Q with x = supn qn , exploit (νQ → νQ )–computability of the restriction h|Q : Q → {0, 1} to obtain pn := h(qn ). Then (pn ) ⊆ Q has supn pn = h(x): In case x ≤ 0, qn ≤ 0 and hence pn = 0 for all n; whereas in case x > 0, qn > 0 and hence pn = 1 for some n. Let x ∈ R be given by a rational double sequence (qi,j ) with x = supi inf j qi,j . By Claim 4.24 below we may w.l.o.g assume that inf j qi+1,j ≥ inf j qi,j . Now compute pi,j := h(qi,j −2−i ). Then in case x ≤ 0, it holds that ∀i∃j : qi,j ≤ 2−i , i.e., pi,j = 0 and thus supi inf j pi,j = 0 = h(x). Similarly, in case x > 0, there is some i0 such that inf j qi0 ,j > x/2 and thus inf j qi,j > x/2 for all i ≥ i0 . For i ≥ i0 with 2−i ≤ x/2, it follows that pi,j = 1 ∀j and therefore supi inf j pi,j = 1 = h(x). Claim 4.24. For real double-sequence (qn,m ) let qñ,m := min{qn,1 , . . . , qn,m } + 2−m and ˜˜qn,m := max{˜ q1,m , . . . , qñ,m } − 2−n . Then it holds a) ˜˜qn,m+1 ≤˜ ˜qn,m − 2−m−1 b) inf m˜ ˜qn+1,m ≥ inf m˜ ˜qn+1,m + 2−n−1 c) supn inf m˜ ˜pn,m = supn inf m qn,m . Proof. See the proof of [ZhWe01, Lemma 3.1]. So real function hypercomputation based on representations weaker than ρ does allow effective evaluation of some discontinuous functions. On the other hand, they still impose well-known restrictions (recall Section 2.2.3): Fact 4.25. Consider f : R → R. a) If f is (ρ → ρ)–computable, then it is continuous. b) If f is (ρ → ρ< )–computable, then it is lower semi-continuous. c) If f is (ρ< → ρ< )–computable, then it is nondecreasing. d) If f is (ρ0 → ρ0 )–computable, then it is continuous. The claims remain valid under oracle-supported computation. Claim a) is the Main Theorem; the others are proven similarly on the basis of discontinuity arguments. For b) see e.g. [WeZh00]. Establishing d) in [BrHe02, Section 6] caused some surprise. We briefly sketch the proofs as a preparation for those of Theorem 4.26 below. Proof.

a) Cf. the proof of Proposition 2.44.

b) As in a), we prove (ρ → ρ< )–uncomputability of the flipped Heaviside Function h = 1 − h as a prototype lacking lower semi-continuity. Consider the ρ–name qn := 2−n for x = 0 which the hypothetical Type-2 Machine transforms into a ρ< –name for y = h(x) = 1, that is, a sequence (pm ) ⊆ Q with supm pm = y; in particular, for some M ∈ N, pM ≥ 32 gets output, having read only (qn )n≤N for some N ∈ N. The latter finite segment is also the initial part of a valid ρ–name for x ˜ = qN > 0, whereas (pm )m≤M has sup ≥ 23 and thus is not the initial part of a valid ρ< –name for y˜ = h(˜ x) = 0; contradiction. This proof for the case h carries over to an arbitrary f : R → R just like in a), that is, by replacing qn = 2−n with rational approximations to a general sequence xn ∈ R witnessing the violation of lower semi-continuity of f in that f (limn xn ) > lim inf n f (xn ).

74

CHAPTER 4. ARITHMETICAL HIERARCHIES IN RECURSIVE ANALYSIS c) As in a) and b), we treat for notational simplicity the case of f : R → R violating monotonicity in that f (0) = 1 and f (1) = 0; the general case can again be handled similarly. Feed the ρ< –name (qn ) = (0, 0, . . .) for x = 0 into a machine which, according to the presumption, produces a sequence (pm ) ⊆ Q with sup pm = 1 and in particular pM ≥ 32 for some M ∈ N. Up to output of pM , only (qn )n≤N has been read for some N ∈ N. Now consider the rational sequence (˜ qn ) consisting of N 0s followed by an infinity of 1s, that is, a valid ρ< –name for x ˜ = 1. This new input will cause the machine to output a sequence (˜ pm ) ⊆ Q coinciding with (pm ) for m ≤ M ; in particular, p˜M ≥ 32 , contradicting the hypothesis that (˜ pm ) is supposed to satisfy supm p˜m = f (˜ x) = 0. d) Suppose that, in spite of its discontinuity at x = 0, h be (ρ0 → ρ0 )–computable by some Type-2 Machine M. (1) (1) Consider the sequence q (1) := qn ⊆ Q, qn :≡ 1, which is by definition a valid ρ0 –name (1) for 1 =: x(1) = limn qn . Thus upon input of q (1) , M will generate a sequence p(1) ⊆ Q as a (1) (1) ρ0 –name for y (1) = h x(1) = 0, that is, satisfying limm pm = 0; in particular, pm1 ≤ 13 for some m1 ∈ N. Up to this output, M has read only a finite initial part of the input q (1) , say, up to n ≤ n1 . (2)

(2)

Next consider the sequence q (2) ⊆ Q defined by qn := 1 for n ≤ n1 and qn := 12 for n > n1 : a valid ρ0 –name for x(2) = 21 which M by presumption transforms into a sequence p(2) ⊆ Q (2) (2) with limm pm = y (2) = h x(2) = 0; in particular, qm2 ≤ 31 for some m2 > m1 . However, due to M’s deterministic behavior and since q (1) and q (2) initially coincide, it still holds (2) pm1 ≤ 13 . Now by repeating the above argument we obtain a sequence of sequences q (k) ⊆ Q, each constant for n ≥ nk of value (and thus a valid ρ0 –name for) x(k) = 2−k+1 and transformed (k) by M into a sequence p(k) ⊆ Q satisfying pmi ≤ 31 for i = 1, . . . k with strictly increasing (k) (ω) (nk ), (mk ) ⊆ N. The ultimate sequence q (ω) ⊆ Q, well-defined by qn := qn for n ≤ nk (k) (and in fact the limit of the sequence of sequences q with respect to Baire’s Topology), k therefore converges to (and is therefore a valid ρ0 –name for) x(ω) = 0; and it is mapped (ω) by M to a sequence q (ω) ⊆ Q satisfying qm ≤ 31 for infinitely many m, contradicting the 0 (ω) hypothesis that a valid ρ –name for y = h x(ω) = 1 should have limm = 1. Being of information-theoretic nature, the above arguments obviously relativize. The main result of the present section is an extension of Fact 4.25 to one level up on the hierarchy of real representations from Definition 4.12. This might not be as surprising any more as Fact 4.25d) in [BrHe02]; nevertheless, even this additional step makes proofs significantly more involved. Theorem 4.26. Consider f : R → R. a) If f is (ρ0 → ρ0< )–computable, then it is lower semi-continuous. b) If f is (ρ0< → ρ0< )–computable, then it is nondecreasing. c) If f is (ρ00 → ρ00 )–computable, then it is continuous. The claims remain valid under oracle-supported computation. We point out that the proofs of Fact 4.25 proceed by constructing an input for which a presumed machine M fails to produce the correct output. They differ, however, in the ‘length’ of these constructions: for Claims a) to c), the counter-example inputs are obtained by running M for a finite number of steps on a single, fixed argument; whereas in the proof of Claim d), M is repeatedly started on an adaptively extended sequence of arguments. The latter argument may thus be considered as of length ω, the first infinite ordinal. Our proof of Theorem 4.26c) will be even longer and is therefore put into the following subsubsection.


75

Proof of Theorem 4.26 As in the proof of Fact 4.25, we treat the special case of the flipped Heaviside Function h for reasons of notational convenience and clarity of presentation; the arguments can be immediately extended to the general case. Claim 4.27. h : R → R is not (ρ0 → ρ0< )–computable. Proof. Suppose a Type-2 Machine M (ρ0 → ρ0< )–computes h. In particular, upon input of x(1) = 1 (1) (1) in the form of the sequence q (1) = (qn ) with qn :≡ 1, M will output a rational double sequence (1) (1) (1) p(1) = (pk,` ) with 0 = y (1) := h(x(1) ) = sup inf pk,` . Observe that p1,`1 ≤ 13 for some `1 . When k

(1)

`

(1)

p1,`1 is written, M has only read a finite part of (qn ), say, up to n1 . (2)

Now consider x(2) := 21 , given by way of the sequence q (2) with qn := 1 for n < n1 (2) and qn := 21 for n ≥ n1 . Then, too, M will output a double sequence p(2) with 0 = y (2) = (2) (2) (2) sup inf pk,` . Observe that, similarly, some p2,`2 ≤ 13 is output when only a finite part of (qn ), k

`

say, up to n2 , has been read. Moreover, as q (1) and q (2) coincide up to n1 and since M operates (2) (1) deterministically, p1,`1 = p1,`1 ≤ 13 .

Figure 4.3: Illustration of the iterative construction employed in the proof of Claim 4.27 Continuing this process with x(k) := 2−k+1 for k = 3, 4, . . . as indicated in Figure 4.3 eventually (ω) yields a rational sequence q (ω) with limn qn =: x(ω) = 0, upon input of which M outputs a double (ω) (ω) sequence p(ω) such that pk,`k ≤ 13 for all k = 1, 2, . . .. In particular, y (ω) := sup inf pk,` ≤ 31 whereas k

`

h(x(ω) ) = 1 : contradiction. Note that the above proof involves one-dimensionally indexed sequences (qn ) for input and twodimensionally indexed ones (pk,` ) for output. We now proceed a step further in proof difficulty, namely involving two-dimensional indices for both input and output in order to establish Item b). Claim 4.28. Let f : R → R violate monotonicity in that f (0) = 1 and f (1) = 0. Then f is not (ρ0< → ρ0< )–computable. Proof. We construct a ρ0< –name for x = 0 from an iteratively defined sequence of initial segments of ρ0< –names for x = 1: (1) (1) Start with qi,j := 1 for all i, j. Then q (1) = (qi,j ) is obviously a ρ0< –name for x = 1 and thus (1)

(1)

yields by presumption, upon input to M, a ρ0< –name pk,` for f (1) = 0, that is, with 0 = sup inf pk,` . k

(1)

1 3 for some (1) p1,`1 , M has

In particular, p1,`1 ≤

`

`1 .

Until output of read only finitely many entries of q (1) ; say, up to i1 and j1 , that is, covered in Figure 4.4 by the light gray rectangle. Now consider q (2) defined as in this figure: (2) (2) (2) Since inf j qi,j = 0 for i ≤ i1 and inf j qi,j = 1 for i > i1 , sup inf qi,j = 1, that is, this is again i

j

a valid ρ0< –name for x = 1; and again, M will, according to the presumption, convert q (2) into

76


Figure 4.4: Illustration of the iterative construction employed in the proof of Claim 4.28 (2)

a ρ0< –name p(2) for f (1) = 0. In particular, p2,`2 ≤ 13 for some `2 ; and, being a deterministic machine, M’s operation on the initial part (dark gray) on which input q (2) coincides with input (2) (1) q (1) will first have generated the same initial output, namely p1,`1 = p1,`1 ≤ 12 . (2)

Again, until output of p2,`2 , M has read only a finite part of q (2) of, say, up to i2 > i1 (light gray). (3)

By now considering input q (3) with inf j qi,j = 0 for i ≤ i2 as in Figure 4.4, we arrive at p(3) and (3)

(3)

(3)

`3 with p1,`1 , p2,`2 , p3,`3 ≤ 13 ; and so on with i3 , q (4) , p(4) , `4 , i4 , . . . Finally observe that continuing these arguments eventually leads to a rational double sequence (ω) (∞) q (ω) = (qi,j ) which has inf j qi,j = 0 for i ≤ i∞ = ∞—and is therefore a valid ρ0< –name for x = 0 (ω)

(∞)

(∞)

(rather than x = 1)—but gets mapped by M to p(ω) = (pk,` ) with inf ` pk,` ≤ pk,`k ≤ 31 for all k. Since f (0) = 1, this contradicts our presumption that M maps ρ0< –names for x to ρ0< –names for f (x). The above proofs involving ρ0 and ρ0< proceeded by constructing an infinite sequence of inputs q (1) , q (2) , . . . , q (ω) (each possibly a multi-indexed sequence of its own). In order to finally assert Claim c) involving ρ00 , we will extend this method from length ω, the first infinite ordinal, to an even longer (but still countable) one. Claim 4.29. h : R → R is not (ρ00 → ρ00 )–computable. Proof. Outwit a Type-2 Machine M, presumed to realize this computation, as follows: (1)

i) Take q (1) to be the constant double sequence 1, i.e., qi,j := 1 for all i, j. Being a ρ00 – name for 1, it is by presumption mapped to a ρ00 –name p(1) for h(1) = 0, that is, satisfying (1) (1) limk lim` pk,` = 0. In particular, almost every column #k contains an entry #` with pk,` ≤ 31 . (1)

Until output of the first such pk1 ,`1 , M has read only a finite part of q (1) —say, up to i1 , j1 . ii) Observe that this Argument i) equally applies to the scaled input sequence 2−m · q (1) for any (2) (1) (2) m. So define qi,j := qi,j for j ≤ j1 (i.e., inherit the initial part of q (1) ) and qi,j :≡ 21 for (2)

j > j1 . Now upon input of this q (2) , M will output p(2) with, again, infinitely many pk,` ≤ 13 , the first one—(k2 , `2 ), say—after having read q (2) only up to some (i2 , j2 ). Furthermore, M’s (2) (1) determinism implies pk1 ,`1 = pk1 ,`1 ≤ 31 . By repeating for m = 2, 3, . . ., we eventually obtain—similarly to the proof of Claim 4.28— (ω) (ω) an input sequence q (ω) with qi,j with limi limj qi,j = 0, that is, a valid ρ00 –name for x = 0


77

Figure 4.5: The first infinitely long iterative construction employed in the proof of Claim 4.29 (ω)

(rather than 1). This is mapped by M to p(ω) with pkm ,`m ≤ 31 for all m. On the other hand, p(ω) is by presumption a ρ00 –name for h(0) = 1. Therefore, there are infinitely many (ω) (ω) m with pm,` ≥ 32 for some ` > `m and pm,`m ≤ 13 ; see the gray columns in the right part of Figure 4.5. iii) Since this gives no contradiction yet, we proceed by considering the first such column m containing an entry ≤ 13 as well as an entry ≥ 23 . Take the initial part of the input q (ω) — up to (iω , jω ), say, depicted in gray in the left part of Figure 4.6 — that M has read until output of both of them; extend it with 12 s in top direction and with 1s to the right. Feed this ρ00 –name for x = 1 into M until output of an entry pk,` ≤ 13 in some column k beyond m. Then repeat extending to the right with 1s replaced by 21 s for a second entry pk,` ≤ 13 .

Figure 4.6: Second infinitely long iterative construction employed in the proof of Claim 4.29 (ω)

More generally, proceed similarly as in ii) and extend qiω ,iω to the right with some ρ00 –name q 0(ω) for x = 0 in such a way as to obtain another column m0 with both entries ≤ 31 and ≥ 23 ; see the middle part of Figure 4.6. Again, M outputs the latter two entries having read only a finite part; say, up to (i0ω , jω0 ). Now extend this part, too, with 12 in top direction and with another q 00(ω) obtained, again, as in ii) for a third column m00 with both entries ≤ 13 and ≥ 32 ; and so on.

78

CHAPTER 4. ARITHMETICAL HIERARCHIES IN RECURSIVE ANALYSIS This eventually leads to an input q (2ω) which, due to the extensions to the top, represents a ρ00 –name for x = 12 and is thus mapped, according to the presumption, to a ρ00 –name p(2ω) for h( 12 ) = 0. In particular, almost every column of p(2ω) has almost every entry ≤ 13 while maintaining infinitely many columns with preceding entries ≤ 13 and ≥ 32 ; see the right part of Figure 4.6. This ensures the existence of infinitely many columns in p(2ω) containing ≤ 13 , ≥ 23 , and ≤ 13 in order. And again, even just a finite initial part of q (2ω) up to some (i2ω , j2ω ) gives rise to the first such triple.

iv) Notice that the arguments in iii) similarly yield the existence of an appropriate, scaled counter-part 12 q 0(2ω) of q (2ω) , of some 41 q 00(2ω) , and so on, all leading to output containing infinitely many columns with alternating triples as above. We now construct input q (3ω) leading to output p(3ω) containing an infinity of columns, each with four entries ≤ 13 , ≥ 32 , ≤ 13 , and ≥ 23 . Take the initial part of q (2ω) leading to output of the first column with alternating triple in the above sense; then extend it with the initial part of the scaled version 12 q 0(2ω) leading to another column with such a triple; and so on. Observing that, due to the scaling, the q (3ω) obtained in this way represents a ρ00 –name for x = 0, and almost every column of the output p(3ω) representing h(0) = 1 contains entries ≥ 23 in addition to the infinitely many columns with triples as above; see the left part of Figure 4.7.

Figure 4.7: Third, fourth, and fifth infinitely long iterative construction employed in the proof of Claim 4.29 v) Our next step is a ρ00 –name q (4ω) for x = 41 giving rise to p(4ω) with infinitely many columns containing alternating quintuples. This is obtained by repeating the arguments in iv) to obtain initial segments of (variants of) q (3ω) , stacking them horizontally—in order to obtain an infinity of columns with alternating quadruples—while extending in top direction with 1 00 (4ω) for h( 14 ) = 0 4 ; see the middle part of Figure 4.7. This forces M to output a ρ –name q 1 having in almost every column almost every entry ≤ 4 , thereby extending the alternating quadruples to quintuples. vi) Noticing that the vertical extension in v) was similar to step iii), we now take a step similar to iv) based on horizontally stacked initial parts of scaled counterparts of q (4ω) in order to obtain a ρ00 –name q (5ω) for x = 0 which M maps to some p(5ω) containing infinitely many alternating six-tuples. Then again construct a ρ00 –name q (6ω) for x = 81 by horizontally stacking initial segments of (variants of) q (5ω) while extending them vertically with 18 and so on.


79

Now for the bottom line: By proceeding with the above construction, one eventually obtains 2 (ω 2 ) a rational double sequence q (ω ) with limj qi,j = 0 for all i — that is, a ρ00 –name for x = 0 2

— mapped by M to some p(ω ) containing (infinitely many) columns #k with infinitely many alternating entries ≤ 31 and ≥ 23 — contradicting the hypothesis that, for ρ00 -names p = (pk,` ), lim` pk,` is required to exist for every k.

4.3.2

Hierarchy of Weakly Evaluable Functions

For every (α → β)–computable function f : A → B, one may obviously replace representation α for A by a stronger one and β for B by a weaker one while yet maintaining the computability of f: f (α → β)–computable ∧ α α ∧ β β ⇒ f (α → β)–computable However, if both α and β are made, say, weaker then the (α → β)–computability of f may in general be violated. For α = β = ρ< , though, we have seen in Example 4.23 that the implication “(ρ< → ρ< ) ⇒ (ρ0< → ρ0< )” does hold at least for the case of f being Heaviside’s function. As the following result shows, it holds in fact for every f : Theorem 4.30. Consider f : R → R. a) If f is (ρ → ρ)–computable, then it is also (ρ0 → ρ0 )–computable. b) If f is (ρ → ρ< )–computable, then it is also (ρ0 → ρ0< )–computable. c) If f is (ρ< → ρ< )–computable, then it is also (ρ0< → ρ0< )–computable. d) If f is (ρ0 → ρ0 )–computable, then it is also (ρ00 → ρ00 )–computable. e) If f is (ρ00 → ρ00 )–computable, then it is also (ρ000 → ρ000 )–computable. The claims remain valid under oracle-supported computation. This shows for example that, for d = 0, 1, . . ., the classes of (ρ(d) → ρ(d) )–computable real functions f : R → R form a hierarchy. According to Equation 4.3, this hierarchy is strict, as can be seen from the constant functions f (x) ≡ c with c ∈ ∆d+1 . As another consequence we conclude that the non-reducibility claims made in Observation 4.13 do not only mean uncomputability, but in fact discontinuity, of the reductions (recall Definition 2.56iv): Corollary 4.31.

It holds

¬t

ρ ≡ ρ< ∧ ρ>

ρ

¬t

ρ0
0 be arbitrary. Since f is lower semi-continuous, its pre-image f −1 (y − , ∞) 3 x is an open set and therefore contains an entire ball around x. In fact, the center of this ball may be chosen as rational and its diameter of the form 2−L for some L ∈ N; formally (see Figure 4.8): ∃K, L, K 0 ∈ N : x ∈ (aK 0 , bK 0 ) ⊆ [aK , bK ] ⊆ f −1 (y − , ∞) ∧ |bK − aK | = 2−L ∧ aK 0 = aK +

3 2

· 2−L−2 ∧ bK 0 = bK −

3 2

· 2−L−2

(4.6)

where we have exploited the fact that every rational pair (a, b) occurs in the list representing the [ρ → ρ< ]–name. Moreover, as it consists of all rational triples (a, b, c) with c < min f [a, b] , (*) ∃M ≥ K : [aK , bK ] = [aM , bM ] ∧ cM ≥ min f [aM , bM ] − ≥ y − 2 where (*) is a consequence of [aK , bK ] ⊆ f −1 (y − , ∞) in Equation (4.6). Finally,

lim qn = x ∈ (aK 0 , bK 0 ) n

⇒

∃N : ∀n ≥ N : qn ∈ (aK 0 , bK 0 ) .

(4.7)

(4.8)

So putting things together, for each n ≥ N , ` ≥ L0 , and k ≥ M , either we have phk,`,ni = +∞ ≥ y − 2; or we are in the first case of Equation (4.5), thus


81

• qn ∈ (ak , bk ) with |bk − ak | ≤ 2−` • qn ∈ (aK 0 , bK 0 ) by Equation (4.8) • hence [ak , bk ] ⊆ [aK , bK ] by Equation (4.6) due to ` ≥ L0 ; cf. Figure 4.8. • So [ak , bk ] ⊆ [aM , bM ] by Equation (4.7) • implying phk,`,ni ≥ cM ≥ y − 2 by Equations (4.5) and (4.7) since k ≥ M . Summarizing, it holds phk,`,ni ≥ y − 2 for all (k, `, n) ∈ N3 not belonging to the finite set {0, 1, . . . , N − 1} × {0, 1, . . . , L0 − 1} × {0, 1, . . . , M − 1} of exceptions. Consequently, lim inf p ≥ y − 2; even lim inf p ≥ y because > 0 was arbitrary.

Figure 4.8: Nesting of some rational intervals of dyadic length contained in f −1 (y − , ∞) . The parameters are chosen in such a way that, whenever (aK 0 , bK 0 ) meets some other (ak , bk ) of length |bk − ak | = 2−` for ` ≥ L0 := L + 2, then [ak , bk ] is entirely contained within the larger [aK , bK ]. – lim inf p ≤ y Indeed: Since the [ρ → ρ< ]–name contains in particular all rational pairs (ak , bk ) and these intervals are dense in R, there exists for every ` ∈ N some k such that |bk − ak | = 2−` and x ∈ (ak , bk ). Furthermore, it holds qn ∈ (ak , bk ) for some sufficiently large n because limn qn = x. We have thus infinitely many triples (k, `, n) for which phk,`,ni is defined by the first case in Equation (4.5) and thus agrees with some cm < min f [am , bm ] ≤ f (x) = y as x ∈ (ak , bk ) ⊆ [am , bm ]. Concluding, we have lim inf m pm = y. Although p may attain the value +∞, this can easily be overcome by proceeding to p˜m := pm for pm 6= ∞ and p˜m := max{0, p˜0 , . . . , p˜m−1 } for pm = ∞ because this transformation p 7→ p˜ on sequences obviously does not affect their lim inf < ∞. This yields a ρ˜0< –name for y which can finally be converted to the desired ρ0< –name due to the easy part of Lemma 4.14b). In order to obtain a similar uniform claim yielding Theorem 4.30c), recall that every (ρ< → ρ< )– computable function f : R → R is necessarily both nondecreasing and lower semi-continuous (Fact 4.25b+c). This suggests Definition 4.34. Let MLSC(R) denote the class of all nondecreasing, lower semi-continuous functions f : R → R. A [ρ< → ρ< ]–name for f ∈ MLSC(R) is an enumeration of the set {(a, c) ∈ Q2 : c < f (a)}. Lemma 4.35. a) Distinct f, g ∈ MLSC(R) have different sets {(a, c) : . . .} according to Definition 4.34; that is, [ρ< → ρ< ] constitutes a well-defined representation. b) A function f ∈ MLSC(R) is (ρ< → ρ< )–computable iff it has a computable [ρ< → ρ< ]–name. c) Let f ∈ MLSC(R), (ak , ck )k with {(a, c) ∈ Q2 : c < f (a)} = {(ak , ck ) : k ∈ N}, x ∈ R, and q = (qn ) ⊆ Q with x = lim inf n qn . Then, the rational sequence p defined by max cm : m ≤ k ∧ am ≥ ak if ak < qn < ak + 2−` phk,n,ì := +∞ otherwise satisfies lim inf p = f (x) =: y.

82

CHAPTER 4. ARITHMETICAL HIERARCHIES IN RECURSIVE ANALYSIS d) Therefore, the apply operator MLSC(R) × R 3 (f, x) 7→ f (x) is ([ρ< → ρ< ] × ρ0< → ρ0< )– computable.

Proof. a) Let f, g ∈ MLSC(R) with f 6= g, that is, w.l.o.g. f (x0 ) < g(x0 ) for some x0 ∈ R. There exists some c0 ∈ Q with f (x0 ) < c0 < g(x0 ). Because they are nondecreasing and lower semi-continuous, their pre-images f −1 (c0 , ∞) 63 x0 and g −1 (c0 , ∞) 3 x0 on open half-interval (c0 , ∞) are again open half-intervals (xf , ∞) and (xg , ∞), respectively. As x0 belongs to the second but not to the first, we have xg < x0 < xf and therefore −1 xg < a0 < xf for some a ∈ Q. Then a ∈ (x , ∞) = g (c , ∞) yields c0 < g(a0 ) whereas 0 0 g 0 −1 a0 6∈ (xf , ∞) = f (c0 , ∞) asserts c0 6< f (a0 ). b) Let M denote a Type-2 Machine (ρ< → ρ< )–computing f ∈ MLSC(R). Evaluating f at a ∈ Q by simulating M on the ρ< –name (a, a, a, . . .) for a thus yields a ρ< –name for f (a) which is (equivalent to) a list of all c ∈ Q with c < f (a) [Weih00, Lemma 4.1.8]. So dove-tailing this simulation for all a ∈ Q yields the desired [ρ< → ρ< ]–name for f . Conversely, knowing a [ρ< → ρ< ]–name (ak , ck )k for f ∈ MLSC(R) and given an increasing sequence (qn ) ⊆ Q with x = supn qn , let pn := cn if an ≤ qn ,

pn := −∞ otherwise.

Then, in the first case, pn = cn < f (an ) ≤ f (qn ) ≤ f (x) =: y by monotonicity, and in the second pn = −∞ ≤ y; hence supn pn ≤ y. To see supn pn ≥ y, fix arbitrary > 0 and consider the open half-interval f −1 (y − , ∞) = (x , ∞) containing x and thus also some rational a = aK ∈ (x , x), K ∈ N. Furthermore qn % x yields some N ∈ N such that qn ∈ (aK , x) for all n ≥ N . And finally there exists M ≥ N with aM = aK and cM ≥ f (aM ) − . Together this ensures qM > aK = aM because M ≥ N and thus pM = cM ≥ f (aK ) − > y − 2 due to ak ∈ f −1 (y − , ∞) . c) Take arbitrary > 0. As f is increasing and lower semi-continuous, the pre-image f −1 (y − , ∞) is an open half-interval (x , ∞) containing x. Therefore there exist K, L ∈ N such that x < aK and aK + 2−L < x; furthermore, since the sequence (ak , ck )k contains all rational pairs (a, c) with c < f (a), there is M ≥ K such that aM = aK and cM ≥ f (aM ) − ; and finally, since lim inf n qn = x > aM + 2−L , it holds qn > aM + 2−L for all n ≥ N with an appropriate N ∈ N. Observe that q > aM + 2−L and a < q < a + 2−L implies a ≥ aM ; so together we have for all n ≥ N , ` ≥ L, and k ≥ M that phk,n,ì is either +∞ or ≥ cM ≥ f (aK ) − ≥ f (x ) − ≥ y − 2 due to the monotonicity of f and by definition of x < aK . This proves lim inf p ≥ y because was arbitrary. To see the reverse inequality “lim inf p ≤ y”, take arbitrary ` ∈ N. There exists k ∈ N with ak < x < ak + 2−` and, because lim inf n qn = x, also n ∈ N with ak < qn < ak + 2−` . We therefore have infinitely many triples (n, k, `) for which phn,k,ì agrees with a certain cm < f (am ) ≤ f (ak ) ≤ f (x) = y. d) Given a ρ0< –name for x, one can obtain a sequence (qn ) ⊆ Q with x = lim inf n qn by virtue of Lemma 4.14b). From this, the sequence p ⊆ Q with lim inf m pm = f (x) according to c) is obviously computable and yields, again by Lemma 4.14b), a ρ0< –name for y = f (x).

4.3.3

Relative Weierstraß Computability versus Weak Evaluation

Section 4.3.2 established the sequence ρ, ρ0 , ρ00 , . . . of increasingly weaker representations for R to yield the strict hierarchy of (ρ → ρ)–computable, (ρ0 → ρ0 )–computable, and (ρ00 → ρ00 )– computable functions f : [0, 1] → R. We now compare these classes with those induced by the other kind of real hypercomputation suggested at the beginning of Section 4.3: relative to the Halting Problem H = ∅0 and its iterated jumps ∅00 , . . . Such a comparison makes sense because both weakly and oracle-computable real functions are necessarily continuous according to Fact 4.25d)/Theorem 4.26c) and Corollary 4.21.


83

The classical Weierstraß Approximation Theorem (Fact 2.9) establishes any continuous real function f : [0, 1] → R to be the uniform limit f = ulimn Pn of a sequence of rational polynomials (Pn ) ⊆ Q[X]. Here we write ‘ulim’ to denote uniform convergence of continuous functions on [0, 1]: kf − Pn k → 0, compare Equation (2.3). The effective Weierstraß Theorem (compare Fact 2.45) relates effectively evaluable real functions to effectively approximable ones: Fact 4.36. Fix f : [0, 1] → R. a) The function f is (ρ → ρ)–computable if and only if it holds [ρ → ρ] ρ]: There exists a computable sequence of (degrees and coefficients of ) rational polynomials (Pn ) ⊆ Q[X] such that kf − Pn k ≤ 2−n . b) For f there exists a ∅0 –computable sequence of polynomials (Pn ) with kf − Pn k ≤ 2−n if and only if it holds [ρ → ρ]0 : There exists a computable sequence (Qm ) ⊆ Q[X] satisfying kf − Pn k → 0 (i.e. such that f = ulim Qm ). m→∞

The notion “[ρ → ρ]” in Claim a) is justified by Fact 2.60d). Claim b) merely repeats Proposition 4.8. It straightforwardly generalizes to Lemma 4.37. For a real function f : [0, 1] → R, there exists a ∅00 –computable sequence of polynomials (Pn ) satisfying kf − Pn k ≤ 2−n if and only if it holds [ρ → ρ]00 : There is a computable sequence (Qm ) ⊆ Q[X] such that f = ulim ulim Qhi,ji . i

j

Proof. If ∅00 –computable (Pn )n ⊆ Q[X] satisfies kf − Pn k ≤ 2−n , then by virtue of the relativization of Fact 4.36b) there exists some ∅0 –computable (Pñ )n ⊆ Q[X] converging to the same f uniformly on [0, 1]. Again by Fact 4.36b), Pñ = ulimm Qn,m for some computable sequence (Qn,m ) ⊆ Q[X]. Conversely, if f = ulimn Pñ with Pñ := ulimm Qn,m for a computable (Qn,m ), then let Pn := Qn,mn where mn := min m ∈ N ∀k, ` ≥ m : kQn,k − Qn,` k ≤ 2−n . (4.9) This sequence (mn )n is well-defined and yields kPn − Pñ k ≤ 2−n , so f = ulimn Pñ = ulimn Pn . Moreover, the minimum in Equation (4.9) is taken over a co-r.e. set; namely r := kQn,k − Qn,` k · 2n being ρ–computable by virtue of [Weih00, Corollary 6.2.5] and the complementary condition “r > 1” being ρ–r.e. open and hence recursive in ∅0 . Similar to Equation (4.9), this ∅0 –computable sequence (Pn )n ⊆ Q[X] converging uniformly (though merely ultimately) to f can be turned into a ∅00 –computable, fast convergent one. We thus have two hierarchies of hypercomputable continuous real functions: • [ρ → ρ], [ρ → ρ]0 , [ρ → ρ]00 , . . . 0 0 • (ρ → ρ), (ρ → ρ ), (ρ00 → ρ00 ), . . . By Fact 4.36a) their respective ground-levels coincide. Our next result compares their respective higher levels. They turn out to lie skew to each other (Claim c). Theorem 4.38. a) Let f : [0, 1] → R be [ρ → ρ]0 –computable (in the sense of Fact 4.36b). 0 Then, f is (ρ → ρ0 )–computable. b) Let f : [0, 1] → R be (ρ0 → ρ0 )–computable. Then, f is [ρ → ρ]00 –computable (in the sense of Lemma 4.37). c) There is a (ρ0 → ρ0 )–computable but not [ρ → ρ]0 –computable f : [0, 1] → R. Underlying c) is the idea that every [ρ → ρ]0 –computable f : [0, 1] → R has a modulus of uniform continuity recursive in ∅0 ; whereas a (ρ0 → ρ0 )–computable f , although also uniformly continuous, in general does not. Before proceeding to the proof, we first provide some generalizations of Observation 2.48:

84


Lemma 4.39.

a) Let f : R → R be (ρ → ρ< )–computable. Then the set (a, b, c) ∈ Q3 ∃x ∈ [a, b] : f (x) > c

(i.e. the question of whether f on [a, b] exceeds c)

is semi-decidable.

b) Let f : R → R be (ρ0 → ρ0< )–computable. Then the set (a, b, c) ∈ Q3 ∃x ∈ [a, b] : f (x) > c (i.e. the question of whether f on [a, b] exceeds c)

is semi-decidable relative to ∅0 .

c) Let f : R → R be (ρ0 → ρ0 )–computable. Then the set (a, b, c, m) ∈ Q3 × N ∀x ∈ [a, b] : c − 2−m ≤ f (x) ≤ c + 2−m is decidable relative to ∅00 . Proof. a) is standard; c) follows from b), which is established as follows: Lower semi-continuity of f due to Theorem 4.26a) implies that, if f exceeds c on the compact interval [a, b], then it does so on some rational x. Feeding, for any such x ∈ [a, b] ∩ Q, the ρ0 –name (x, x, x, x, . . .) for x into the Type-2 Machine computing f reveals the mapping Q 3 x 7→ f (x) to be (νQ → ρ0< )–computable. With ∅0 –oracle, it thus becomes (νQ → ρ< )–computable by virtue of [ZhWe01, Lemma 4.2]. Since {(y, c) : y > c} is (ρ< × νQ )–semi-decidable, the claim follows.

Figure 4.9: A piecewise linear and a smooth unit pulse, and a non-overlapping superposition by scaled shifts used in the Proof of Theorem 4.38c) Proof (Theorem 4.38). a) Let (Pn ) ⊆ Q[X] denote a computable sequence converging uniformly (yet not necessarily fast) to f . Let x ∈ [0, 1] be given as the limit of a sequence (qn ) ⊆ Q. Then, pn := Pn (qn ) eventually converges to f (x). b) Let x ∈ [0, 1] be given by (an T equivalent to) its ρ–name in the form of two rational sequences (an ) and (bn ) with {x} = n [an , bn ]. There exists a rational sequence (cm ) forming a ρ– name for f (x), that is, satisfying c − 2−m ≤ f (x) ≤ c + 2−m for all m; and by virtue of Lemma 4.39c), such a sequence can be found with the help of a ∅00 –oracle. This reveals that f is ∅00 –recursive in the sense of [Ho99, Section 4] and thus, similarly to [Ho99, Corollary 17], [ρ → ρ]00 –computable. c) Let h : N → N denote a ∅0 –computable injective total enumeration of some subset H = h[N] ∈ Σ2 \ ∆2 . Observe that am := 2−h(m) is a ρ0 –computable real sequence converging to 0 with modulus of convergence [Weih00, Definition 4.2.2] lacking ∅0 –recursivity; compare [Weih00, Exercise 4.2.4c)]. Let ϕ : R → R denote some (ρ → ρ)–computable unit pulse, that is, vanishing outside [0, 1] and having height maxx ϕ(x) = ϕ( 21 ) = 1; a piecewise


85

linear ‘hat’ function for instance will do fine but we can even choose ϕ as in [PERi89, Theorem 1.1.1] to obtain the counter-example f (x) :=

lim fM ,

M →∞

fM (x) :=

M X

am · ϕ(2m x − 1) ,

(4.10)

m=1

(that is, a non-overlapping superposition of scaled shifts of such pulses) to be C ∞ ; compare Figure 4.9. By Theorem 4.30a), x 7→ am · ϕ(2m x − 1) is (ρ0 → ρ0 )–computable; in fact even uniformly in m: Given (qn )n ⊆ Q with x = limn qn and M ∈ N, one can obtain a sequence (pk,M )k ⊆ Q with limk pk,M = fM . The functions fM converge uniformly (though not effectively) to f because of the disjoint supports of the terms ϕ(2m x − 1) in Equation (4.10). Therefore lim pM,M = f (x), thus establishing the (ρ0 → ρ0 )–computability of f . M

Suppose f were [ρ → ρ]0 –computable. Then, by virtue of [Ho99, Lemma 15], it has a ∅0 –recursive modulus of uniform continuity; cf. [Weih00, Definition 6.2.6.2]. In particular given n ∈ N, one can ∅0 –compute m ∈ N such that x := 2−m and y := 23 x satisfy 2−n ≥ |f (x) − f (y)| = |0 − am | contradicting the hypothesis that (am ) has no ∅0 –recursive modulus of continuity.

4.3.4

Weak Function Evaluation and Borel’s Hierarchy

According to Fact 2.45 and for r.e. open X ⊆ Rk , (ρ → ρ)–computability of a real function f : X → R amounts to effective continuity: its pre-images f −1 [(q, ∞)] and f −1 [(−∞, p)] are open again (that is in Σ1 (X)) and δΣ1–computable uniformly in q, p ∈ Q. More precisely, a [ρ → ρ]– name of f is equivalent to a δΣ12ω –name of the two families of open sets f −1 [(q, ∞)] q∈Q and f −1 [(−∞, p)] p∈Q [Weih00, Lemma 6.1.7]. S S Observation 4.40. Indeed it follows that f −1 [ n (qn , pn )] = n f −1 [(qn , ∞)] ∩ f −1 [(−∞, pn )] is δΣ1–computable according to [Weih00, Corollary 5.1.18.1 and Example 5.1.19.1]. A weakening of the above notion is (ρ → ρ< )–computability; equivalently [WeZh00, Theorem 4.5(1) and Corollary 5.1(2)]: pre-images f −1 [(q, ∞)] (but not necessarily f −1 [(−∞, q)]) are in Σ1 (X) and δΣ1–computable uniformly in q ∈ Q. It is natural to call this condition effective lower semicontinuity, and it has led to the representation of the class LSC(X) of lower semi-continuous ω functions on X we already mentioned in Lemma 4.33: encode f as the join of a (δΣ1) –name of −1 the family of open sets f [(q, ∞)] q∈Q . We shall write [ρ → ρ< ] for this representation because [WeZh00] actually establishes it as equivalent to the one from Fact 2.60c) when A = R and B = R are chosen, the latter equipped with topology {(q, ∞) : q ∈ Q}. A different weakening due to Brattka considers, equally naturally, functions f : X → R for which both pre-images f −1 [(q, ∞)] and f −1 [(−∞, p)] belong to the Borel class Σd (X) and are δΣd–computable uniformly in p, q ∈ Q. He terms this condition effective Σd –measurability because it effectivizes the well-known notion in classical measure theory which calls a function f Σd –measurable if f −1 [V ] belongs to Σd (X) for every open V . (The comprehensive paper [Brat05] studies this notion and its consequences thoroughly. It is as general as to also include partial and multi-valued functions on arbitrary computable metric spaces, but in that respect goes beyond our purpose.) [Brat05] also introduces a natural representation δΣd (X→R) for the class of Σd (X)–measurable functions f : X → R as a (δΣd)2ω –name of the sequences f −1 [(q, ∞)] q∈Q and f −1 [(−∞, p)] p∈Q It is a simple matter to unify these approaches: Definition 4.41. Call f : X → R Σd –lower semi-measurable if f −1 (q, ∞) ∈ Σd (X)for all q ∈ R. This function is effectively Σd lower semi-measurable if the sequence f −1 (q, ∞) q∈Q is (δΣd)ω –computable. The representation δΣd (X→R< ) of all Σd lower semi-measurable functions is defined to encode f : X → R as a δΣdω –name of this sequence.

86


Effective lower semi-continuity thus amounts to effective Σ1 lower semi-measurability; and Observation 4.40 of course generalizes: [Brat05, Proposition 3.2(4)] and Lemma 4.14a) imply δΣd (X→R) ≡ δΣd (X→R< ) ∧ δΣd (X→R> ) ; where the latter representation of all Σd upper semimeasurable functions encodes f : X → R as a δΣd (X→R< ) –name of the Σd lower semi-measurable function −f . The main result of the present section connects these notions and representations to weak (d) function evaluation (ρ → ρ(d) ) and (ρ → ρ< ) in the sense of Section 4.3.1: Theorem 4.42. Fix r.e. open X ⊆ Rk and d ∈ N. a) The uniformly characteristic function χ : Σd (X) × X → {0, 1}, (d−1)

is (δΣd × ρ → ρ
q and to (d−1) zq := −∞ in case f (~x) ≤ q. We finally obtain a ρ< –name of supq zq = f (~x) because (d) N (d) N ¯ ¯ R 3 (zn ) 7→ sup zn ∈ R is obviously (ρ< ) → ρ< –computable. n

n


87

c) To start with, recall the proof of the classical case as in [WeZh00, Theorem 3.7]: d = 0: Evaluate f simultaneously on all ~x ∈ X to obtain rational sequences p~x,n with f (~x) = supn px,n . More precisely, using feasible countable (as opposed to infeasible uncountable) dove-tailing, simulate the machine evaluating f on all initial parts of ρ– names of ~x ∈ X, that is on all finite rational sequences q¯ = (~q1 , ~q2 , . . . , ~qN ) in Qk with N ∈ N and |~qn − ~qk | ≤ 2−n ∀n ≤ k ≤ N . For each q¯, we obtain as output a finite rational sequence (pq¯,m )m≤M . Observe that q¯ is the initial segment of a ρ–name to any SN (¯q) x ∈ B q¯ := n=1 B(qn , 2−n ), B q¯ having a non-empty interior. Hence ◦

∃m : pq¯,m > a

⇔

∀~x ∈ B q¯ : f (~x) > a

⇔

∃~x ∈ B q¯ : f (~x) > a

which implies f

−1

(a, ∞)

[ B :: p q¯ q¯,m > a ∅ : pq¯,m ≤ a

=

◦ [ B :: p q¯ q¯,m > a ∅ : pq¯,m ≤ a q¯,m {z } |

=

q¯,m

∈Σ1

and immediately yields δΣ1–computability of f −1 [(a, ∞)] for a given a ∈ Q. d = 1: Similarly, evaluate f on all ~x ∈ X to obtain sequences p~x,n,m with f (~x) = sup inf p~x,n,m . n m More precisely countable dove-tailing yields, for each finite ρ–initial segment q¯, a finite sequence (~ pq¯,m,n )m,n in Qk with ◦

∃m∀n : pq¯,m,n > a

⇔

∀~x ∈ B q¯ : f (~x) > a

⇔

∃x ∈ B q¯ : f (x) > a

which, because of =:Aq,m ∈Π1 ¯

}| z ◦ { [ \ B : : p [ \ B : : p > a q¯ q¯,m,n q¯ q¯,m,n > a −1 f (a, ∞) = = , ∅ : pq¯,m,n ≤ a ∅ : pq¯,m,n ≤ a q¯,m n q¯,m n | {z } ∈Σ2

constitutes a δΣ2–name of f −1 [(a, ∞)] as the δΣ1–names of all open X \ Aq¯,m . d = 2: Compute finite rational sequences (pq¯,m,n,k )m,n,k with ◦

∃m∀n∃k : pq¯,m,n,k > a

f

−1

(a, ∞)

=

⇔

⇔

∀~x ∈ B q¯ : f (~x) > a

∃~x ∈ B q¯ : f (~x) > a ,

[ \[ B : p q¯ q¯,m,n,k > a ∅ : pq¯,m,n,k ≤ a

q¯,m n

k

∈Σ1

=

z ( [ \[ q¯,m n

|

k

◦

}|

){

B q¯ : pq¯,m,n,k > a ∅ : pq¯,m,n,k ≤ a {z } ∈Π2

d ≥ 3: analogously. (d−1)

d) “δΣd (X→R< ) ≡ [ρ → ρ< ]” follows from Claims a+b) and [Weih00, Lemma 3.3.14]. The second claim is a consequence of δΣd (X→R) ≡ δΣd (X→R< ) ∧ δΣd (X→R> ) .

88


4.3.5

Are Discontinuous Real Functions Hyper computable?

Let us return to Question 4.22 (which is not to be confused with the question of physical realizability we discussed in Section 3.4.1). It is easy to define a reasonable representation (and thus also a notion of computability) on a subclass of real functions including discontinuous ones. For instance, [ZhWe03] do so for the space of generalized functions in order to study the effective solvability of partial differential equations. But for these objects f , evaluation x 7→ f (x) on real points x does not even make sense mathematically. Let us therefore be more specific about Question 4.22 and understand it in the sense of (hyper-) effective function evaluability. Here, ordinary oracle access does not help (Corollary 4.21); but hypercomputation in the form of weakened real number representations does: Heaviside’s function, for instance, has turned out to be (ρ → ρ0 )–computable. Note that this example relies on a weaker encoding ρ0 for the output value y = f (x) than that of the input argument x, namely ρ. However, a notion of computability involving weaker output encoding than input obviously lacks closure under composition: Example 4.45. Let f : R → R, f (0) := 0 and f (x) := 1 for x = 6 0. Let g(x) := −x. Then both f and g are (ρ → ρ< )–computable but their composition g ◦ f : 0 7→ 0, 0 6= x 7→ −1 fails lower semi-continuity On the other hand if input and output encoding are required to coincide, nothing is gained topologically by weakening them both: compare Fact 4.25 and Theorem 4.26. Under these reasonable conditions, the answer to Question 4.22 thus still seems to be negative. Only in Section 4.5 shall we see that nondeterministic computation does allow for the evaluation of certain discontinuous functions while maintaining closure under composition. In the meantime, let us study

4.3.6

Markov with Oracles

In Section 2.6.3 we considered the Markov computability of real functions and saw that (at least every total) such f is necessarily continuous. Observe how the proof of Fact 2.67 proceeded by reducing the evaluation of Heaviside’s function at a point of discontinuity to the task of deciding “e ∈ H?”; it is thus justified for Chee Yap to call the real zero test “x = 0?” the numerical Halting problem. On the other hand, when oracle access to H is granted, this test becomes easily decidable in the Markov setting (as opposed to relativized evaluability, recall Corollary 4.21): Example 4.46. Given a G¨ odel index e of some machine Me computing x, modify Me slightly to detect, and to abort (only) in the case that x 6= 0. Feed this new machine’s index e˜ into the Halting oracle. A negative answer is equivalent to x = 0. Observe that this example shows only a certain partial relativization of Fact 2.67 to fail: the operations on G¨ odel indices are permitted oracle access, whereas the indices input and output themselves still refer to oracle-free machines. It seems useful to generalize Tseitin’s Theorem to Markov hypercomputation by answering the following Question 4.47. Fix a total function f : Rc → R. a) If f is Markov–computable relative to ∅0 , is it then Σ2 –measurable? b) Consider naive Markov hypercomputation of f in the following sense: the input consists of a G¨ odel index e of some Turing machine Me generating (qn ) ⊆ Q with x = limn qn , and the output is a similar integer e0 such that Me0 produces (pm ) ⊆ Q with f (x) = limm pm . If the transition e 7→ e0 is effective, does this require f to be continuous, again? c) Characterize the class of total functions tht are Markov–computable relative to ∅0 .

4.4. REVISING TYPE-2 COMPUTATION

89

d) How about higher degrees? Item b) amounts to a full relativization of Fact 2.67.

4.4

Revising Type-2 Computation

Definition 4.12 has turned the two real representations ρ< ¬ ρ into a countably infinite hierarchy (d) (d+1) of weakened representations ρ< ¬ ρ(d) ¬ ρ< related to the classical Arithmetical Hierarchy in (d) (d) the sense that a real number x is ρ –computable (ρ< –computable) iff it is ρ–computable (ρ< – computable) relative to ∅(d) . The construction of ρ0 from ρ (ρ0< from ρ< ) was ad-hoc and based on Theorems 4.9 and 4.10. Similarly, Fact 4.36a) and Lemma 4.37 provided the justifications for the representations [ρ → ρ]0 and [ρ → ρ]00 for C([0, 1]) implicitly defined in them. The present section proposes as a natural generalization a generic way of turning a given representation α :⊆ {0, 1}ω → A into a weakened one α0 :⊆ {0, 1}ω → A such that it holds: a ∈ A is α0 –computable iff a is α–computable relative to ∅0 .

4.4.1

Motivation

An important point in the definition of a Type-2 machine is that its output tape must be oneway; compare e.g. [Weih00, top of p.15]. This condition models and reflects the practitioner’s reasonable requirement an (infinite) real number computation be aborted as soon as the desired precision is reached, knowing that this preliminary approximation will not be reverted. It also enters crucially in the Main Theorem (e.g. in the form of Proposition 2.44 or Fact 2.57). In the Type-1 setting with finite results appearing after finite time, every machine with rewritable output tape can be simulated by one with one-way output tape. However, when finite results appearing after possibly infinite time are considered, we enter the realm of revising computations; recall Section 3.6. Its Definition 3.6 for finite strings leads to two natural ones for infinite ones (recall Requirement 3.8): • either require sequence (¯ τm )m in {0, 1}ω to converge to σ ¯ in Cantor topology, i.e. symbolwise; • or demand that (¯ τm )m ‘stabilizes’ to σ ¯ in the following sense: ∃M : ∀m ≥ M :

τ¯m = σ ¯.

Section 4.4.2 will focus on the notion of revising Type-2 computation induced by the first condition while consideration of the second is postponed to Section 6.4. Both are based on the observation that preliminary information in the sense of strings with embedded ‘editing’ commands as in Example 3.7 occur not only as output from but also as input to programs: Data Stream Algorithms Many practical applications are desired to run ‘forever’: a scheduler, a router, a monitor are all not supposed to terminate but to continue processing the stream of data presented to them. This has led to the prospering field of Data Stream Algorithms16 . It distinguishes various ways in which the input can be presented to the program [Muth05, Section 4.1]: • In the Time Series Model, all data items (binary digits, say) are enumerated in order; in particular, they must not later be reverted. This corresponds to classical, i.e. non-revising input. • The Turnstile Model on the other hand permits (finitely many) later updates to previously enumerated items. This corresponds to revising input (subject to Requirement 3.8). 16 Which

usually focuses on the (space) complexity of randomized approximations of discrete problems, however

90


4.4.2

The Jump of a Representation

We start with Cantor space {0, 1}ω which is usually and canonically represented by the identity ı [Weih00, Definition 3.1.2.1]. Definition 4.48. Let the revising representation ı0 :⊆ {0, 1}ω → {0, 1}ω encode (via some computable standard pairing h · i : {0, 1}ω×ω → {0, 1}ω ) an infinite string σ ¯ ∈ {0, 1}ω as a sequence of infinite strings ultimately converging to σ ¯. This amounts to the naive Cauchy representation of the effective metric Cantor space [BrHe02, Section 6]. An ı0 –name for (σn )n is thus (an ı–name for) some (τhn,mi )n m ∈ {0, 1}ω such that, for each n ∈ N, σn = limm→∞ τhn,mi . We state for the records and for later application the following easy Claim 4.49. For each n ∈ N let (¯ τn,m )m be a sequence in {0, 1}ω and σ ¯n ∈ {0, 1}ω . Then m→∞

∀n : τ¯n,m −→ σ ¯n

⇔

m→∞

h(¯ τn,m )n i −→ h(¯ σn )n i .

The name ı0 , reminiscent of the recursion-theoretic jump, is justified because Shoenfield’s Limit Lemma 3.5 immediately yields Remark 4.50. Let O denote an arbitrary oracle. An infinite string is (ı–) computable relative to O0 if and only if it is ı0 –computable relative to O. Lemma 4.51. a) Every ((ı → ı)–) computable string function F :⊆ {0, 1}ω → {0, 1}ω is also 0 0 (ı → ı )–computable. b) More precisely, the apply operator (F, σ ¯ ) 7→ F (¯ σ ) is (η ωω × ı0 → ı0 )–computable on (F, σ ¯ ) F :⊆ {0, 1}ω → {0, 1}ω continuous, σ ¯ ∈ dom(F ) . c) Every (ı0 → ı0 )–continuous string function F :⊆ {0, 1}ω → {0, 1}ω is (Cantor–)continuous. In b), η ωω denotes a natural representation for continuous string functions [Weih00, Section 2.3]. Proof.

a) follows from b) and Fact 2.57.

b) Let τ˜m := F (¯ τm ) where F :⊆ {0, 1}ω → {0, 1}ω is continuous. Then limm τ˜m = F limm τ¯m . c) Similar to Fact 4.25d); see also [BrHe02, Section 6] which applies to any effective metric space. We now extend Lemma 4.51c) in five steps to multi valued functions: S Lemma 4.52. Fix X ⊆ {0, 1}ω and write X 0 := x¯∈X x ¯0 where x ¯0 := {(¯ σm )m ⊆ {0, 1}ω : 0 limm σ ¯m = x ¯}. For each n ∈ N, let Gn : X → {0, 1} be continuous (i.e. locally constant). Furthermore suppose that, for every (¯ σm )m ∈ X 0 , limn Gn (¯ σm )m exists. a) To x ¯ ∈ X there exist b ∈ {0, 1}, M, K ∈ N, and σ ¯1 , . . . , σ ¯M ∈ {0, 1}ω such that every 0 (¯ σm )m>M ∈ x ¯ with σm,k = xk (for all k ≤ K and all m > M ) satisfies limn Gn (¯ σm )m = b. b) For x ¯ ∈ X and b ∈ {0, 1} as in a), there exists L ∈ N such that, to every y¯ ∈ X with x` = y` for ` ≤ L, there is some (¯ τm )m ∈ y¯0 with limn Gn (¯ τm )m = b. c) Let (Ui )i∈I denote a family of open balls in Cantor space {0, 1}ω , i.e. Ui = u ¯i ◦ {0, 1}ω ∗ with ¯i ∈S{0, 1} . Then there exists a subfamily (Uin )n which is pairwise disjoint such that S u U = n Uin . i i d) Let g :⊆ {0, 1}ω ⇒ {0, 1} denote a multivalued mapping. If g is (ı0 → ı0 )–continuous, then it admits a locally constant selection.


91

e) Every (ı0 → ı0 )–continuous multivalued mapping f :⊆ {0, 1}ω ⇒ {0, 1}ω admits a (Cantor)continuous selection. Proof. a) W.l.o.g. let x ¯=¯ 0 and b := lim Gn x ¯, x ¯, . . . = 0. Start off with M, K := 0: if every sequence (¯ σm )m converging to ¯ 0 satisfies lim Gn (¯ σm )m = 0, then we are done. (1) (1) Otherwise lim Gn (¯ σm )m = 1 =: b1 for some (¯ σm )m with limm σ ¯m = ¯0. In particular (1) (1) ˜ 1 ∈ N such that Gn (¯ ˜ 1 . By there is some n1 , M σm )m = 1 and σm,1 = 0 for all m > M 1 ˜ continuity of Gn1 , there exists some M1 ≥ M1 such that Gn1 (¯ σm )m = 1 holds whenever (1) (¯ σm )m satisfies σ ¯m = σ ¯m for all m ≤ M1 . Now if even limn Gn (¯ σm )m = 1 holds for every (¯ σm )m ∈ ¯ 00 as above with σ ¯m,1 = 0, then we can choose (K, M, b) := (1, M1 , b1 ) and are done again. (2) (2) (2) (2) Otherwise lim Gn (¯ σm )m = 0 =: b2 for some (¯ σm )m with limm σ ¯m = ¯0 and σm,1 = 0 for (2)

(1)

m > M1 and σ ¯m = σ ¯m for m ≤ M1 . Similarly as above, there are n2 ≥ n1 and M2 ≥ M1 (2) such that Gn2 (¯ σm )m = 0 holds whenever (¯ σm )m satisfies σ ¯m = σ ¯m for all m ≤ M2 ; and either we are done with (K, M, b) := (2, M2 , b2 ), or we proceed to K := 3 and b3 := 1 − b2 and M3 ≥ M2 ; and so on. Now the crucial observation asserts this iterative procedure to indeed terminate for some K: (ω) because otherwise it would yield in the limit sequences (¯ σm )m ∈ ¯00 and (nK )K such that (ω) GnK (¯ σm )m σm )m is = 0 for K even and = 1 for K odd: contradicting that limn Gn (¯ was supposed to exist for every (¯ σm )m ∈ ¯00 . b) Again presume w.l.o.g. x ¯ = ¯ 0 and b = 0. Take K and σ ¯1 , . . . , σ ¯M from a) and argue indirectly: Failure of L = K means there is some y¯ ∈ X starting with K 0s such that the se (K) (K) σ1 , . . . , σ ¯M , y¯, y¯, . . .) ∈ y¯0 has limn Gn (¯ τm )m = 1; in particular quence (¯ τm )m := (¯ (K) GnK (¯ τm )m = 1 for some nK , MK ∈ N and any (¯ τm )m satisfying τ¯m = τ¯m for all m ≤ MK . Similarly, failure of L = K + 1 means that, for some y¯ ∈ X starting with K + 1 0s, the (K+1) (K) (K+1) (K+1) = y¯ for m > MK , = τ¯m for m ≤ MK and τ¯m )m , defined as τ¯m sequence (¯ τm (K+1) )m = 1 and GnK+1 (¯ τm )m = 1 for some nK+1 , MK+1 ∈ N and any has limn Gn (¯ τm (K+1) for all m ≤ MK+1 . (¯ τm )m satisfying τ¯m = τ¯m Again we argue that, for some finite L, this iteration of successive failures terminates and (ω) hence leads to success: otherwise it would yield sequences (¯ τm )m , (nL )L , and (ML )L such that it holds (ω)

• τ¯m = σm for m ≤ M • τ¯m,` = 0 for all m ≥ ML and all ` ≥ L; (ω)

• (in particular with (¯ τm )m ∈ ¯00 ) (ω) • limn Gn (¯ τm )m = 1 (ω) contradicting a) that limn Gn (¯ τm )m = 0. c) Notice that any two open balls Ui , Uj are either disjoint Ui ∩ Uj = ∅, subsets Ui ⊆ Uj or Uj ⊆ Ui , or equal Ui = Uj . So take17 some maximal set J ⊆ I of indices satisfying ∀j, k ∈ J, j 6= k : Uj 6⊆ Uk . Then (Uj )j∈J has the desired properties. 17 using

the axiom of countable choice

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CHAPTER 4. ARITHMETICAL HIERARCHIES IN RECURSIVE ANALYSIS L L d) Let X := dom(g) and G = n Gn : X 0 → n {0, 1} = {0, 1}ω denote a continuous (ı0 → ı0 )– realizer of g; that is such that, for every (¯ τm )m ∈ X 0 , limn Gn (¯ τm )m exists and belongs to G(limm τ¯m ). Now Claim b) means that there exists, to every x ¯ ∈ X, some Lx¯ ∈ N such that g, restricted to the open ball Ux¯ := (x1 , . . . , xLx¯ ) ◦ {0, 1}ω , admits a constant selection. According to c), these open restrictions Ux¯ can be attained as pairwise disjoint, i.e. yield a locally constant selection of g. e) Let X := dom(f ) and consider the mapping g : X ⇒ {0, 1} defined by g(¯ x) 3 0 if there is some ¯b ∈ {0, 1}ω such that 0 ◦ ¯b ∈ f (¯ x) and g(¯ x) 3 1 if 1 ◦ ¯b ∈ f (¯ x) for some ¯b ∈ {0, 1}ω . Performing this operation on the continuous (ı0 → ı0 )–realization of f yields a continuous (ı0 → ı0 )–realization of g. So apply d) to get a locally constant selection h1 : X → {0, 1} of g; that is such that, for every ~x ∈ X, h1 (~x) ∈ {0, 1} has an extension to an infinite string in f (~x). −1 Let U := h−1 1 (0) ⊆ X and V := h1 (1) ⊆ X. Now consider the restriction f(0) := sh ◦f U ω ω where sh : {0, 1} → {0, 1} , (σ1 , σ2 , σ3 , . . .) 7→ (σ2 , σ3 , . . .) denotes the left-shift operator. Repeating the above argument, separately for f(0) and for f(1) := sh ◦f V instead of f , yields a joint, locally constant function h2 : U ] V = X → {00, 01, 10, 11} which • refines h1 in the sense that, for every ~x ∈ X, h1 (~x) is a prefix of h2 (~x) • which in turn admits an extension to an infinite string in f (~x), that is to a (not necessarily continuous) selection of f . Iteration yields for every k a locally constant refinement hk : X → {0, 1}k of hk−1 (and thus also of h` for each ` < k) which admits an extension to a selection of f (~x). This process yields a function hω : X → {0, 1}ω which • refines all locally constant hk , and thus is continuous; • constitutes a selection of f .

Note that α ◦ ı0 :⊆ {0, 1}ω → A is a representation for A (called the jump of α) whenever ¨ der has α :⊆ {0, 1}ω → A is. This new representation need not be admissible even if α is. Schro extended the Fact 2.62 to a generalized notion of admissibility covering certain topological spaces lacking a countable base [Schr06]. However, it turns out that we can do without his contributions in proving the following Lemma 4.53. a) Let α and β denote representations for A and B, respectively. Every (α◦ı0 → 0 β ◦ ı )–continuous function g :⊆ A → B is (α → β)–continuous. b) α β implies α ◦ ı0 β ◦ ı0 . c) For a representation for Ai , i ∈ I. Then the representations Q I = N, or for IQfinite, let αi be Q ( i∈I αi ) ◦ ı0 and i∈I (αi ◦ ı0 ) for i∈I Ai coincide. Proof. b) Let F denote a computable string function converting α–names to β–names. By Lemma 4.51a), F has a computable (ı0 → ı0 )–realization G :⊆ {0, 1}ω → {0, 1}ω . This G converts (α ◦ ı0 )–names to (β ◦ ı0 )–names. Q Q c) It suffices to treat the case Ai = {0, 1}ω and αi = ı and I = N. For “ i∈I (ı0 ) ( i∈I ı) ◦ ı0 ” let σ ¯m ∈ {0, 1}ω be a sequence of infinite strings with respective ı0 –names τ¯n,m , i.e. σ ¯m = limn τ¯n,m ; then apply Claim 4.49. Proceed similarly for the converse reduction. a) Let G :⊆ {0, 1}ω → {0, 1}ω be a continuous (α ◦ ı0 → β ◦ ı0 )–realization of g; that is G maps (encodings as infinite strings of) sequences (¯ τm )m of infinite strings to (encodings as infinite strings of) sequences Gn (¯ τm )m of infinite strings. By continuity of G (and due n

to Claim 4.49 above), F :⊆ {0, 1}ω → {0, 1}ω ,

F (lim τ¯m ) := lim Gn (¯ τm )m m

n


93

is well-defined multivalued18 and (ı0 → ı0 )–continuous (G being a realization). Hence by Lemma 4.52e), F admits a continuous selection. This constitutes an (α → β)–realization of g. Remark 4.54. Notice how Lemma 4.52e) entered in the proof to Lemma 4.53a); conversely, Lemma 4.53a) implies Lemma 4.52e) as follows: Let multivalued f :⊆ {0, 1}ω ⇒ {0, 1}ω be (ı0 → ı0 )–continuous. Equip B := dom(f ) with the following artificial representation x, y¯) is a β–name of x ¯ iff x ¯ ∈ dom(f ) ∧ y¯ ∈ f (¯ x) β: A pair (¯ holds. Then g : x ¯ 7→ β x ¯, f (¯ x) is single-valued (!), namely the identity on B; and, by (ı0 → ı0 )– continuity of f , g is (ı0 → β ◦ ı0 )–continuous. So take an (ı → β)–continuous realization h according to Lemma 4.53)a). This h thus has the form h : B 3 x ¯ 7→ x ¯, h2 (¯ x) with continuous single-valued h2 : B → {0, 1}ω satisfying h2 (¯ x) ∈ f (¯ x): the desired selection. For concrete representations, the next remark may provide better intuition about their respective jumps in terms of ‘stabilizing’ double sequences and sequences with ‘revocations’. Remark 4.55. Let representation α :⊆ {0, 1}ω → A arise19 from a notation ν :⊆ {0, 1}∗ → X ω in that every α–name is (a ν –name of ) some sequence (xn )n in X. Also suppose that the ν– equivalence problem (u, v) u, v ∈ dom(ν) ∧ ν(u) = ν(v) is recursive. Qω a) An α ◦ ı0 –name for a ∈ A is uniformly equivalent to a ν–name of a double sequence (xn,m ) in X satisfying i) ∀n ∃M xn,M = xn,M +1 = . . . =: xn,∞

((xn,m )m stabilizes for every n.)

ii) xn,∞ = xn for each n ∈ N. ˜ := X ] N. Define α b) Let ν˜ denote an extension of ν to X ˜ :⊆ {0, 1}ω → A to encode a ∈ A as ˜ where x (any ν˜–name of ) some sequence (˜ xn )n in X ñ = m ∈ N means (in informal terms): Revoke member x ˜m ( provided20 that it belongs to X rather than to N). We say that a member x ñ of this sequence survives if x Ñ 6= n for all N ≥ n ∈ N and require that (a ν ω –encoding of ) the subsequence of surviving elements is an α–name of a. Then α ˜ constitutes a representation of A which is also uniformly equivalent to α ◦ ı0 . Proof. a) We begin with the reduction to α ◦ ı0 . For m ∈ N, let τ¯m denote a ν ω –name of (xn,m )n . Since ν-equivalence is decidable, one can effectively obtain an encoding of this sequence in which identical elements x have identical ν-names. Property i) then implies that (¯ τm )m converges; and by ii), its limit is an α–name of a. Concerning the converse reduction from α ◦ ı0 , suppose that limm τ¯m =: σ ¯ is a ν ω –name of (xn )n . Since the decidability of ν-equivalence implies the decidability of dom(ν), one can, for each m ∈ N, start decoding τ¯m (which is in general not a valid ν ω –name) into a finite sequence (x1,m , . . . , xNm ,m ) in X up to the maximal Nm ≤ m, for which the corresponding initial segment of τ¯m is a valid ν Nm –name. Since m 7→ Nm is computable, we may furthermore effectively extend and output arbitrary values for xn,m , n > Nm . For n ∈ N, let sn ≤ tn ∈ N denote the respective starting and ending positions of the substring of σ ¯ ν–encoding sequence member xn . From σ ¯ = limm τ¯m it follows that, for every n, there is some M ∈ N such that the initial segment of τ¯m up to tn coincides with that of σ ¯ for all m ≥ M . In particular, this initial segment of τ¯m constitutes a valid ν n –name for (x1 , . . . , xn ); hence, according to the above algorithm, it holds xn,m = xn for m ≥ M , i.e. the output satisfies both i) and ii). 18 I

am grateful to Vasco Brattka for pointing out that gap in a previous version of this proof is the case for example for many standard representations of a computable topological space [Weih00, Definitions 3.2.1 and 3.2.2]. Strictly speaking, every representation α arises in the above way from some—more or less artificial—notation ν. . . 20 Without this condition, that is when revocations of revocations are permitted, we arrive at representations equivalent to iterated jumps; cf. Section 4.4.5 19 This

94


Figure 4.10: Converting a sequence of stabilizing sequences into a sequence with revocations. b) We first show “˜ α α ◦ ı0 ”: For m ∈ N, consider (and ν ∗ –compute) the finite subsequence xn )n≤m of ‘tentatively surviving elements’, i.e. where all revoke-commands (xm,n )n≤Nm of (˜ have been applied and removed. Notice that the length Nm of this subsequence is computable. This permits us to effectively extend it (in fact arbitrarily) to an infinite sequence (xm,n )n . We claim that this satisfies i) and ii) in a) and is therefore computably convertible to an α ◦ ı00 -name of the same element. Indeed, by the above condition on survival, it holds Nm → ∞ for m → ∞ because the ultimately prevailing elements must enumerate an infinite subset of X. Moreover, precisely every surviving element eventually becomes and remains tentatively surviving as soon as all its (ultimately revoked) predecessors have been removed, that is beyond some finite m. For “α ◦ ı0 α ˜ ”, again invoke a) to obtain (a ν ω –name of) a double sequence (xn,m ) in X satisfying i) and ii). For each m, let Nm+1 := min{n : xn,m 6= xn,m+1 } denote the first position n where (xn,m )n differs from (xn,m+1 )n ; N1 := 0, see also Figure 4.10. The integer sequence (Nm )m is computable because we required the ν–equivalence problem to be decidable. Now apply the following algorithm: For m = 1, 2, . . . I) in case Nm+1 > Nm , append (ν–names of) xNm ,m+1 , . . . , xNm+1 −1,m+1 to the output; II) in case Nm+1 < Nm , revoke (the indices in the output stream of) xNm+1 ,m , . . . , xNm −1,m . Since every xn,∞ stabilizes beyond some M (n), (Nm )m is unbounded; hence xn,∞ eventually gets output in I) at m = M (n) and not revoked by II). Conversely, to every xn,m 6= xn,∞ , if output at I), there corresponds some M at which it changes and is therefore revoked in II).

4.4.3

Examples of Known Jumps

We now reveal several known representations to be of the form α ◦ ı0 for some (also well-known) α. Proposition 4.56.

ρ ◦ ı 0 ≡ ρ0 .

In combination with Remark 4.50, this implies Proposition 4.8. Proof. By Remark 4.55a), a (ρ ◦ ı0 )–name for x ∈ R can be regarded as a sequence of rational sequences stabilizing (elementwise) to a fast converging Cauchy sequence (q(n,∞) )n ; that is a double


95

sequence (q(n,m) ) in Q such that ∀n ∃m0 ∀m ≥ m0 :

q(n,m) = q(n,m0 ) ∧ |x − q(n,m0 ) | ≤ 2−n .

“”: For each m, let (q(1,m) , q(2,m) , . . . , q(Nm ,m) ) denote the longest initial part of (q(1,m) , . . . , q(m,m) ) satisfying |q(n,m) − q(n0 ,m) | ≤ 21−n ∀1 ≤ n ≤ n0 ≤ Nm . (4.11) Since (q(n,∞) )n is a ρ–name and due to the eventual stabilization, Nm → ∞ as m → ∞. Also, the sequence (Nm )m is computable from the above input. Consider the following algorithm, starting with empty output tape: For each m = 1, 2, . . ., test whether the initial parts of q(·,m) and q(·,m−1) up to Nm coincide: (q(1,m) , . . . , q(Nm ,m) ) = (q(1,m−1) , . . . , q(Nm ,m−1) ). (For notational convenience, set qn,0 :≡ ∞ and N0 := 0.) If so, then obviously Nm ≥ Nm−1 ; so append (the possibly empty sequence) (q(Nm−1 ,m) , . . . , q(Nm ,m) ) to the output. Otherwise let nm be maximal with (q(1,m) , . . . , q(nm ,m) ) = (q(1,m−1) , . . . , q(nm ,m−1) ); obviously nm < Nm , so append (q(nm ,m) , . . . , q(Nm ,m) ) to the output in this case. It remains to be shown that this yields a valid ρ0 –name for x. Let = 21−n . Then |q(n,∞) − q(n0 ,∞) | ≤ for all n0 ≥ n because q(n,∞) constitutes a ρ–name. Moreover, due to stabilization, there exists some maximal m with q(n,m) 6= q(n,m−1) . During phase no. m corresponding to that last change, the above algorithm will detect nm < Nm and thus output (a finite sequence beginning with) q(n,m) . Moreover, as q(n,·) afterwards no longer changes, all elements q(n0 ,m0 ) appended subsequently will have n0 ≥ n and m0 ≥ m; in fact Nm0 ≥ n0 ≥ nm0 ≥ n, hence |q(n0 ,m0 ) − q(n,m) | ≤ because q(n,m0 ) = q(n,m) and due to Equation (4.11). Therefore the output constitutes a (naive) Cauchy sequence converging to x. “”: Let (qn )n be a sequence in Q ultimately converging to x. There exists an increasing sequence (nm )m in N such that ∀k ≥ nm : |qnm − qk | ≤ 2−m−1 . (4.12) The subsequence (qnm )m constitutes a ρ–name for x. For each single m, Condition (4.12) can be falsified (formally: is co-r.e. in the input). A Turing machine is therefore able to iteratively try for nm all integer values from nm−1 on and fail only finitely often for each m. Trial no.` thus yields a sequence (n0(`,m) )m≤` of length ` such that, for each m, n0(·,m) eventually stabilizes to nm satisfying (4.12). By artificially extending each finite sequence to an infinite one, we obtain a ρ ◦ ı0 –name for x. Proposition 4.57.

ρ< ◦ ı0 ≡ ρ0< .

Proof. A (ρ< ◦ ı0 )–name for x ∈ R amounts according to Remark 4.55a) to a sequence of rational sequences eventually stabilizing (elementwise) to a sequence approaching x from below, that is a double sequence (q(n,m) ) in Q such that ∀n ∃m0 = m0 (n) ∀m ≥ m0 :

q(n,m) = q(n,m0 ) ∧ x = sup q(n,m0 (n)) . n

“”: Since the limit (which exists) coincides with the least accumulation point, we have q(n,m) : m ≥ j x = sup lim q(n,m) = sup sup inf q(n,m) = sup inf ∞ : m b” are semi-decidable: by continuity it suffices to evaluate Q on a countable dense subset like the rationals. It remains to be decided whether Q−b admits of a real root in [0, 1]n . In the univariate case n = 1, this can be performed by means of root bounds [BFMS00], recall Section 2.7. In the multivariate case, we content ourselves with a less constructive argument: Quantifier Elimination (Fact 2.13) yields a quantifier-free formula Ψ in (q~0 , . . . , qd~, b) with constants from F := Q equivalent to Φ(q~0 , . . . , qd~, b).

4.4.4

Jump of Set Representations θ< and ψ>

We now establish a nice characterization of the jump θ< ◦ ı0 of the inner representation of open sets (Definitionx:Representationf). For reasons of notational convenience we turn to complements and consider closed real subsets with the representation ψ> (Section 4.1.2). Definition 4.60. Let ψ>0 denote the representation of closed real sets encoding A ⊆ Rk as a sequence (~qn ) of rational vectors whose set of accumulation points coincides with A. Let ψ˜>0 encode A ⊆ Rk as an enumeration of a set Q of rational vectors whose accumulation points are precisely the elements of A. Recall that a sequence may repeat elements in order to establish an accumulation point but a set cannot. Theorem 4.61. It holds ψ> ◦ ı0 ≡ ψ>0 ≡ ψ˜>0 . Proof. In view of Remark 4.55a), a ψ> ◦ ı0 –name for A ⊆ Rd is (equivalent to) a double sequence of open rational balls S Bn,m such that, for every n and some M = M (n), Bn,M = Bn,M +1 = . . . =: Bn,∞ and A = R \ n Bn,∞ . ˜ (m) := max{N ≤ m : Bn,m = Bn,m+1 ∀n ≤ N } ψ> ◦ ı0 ψ>0 : Given (Bn,m ) as above, calculate N SN˜ (m) and let (~qhm,ki )k enumerate (without repetition) the set Qd \ n=1 Bn,m for each m ∈ N. Then this sequence (~q` )` is a ψ>0 –name for A: ˜ (m) ≥ N for • For ~x 6∈ A, there exists N such that ~x ∈ BN,∞ . By finite stabilization, N m ≥ M (N ). Therefore ~qhm,ki ∈ BN,∞ can hold only for m < M (N ), i.e., finitely often; hence ~x is not an accumulation point of (~q` ).


97

SN Bn,∞ ⊇ n=1 Bn,m for every N and m ≥ M (N ). In SN˜ (m) particular for m ≥ M (N ), it holds x 6∈ n=1 Bn,m and by construction plus Claim 4.62 there is some km with |~qhm,km i − ~x| ≤ 2−m . So ~x is an accumulation point of (~q` ).

• Suppose ~x ∈ A, i.e. ~x 6∈

S

n

ψ>0 ψ˜>0 : We are given a rational sequence (~q` )` . Starting with Q = {}, add, inductively for each −n ` ∈ N, a rational vector not yet in Q and closer to ~q` than 2 . Indeed, the finiteness of Q d −n −n at each step ensures: ∃~ p ∈ Q ∩ (~q` − 2 , ~q` + 2 ) \ Q . ψ˜>0 ψ> ◦ ı0 : Let (Bk,0 )k denote an effective enumeration of all open rational balls. Given (~q` )` , calculate inductively for m ∈ N the subsequence (Bn,m+1 )n of (Bn,m )n containing those balls disjoint to {~q1 , . . . , ~qm }. • For ~x accumulation point of (~q` ) and Bk,0 an arbitrary open rational ball containing ~x, there is some ~qM ∈ Bk,0 S. By construction, this Bk,0 will not occur in (Bn,m+1 )n for m ≥ M . Hence ~x ∈ Rd \ n Bn,∞ = A. • If ~x is contained in some ball Bk,0 which ‘prevails’ as Bn,∞ , it cannot contain any ~qm by construction. Therefore ~x is not an accumulation point. SN Claim 4.62. Let N ∈ N, ~un , ~vn ∈ Qd , and ~x ∈ Rd with ~x 6∈ n=1 (~un , ~vn ). Then, for every > 0, S N there is some ~q ∈ Qd \ n=1 (~un , ~vn ) such that k~x − ~qk ≤ . Proof. If ~x ∈ Qd then let ~q := ~x. Otherwise ~x belongs to the open set Rd \ rational numbers lie dense.

SN

un , ~vn ] n=1 [~

in which

Corollary 4.63. A subset of X = Rk belongs to the relativized class Π1 [∅0 ] iff it is the set of accumulation points of a recursive sequence in Qk . When the representation ψ 0 from Definition 4.60 is restricted to subsets of [0, 1]k (rather than of Rk ), one may in general not require the sequence (~qn ) of rational vectors to remain in [0, 1]k : otherwise the empty set has no encoding. (Technically speaking, in our proof of “ψ> ◦ ı0 ψ>0 ”, SN˜ (m) the set [0, 1]d ∩ Qd \ n=1 Bn,m may be empty. . . ) For non-empty closed subsets of [0, 1]k , on the other hand, this additional requirement can be achieved effectively: Lemma 4.64. Given a sequence (~qn )n in Qd with at least one accumulation point in [0, 1]d , one can effectively obtain a sequence (~ pm )m in Qd ∩ [0, 1]d whose accumulation points coincide with d those of (~qn )n contained in [0, 1] . Proof. Simply dropping from (~qn )n all elements not in the cube [0, 1]k maintains all accumulation points in its interior but may miss some on its boundary. We show how to effectively intersect both the sequence (~qn )n and its accumulation points with the halfspace [0, ∞) × Rd−1 . Further similar application to (−∞, 1] × Rd−1 , R × [0, ∞) × Rd−2 , R × (−∞, 1] × Rd−2 , . . . , Rd−1 × [0, ∞), Rd−1 × (−∞, 1] yields the claim. Start (phase #0) by passing through from input to output all qn ≥ 0 and dropping all qn < −1 until encountering some −1 ≤ qn < 0; in this case, instead of this element, append 0 to the output and enter phase #1: again, all qn ≥ 0 are simply transmitted, but now qn < − 21 are dropped until encountering some − 12 ≤ qn < 0 which is transformed to 0 and initiates phase #2. Generally phase #j passes on qn ≥ 0, skips qn < −2−j , and waits for some −2−j ≤ qn < 0.

4.4.5

Iterated Jumps and Applications

Climbing up the Arithmetical Hierarchy corresponds to iterated jumps of the Halting problem. We proceed similarly with our hierarchy of representations: Definition 4.65. Let ı(d+1) := ı(d) ◦ ı0 = ı0 ◦ ı(d) .

98


Straightforward inductive application of Remark 4.50 shows that ı(d) –computability is equivalent to ı–computability relative to ∅(d) . If F and G are partial (ı → ı0 )–computable string functions, then their composition G ◦ F is (ı → ı00 )–computable by Lemma 4.51a). (d)

Theorem 4.66. For each d ∈ N, it holds ρ(d) ≡ ρ ◦ ı(d) and ρ< ≡ ρ< ◦ ı(d) . Proof. The induction start d = 1 has been treated in Propositions 4.56 and 4.57. Since a ρ(d+1) – name of x ∈ R is the join of ρ(d) –names of elements xn with x = limn xn , Proposition 4.56 together (d+1) with Lemma 4.53b) also provides the induction step; similarly for ρ< . As a consequence we obtain by inductive application of Lemmas 4.51a+c) and Lemma 4.53a) the following extension (and simplification of the proofs) of Theorems 4.26 and 4.30 which also includes Corollary 4.43: Corollary 4.67. Fix f : X → R. a) If f is (ρ(d) → ρ(d) )–continuous, then it is continuous. If f is (ρ(d) → ρ(d) )–computable, then it is also (ρ(d+1) → ρ(d+1) )–computable, If f is (ρ(k) → ρ(k+d) )–continuous, then it is Σd+1 –measurable. (d)

b) If f is (ρ(d) → ρ< )–continuous, then it is lower semi-continuous. (d) (d+1) If f is (ρ(d) → ρ< )–computable, then it is also (ρ(d+1) → ρ< )–computable. (k+d) If f is (ρ(k) → ρ< )–continuous, then it is Σd+1 lower semi-measurable. (d)

(d)

c) If f is (ρ< → ρ< )–continuous, then it is nondecreasing. (d) (d) (d+1) If f is (ρ< → ρ< )–computable, then it is also (ρ(d+1) → ρ< )–computable. We finally turn to uniform variants of the claims considered in Theorem 4.4: Theorem 4.68. For X = Rk it holds Σ1 k a) θ< ◦ ı0 ¬ δΣ2 Σ 1 k b) θ< ◦ ı0 6 δΠ2 Σ1 k c) δΠ2 θ< ◦ ı00 k Recall that Σ1 denotes the class of open subsets of X for which θ< ≡ δΣ1 is a representation. Of course the restriction of δΣ2 and δΠ2 is thus necessary for the reductions to make sense.

Σ1 k Proof. a) Non-reducibility from δΣ2 to θ< ◦ ı0 follows from Proposition 4.7, so let us turn to the reverse positive reducibility claim. A (θ< ◦ı0 )–name for U ∈ O(X) consists of two rational double sequences (c(n,m) ) and (r(n,m) ) such that each single S sequence c(n,·) and r(n,·) , n ∈ N, eventually stabilizes to some c(·,∞) and r(·,∞) where U = n∈N B(c(n,∞) , r(n,∞) ) and B(c, r) denotes the open ball with center c and radius r. Both representations admit of effective countable unions; it therefore suffices to show the reduction on open rational balls, that is, we may assume w.l.o.g. U = B(cm , rm ) for all m ≥ m0 . So let ( B cm , rm · (1 − 2−n ) : (ck , rk ) = (ck+1 , rk+1 )∀k ≥ m An := , ∅ : otherwise S hence U = n An . Moreover the closed set An can be ψ> –computed, uniformly in n and the given sequences (cm ) and (rm ): start generating B(· · · ); if the co-r.e. condition “∀k ≥ m” eventually turns out to fail, the machine may still revert to a ψ> –name for ∅ by adding further negative information to the output. We thus obtain a δΣ2–name for U .

4.5. TYPE-2 NONDETERMINISM

99

b) Suppose a Type-2 Machine M performs the reduction described above. Feed into it the constant double-sequence Bn,m := (0, 1) ∈ Σ1 , obviously a valid θ< ◦ ı0 –name for U1 = S 0 B sequence sequence (Bn,m ) of open n n,∞ = (0, 1). Due to the presumption, the double T S 1 2 0 rational intervals output satisfies A := [ 3 , 3 ] ⊆ U1 = n m Bn,m . In particular, by comS M1 0 B1,m for some M1 ∈ N. Up to this output, M has read only finitely pactness, A ⊆ m=1 many Bn,m ; up to (n1 , m1 ), say. Now modify (some of the unread part of) the input to Bn,m := ∅ for n ≤ n1 and m > m1 . Observe that, for each n, the sequence (Bn,m )m is still finitely stabilizing and hence a valid 0 θ< ◦ ı0 –name for (0, 1) ⊇ A. By hypothesis, it leads M to extend its output (Bn,m ) such that S M2 0 additionally A ⊆ m=1 B2,m for some M2 ∈ N. Continuing iteratively, we ultimately obtain a (Bn,m ) with Bn,mn = ∅ for some mn ∈ N SMn 0 0 Bn,m for (i.e. a θ< ◦ ı0 –name for ∅) whose input leads M (Bn,m ) with A ⊆ m=1 T toSoutput 0 each n and some Mn ; in particular, ∅ 6= A ⊆ n m Bn,m , that is the output cannot be a δΠ2–name for ∅: contradiction. T S c) Given a double-sequence (Bn,m ) of open rational balls with U = n m Bn,m open. Let (BK ) denote an enumeration of all (non-empty) open rational balls. Define the triplesequence ( SM BK : ∀n ≤ N : B K ⊆ m=1 Bn,m 0 BK,N,M := . ∅ : otherwise Firstly, this is easy to compute from the input. Secondly, we claim that this SMis a valid (θ< ◦ ı0 ◦ ı0 )–name: For fixed (K, N ) either there exists some M such that B K ⊆ m=1 Bn,m : 0 )M stabilizes to BK beyond M ; or there is no such M , in which then the sequence (BK,N,M 0 denote this (finitely attained) limit and case the sequence is constantly ∅. Let BK,N,∞ remember that it equals either ∅ or, independently of N , BK . Then, similarly, the sequence 0 )N is either constantly BK (namely in case that, for every N , there exists some M (BK,N,∞ SM such that B K ⊆ m=1 BN,m ); or eventually stabilizes to ∅. S 0 Thirdly, this (θ< ◦ ı0 ◦ ı0 )–name encodes (i.e. K BK,∞,∞ coincides with) the set U : If 0 BK,∞,∞ 6= ∅ then, by construction, it equals BK where for all N there exists some B K ⊆ SM T S ∈ BK ⊆ B K ⊆ U m=1 BN,m ; i.e. BK ⊆ N m BN,m = U . Conversely, if x ∈ U , then xS for some K because U is open; moreover, as B K is compact, B K ⊆ U ⊆ M BN,M implies: SM 0 ∀N ∃M : B K ⊆ m=1 BN,M . Therefore x ∈ BK = BK,∞,∞ by construction.

4.5

Type-2 Nondeterminism

Our failure sofar to hypercompute discontinuous functions as discussed in Section 4.3.5 hinges on Fact 4.25 and Theorem 4.26. Closer examination of their proofs in turn reveals them to crucially rely on the underlying Turing Machines to behave deterministically. It is therefore an obvious step to consider nondeterminism as a putative means of effective evaluation of discontinuous real functions like Heaviside’s. In the discrete (i.e., Type-1) setting, where any computation is required to terminate, the finitely many possible choices of a nondeterministic machine can of course be simulated by a deterministic one—though even here, this is subject to the important condition that all paths of the nondeterministic computation indeed terminate, cf. [BST89]. In contrast, a Type-2 computation realizes a transformation from/to infinite strings and is therefore a generally non-terminating process. Therefore, nondeterminism here involves an infinite number of guesses which it turns out cannot be simulated by a deterministic Type-2 machine. We also point out that nondeterminism was already revealed to be not only a useful but ¨chi extended automaindeed the most natural concept of computation on Σω . More precisely, Bu ton theory from finite to infinite strings and proved that here, as opposed to deterministic ones,

100


nondeterministic finite automata are closed under complement [Thom90] and thus the more appropriate model of computation. Since automata and Turing Machines constitute the top and bottom Chomsky Level 3: regular 2: context-free 1: context-sensitive 0: unrestricted

Σ∗ Finite Automata

Σω B¨ uchi Automata (nondeterministic)

(Type-1) Turing Machines nondeterministic Type-2 Machines

Figure 4.11: Models of Computation in Chomsky’s Hierarchies over finite/infinite strings levels, respectively, of Chomsky’s Hierarchy of classical languages L ⊆ Σ∗ (Type-1 setting), we suggest that over infinite strings Σω (Type-2 setting) both their respective counterparts, that is B¨ uchi Automata and Type-2 Machines be considered nondeterministically; compare Figure 4.11. The concept of nondeterministic computation of a function f :⊆ Σ∗ → Σ∗ (as opposed to a ńyi Theorem in computational decision problem) is taken from the famous Immerman-Szelepsce complexity; cf. for instance the paragraph preceding Theorem 7.6 in [Papa94]: For x ¯ ∈ dom(f ), some computing paths of the machine M may fail by leading to rejecting states, as long as 1) there is an accepting computation of M on x ¯

and

2) every accepting computation of M on x ¯ yields the correct output f (¯ x). This notion extends straightforwardly from Type-1 to the Type-2 setting: Definition 4.69. Let A and B be sets with respective representations α :⊆ Σω → A and β :⊆ Σω → B. A function f :⊆ A → B is called nondeterministically (α → β)–computable if some nondeterministic one-way Turing Machine M, – upon input of any α–name σ ¯ ∈ Σω for some a ∈ dom(f ), – has a computation which outputs a β–name for b = f (a)

and

– every infinite computation of M on σ ¯ outputs a β–name for b = f (a). This definition is sensible insofar as it leads to closure under composition: Observation 4.70. Let f :⊆ A → B be nondeterministically (α → β)–computable and g :⊆ B → C be nondeterministically (β → γ)–computable. Then, g ◦ f :⊆ A → C is nondeterministically (α → γ)–computable. While admittedly even less realistic than a classical N P–machine, Type-2 nondeterminism turns out in Section 4.5.2 to benefit from particular structural elegance (in addition to closure under composition). Remark 4.71. [20, Definition 14] defined nondeterministic computability deviating from Definition 4.69 in the third condition: there we required any infinite output of M on σ ¯ to constitute a β–name for b = f (a). Since any infinite output requires infinite computation but not vice versa, this may seem to lead to a different notion. However, both do coincide: M may additionally guess and verify a function F : N → N such that the n–th symbol is output after F (n) steps. If F has been guessed incorrectly (and in particular if, for the given input σ ¯ , no such F exists at all), then this can be detected within finite time and the computation then be aborted, thus complying with the (only seemingly stronger) Definition 4.69.


4.5.1

101

Computational Power on Finite Inputs

A subtle point in Definition 4.69: the nondeterministic machine may ‘withdraw’ from a guess as long as it does so within finite time. This gives it the surprising power of ‘deciding’ all hyperarithmetical sets: Theorem 4.72. For L ⊆ N, the ‘padded’ characteristic function χL : N → {0, 1} × { }ω is nondeterministically computable if and only if L ∈ ∆11 (2N ). In particular, the power of Type-2 nondeterminism goes strictly beyond the finite Effective Borel Hierarchy (Section 4.2.2); see also Corollary 4.75 below. The following notion turns out to be both natural and useful in the proof of Theorem 4.72: Definition 4.73. A set L ⊆ N is nondeterministically semi-decidable if there exists a nondeterministic Turing machine M which, upon input of x ∈ N, – has a computational path which outputs an infinite string

in case x ∈ L;

– in case x 6∈ L aborts, on each computational path, after finite time. L is nondeterministically enumerable if a nondeterministic Turing machine M without input – has a computational path which outputs a list (xn )n of integers with L = {xn : n ∈ N}; – every infinite computation of M prints a list (xn )n of integers with L = {xn : n ∈ N}. Nondeterministic enumerability thus amounts to nondeterministic computability of an En–name, cf. [Weih00, Definition 3.1.2.5]. Surprisingly, it turns out to be equivalent not to nondeterministic semi-decidability but to nondeterministic decidability: Proposition 4.74. With respect to Type-2 nondeterminism and nondeterministic reducibility “n ”, it holds: a) En ≡n Cf, where the latter refers to the representation of the powerset of N enumerating a set’s members in order [Weih00, Definition 3.1.2.6]. b) L ⊆ N is nondeterministically decidable if and only if it has a computable Cf–name; equivalently: both L and its complement are nondeterministically semi-decidable. c) L ⊆ N is nondeterministically semi-decidable if and only if L ∈ Σ11 (2N ). Proof. a) “Cf En” holds even deterministically. For the converse we are given a list (xn )n of integers enumerating L. Let the machine guess a function F : N → N with xn ≥ m ∀n ≥ F (m): Such obviously F exists; and an incorrect guess can be detected within finite time. Knowing F , we can determine and sort all restrictions L ∩ [1, m], m ∈ N. b) Immediate. c) Let L ∈ Σ11 . By Equation (3.8), a nondeterministic Type-2 machine M may, given x, iteratively guess entries of ¯b and check “hx, n; ¯b|≤n i ∈ R” for each n ∈ N: if this condition is violated, abort (within finite time); otherwise output a dummy symbol and continue with n + 1. This yields nondeterministic semi-decision of L. Conversely let L be semi-decided by M. Then x ∈ N belongs to L if and only if there exists a sequence (bn )n of guesses bn ∈ {0, 1}ω such that M, on input x and ¯b, “performs at last n steps”. The latter predicate on hx, n; b1 , . . . , bn i being decidable, L is of the form (3.8). Claims b) and c) together yield Theorem 4.72.

102


Corollary 4.75. There is nondeterministically computable real c ∈ R which does not belong to (any finite level of ) the Arithmetical Hierarchy of real numbers ∆d (R) from Section 4.2. (The number c will, however, belong to some transfinite ∆α (R), recall Section 4.2.2.) Proof. Take some hyperarithmetical but not finitely arithmetical L ⊆ N, that is, L ∈ ∆11 \ Σ nk−1 and check whether it complies with Inequality (4.13) for the (finitely many) ` ≤ k; if it does not, we may abort this computation in accordance with Definition 4.69. For d = 1, let x = limn xn with xn = limm qn,m . Then apply the case d = 0 to convert for each n the ρ0 –name (qn,m )m of xn ∈ R into an according ρ–name, that is, a sequence pn,m satisfying |xn − pn,m | ≤ 2−m . Its diagonal (pn,n )n then has |x − pn,n | ≤ |x − xn | + 2−n → 0 and is thus a ρ0 –name for x. Higher levels d can be treated similarly by induction. For (ρ → ρb,2 )–computability, let x ∈ (0, 2) be given by a fast convergent sequence (qn ) ⊆ Q. We guess the leading digit b ∈ {0, 1} for x’s binary expansion b.∗; in case b = 0, check whether x > 1—a ρ–semi decidable property—and if so, abort; similarly in case b = 1, abort if it turns out that x < 1. Otherwise (that is, proceeding while simultaneously continuing the above semi-decision process via dove-tailing) replace x by 2(x − b) and repeat guessing the next bit. It is instructive to observe how, in the case of non-unique binary expansion (i.e., for dyadic x), nondeterminism in the above (ρ → ρb,2 )–computation generates, in accordance with the third requirement of Definition 4.69, both possible expansions. Theorem 4.76 implies that the nondeterministic computability of real functions is largely independent of the representation under consideration—in striking contrast to the deterministic case (Observation 4.13) where the effectivity subtleties arising from different encodings had already confused Turing himself [Turi37]. Corollary 4.77. a) With respect to nondeterministic reduction “n ” , it holds that ρb,2 ≡n n n ρ ≡ ρ< ≡ ρ0 ≡n ρ0< ≡n ρ00 ≡n . . .. b) The entire Real Arithmetical Hierarchy from Section 4.2 is nondeterministically computable. Proof.

a) follows from Theorem 4.76.

b) Let x ∈ ∆d+1 for some d ∈ N. Then x ∈ R is ρ(d) –computable, hence nondeterministically also ρb,2 –computable by a). Alternatively or for d ≥ ω combine Theorem 4.72 with Theorem 4.9.


103

In particular, this kind of hypercomputation allows for nondeterministic (ρ → ρ)–evaluation of Heaviside’s function by appending to the (ρ → ρ< )–computation in Example 4.23 a conversion from ρ< n ρ0 back to ρ. Section 6.4 establishes many more real functions, both continuous and discontinuous, to be nondeterministically computable, too.

104


Chapter 5

BCSS Hypercomputation Unlike the Turing machine, the BCSS machine over (R, +, −, ×, ÷, D := dn+m · max{n − 1, m − 1}. Notice that p and q themselves in general do not satisfy claim a); e.g. p = π · p˜ and q = π · q˜. Proof. a) Without loss of generality, take p and q to be co-prime. Let yi := f (ai ). The idea is to solve the rational interpolation problem for (ai , yi ). Already knowing that it has a solution (namely p, q) avoids many of the difficulties discussed in [MaDu62]. More precisely, observe that the coefficients p0 , . . . , pn−1 , q0 , . . . , qm−1 ∈ R of the homogeneous (n + m) × (n + m)-size system of linear equations  p0   . m−1  n−1 2 −y1 −y1 a1 . . . −y1 a1 1 a1 a1 . . . a1  ..  m−1   1 a2 a22 . . . an−1 −y2 −y2 a2 . . . −y2 a2  pn−1 2    1 a3 a2 . . . an−1 −y3 −y3 a3 . . . −y3 am−1  ·   q0 3 3 3    .. .. .. . . .. .. .. .. ..  . . . . . . . . . .  ..

p and q satisfy          

=

0 .

qm−1 In particular, this system has (p0 , . . . , qm−1 ) ∈ Rn+m as a non-zero solution. The coefficients of the matrix live in Q(a1 , . . . , an+m ). Therefore, Gaussian Elimination yields a (possibly different) non-zero solution (¯ p0 , . . . , q¯m−1 ), also with entries in Q(a1 , . . . , an+m ). Now apply the Euclidean Algorithm to the polynomials p¯, q¯ obtained in this way and calcu¯ which, again, has coefficients in Q(a1 , . . . , an+m ). late their greatest common divisor h ¯ and q˜ := q¯/h ¯ are co-prime polynomials over Q(a1 , . . . , an+m ) of deg(˜ Thus, p˜ := p¯/h p) < n and deg(˜ q ) < m such that p˜ · q coincides with p · q˜ on arguments a1 , . . . , an+m . This implies that the latter polynomials of degree less than n + m are identical: p˜ · q = p · q˜. It follows that q divides both sides; and coprimality of (p, q) in the factorial ring R[X] requires that q divides q˜. Similarly, q˜ divides q, yielding q˜ = λq for some λ ∈ R. Analogously, p˜ = λp for the same λ. b) Consider x ∈ R with y := f (x) ∈ Q and suppose x is algebraic of deg(x) > dn+m · max{n − 1, m−1} or transcendental. Since f is non-constant, the polynomial p˜−y · q˜ is not identically zero. Being, by virtue of a), a zero of this polynomial with coefficients from Q(a1 , . . . , an ), x lies in an algebraic extension of the latter field, hence ruling out the possibility that it is

5.3. POST’S PROBLEM OVER THE REALS

111

transcendental. More precisely, the degree of x over Q(a1 , . . . , an ) is bounded by deg(˜ p −y· q˜); and deg(x), its degree over Q, is at most deg(˜ p − y · q˜) · deg(a1 ) · · · deg(an+m ) ≤ max{n − 1, m − 1} · dn+m by Fact 2.2a)—contradiction. Proof (Theorem 5.13b). Consider the 6-ary tree obtained by unrolling a putative BCSS algorithm semi-deciding R \ A relative to Q. Since this tree has at most countably many leafs v but the set R \ A of terminating inputs x is uncountable, some Xv must be uncountable as well. Consider the path u1 , . . . , um = v of nodes to that leaf: Each such u is associated some quolynomial gu ∈ R(X) with Xv ⊆ dom(gu ) and gives rise to a branch based on sgn gu (x) and/or on oracle query “gu (x) ∈ Q?”. W.l.o.g. no branch tests “0 = gu (x)” or (positively answered) queries “gu (x) ∈ Q” occur on u1 , . . . , um : for, otherwise, the uncountable set Xv of transcendentals x passing through this branch would require gu to be constant (Lemma 5.15) and node u be thus dispensable. Now take some t ∈ Xv ⊆ R \ A. Due to the continuity of rational functions, there exists ε > 0 such that sgn gui (x) = sgn gui (t) 6= 0 for all i = 1, . . . , m and all x ∈ R satisfying |x − t| < ε. In particular, gu (a) and gu (t) have the same sign for algebraic numbers a of unbounded degree according to Lemma 5.14. From the presumption, we conclude that none of them completes the (terminating) computational path to leaf v: they must branch off somewhere, that is, satisfy gui (a) ∈ Q for some i = 1, . . . , m. However by Lemma 5.16b), each single gu can sort out only algebraics of degree up to some finite D = D(u)—contradiction.

5.3.1

Embedding any Countable Lattice as BCSS Degrees

A further achievement of the works of Friedberg and Muchnik was the existence of incomparable semi-decidable degrees below the Halting problem. We now extend Theorem 5.13 to establish the same for the real case—again explicitly. Consider the following type of algebraic extensions: Definition 5.17. For fields Q ⊆ F ⊆ R and 0 < r ∈ Q, let 1 √ F ( ∗ r) := F r n : n ∈ N where the fractional powers are understood as positive real numbers. √ of 2, n ∈ N. The ancient Q( ∗ 2) for √ √ instance results from Q by field adjunction of all n-th roots ∗ proof of 2’s irrationality immediately generalizes to reveal that [Q( 2) :√ Q] is indeed infinite. √ √ By Lemma 5.18a) below this extends to show, for example, [Q( ∗ 2, ∗ 3) : Q( ∗ 2)]. In combination with Lemma 5.18a) it generalizes Lemma 5.14. Lemma 5.18.

a) For distinct prime numbers p1 , p2 , . . . , pd , pd+1 and n ∈ N, it holds √ √ √ √ √ √ √ Q ∗ p1 , ∗ p2 , . . . , ∗ pd , ∗ pd+1 : Q ∗ p1 , ∗ p2 , . . . , ∗ pd = ∞ .

√ √ b) For any n ∈ N, > 0, and x ∈ R, there exists y ∈ Q( ∗ 2) of degree at least n over Q( ∗ 3) such that |x − y| < . Proof.

a) Combine Lemma 2.3a+c) with Fact 2.2a). √ b) By a), b := 21/n has degree n over Q( ∗ 3); and so has y := b + r for any r ∈ Q. Q being dense, take r close to x − b. √ √ Theorem 5.19. Both sets Q( ∗ 2) and Q( ∗ 3) √ are BCSS √ semi-decidable—yet both √ √ are incomparable with respect to BCSS reducibility: It holds Q( ∗ 2) < 6 Q( ∗ 3) and Q( ∗ 3) < 6 Q( ∗ 2). The proof is based on the following immediate generalization of Lemma 5.16. Lemma 5.20. Let f ∈ R(X), f = pq with polynomials p, q of degree less than n and m, respectively. √ √ Let a1 , . . . , an+m ∈ Q( ∗ 2) ∩ dom(f ) be distinct with f (ai ) ∈ Q( ∗ 3).

112

CHAPTER 5. BCSS HYPERCOMPUTATION

a) There are co-prime polynomials p˜, q˜ of deg(˜ p) < n, deg(˜ q ) < m with coefficients in the √ algebraic field extension Q( ∗ 3; a1 , . . . , an+m ) such that, for all x ∈ dom(f ) = {x : q(x) 6= 0} ⊆ R, it holds f (x) = f˜(x) := p˜(x)/˜ q (x). √ ∗ ∗ b) Let d := maxi degQ( √ 3) (ai ). Then f (x) 6∈ Q( 3) for all transcendental x ∈ dom(f ) as well √ n+m ∗ · max{n − 1, m − 1}. as for all x ∈ Q( ∗ 2) of degQ( √ 3) (x) > D := d Proof (Theorem 5.19). For semi-decidability observe that, by virtue of Lemma 2.3a) and Fact 2.2c), [ √ √ ∗ n Q( 2) = Q[ 2] = x ∈ Q ∃n ∈ N ∃a0 , . . . , an−1 ∈ Q n∈N

∃y ∈ R : x = a0 + ya1 + . . . + y n−1 an−1 ∧ y n = 2 | {z }

.

=:Φ(n;a0 ,...,an−1 ;x)

Now existential quantification with respect to y can be eliminated from Φ (Fact 2.13); hence √ membership of x in Q( ∗ 2) can be semi-decided by searching for n ∈ N and a0√ , . . . , an−1 ∈ Q. √ Consider a putative machine semi-deciding R \ Q( ∗ 2) by means of an Q( ∗ 3)-oracle. Follow the proof of Theorem 5.13 and apply Lemma 5.15 to obtain in just the same way a leaf v together √ with the related path set Xv ⊆ R \ Q( ∗ 2). Since Xv is uncountable, it contains a transcendental t √ ∗ and, in each neighborhood of t by virtue of Lemma 5.18b), elements of Q( 2) of arbitrarily high p √ degree over the field Q( ∗ 3). Thus, applying Lemma 5.20, there exist elements in Q( ∗ [2]) that √ ∗ are branched along v, contradicting√the assumption that the machine semi-decides R \ Q( 2). √ The semi-decidability of R \ Q( ∗ 3) relative to Q( ∗ 2) is ruled out similarly. The numbers 2 and 3 in the above proof can obviously be replaced by any two distinct primes; that √ √ is, the sets Q( ∗ p) and Q( ∗ q) are incomparable for any two p, q ∈ P = {2, 3, 5, 7, 11, 13, 17, . . .}. In particular, we have explicitly an infinite number of incomparable degrees. Moreover, the argument √ √ immediately extends to reveal that, for P, Q ⊆ P, Q({ ∗ p : p ∈ P }) < Q({ ∗ q : q ∈ Q}) iff P is contained in Q. Since the collection of subsets with inclusion is the prototype of a poset, we have thus arrived at the following Scholium 5.21. Every countable poset can be embedded in the semi-decidable BCSS-degrees. The latter parallels classical results in discrete recursion theory; see for instance [Soar87, Exercise §VII.2.2(b) and Exercise §VIII.4.10].

5.3.2

The Case of Linear BCSS Machines

We have so far considered the full BCSS model over the reals, that is the structure (R, +, −, ×, ÷, R, < ). Over the last decade, its linearly restricted version (R, +, −, 0, 1, 2 require some more sophistication, to which end we generalize ii) as follows: iii) Either ~gu (~x) ∈ Ld−1 ∀~x ∈ Rd or ~gu (~π ) 6∈ Ld−1 . iv) For finitely many ~gui , i = 1, . . . , k, the subset ~x ∈ Ld : ~gui (~x) 6∈ Ld−1 ∀i = 1, . . . , k of Ld is dense in Rd . Indeed, this follows from applying Lemma 5.25a+b+d) below to the homogeneously linear f~0 (x0 , ~x) := 1, (x0 −1)~gu (0)+~gu (~x) and ~x0 := (1, π, π 2 , . . . , π d ); linear dependence of the component functions of ~gu can, according to the first case of iii), without loss of generality be ruled out as redundant an oracle query. By i) and iv), there exist inputs ~π 0 ∈ Ld for which sign tests and oracle queries give the very same results as for input ~π 6∈ Ld and in particular lead to termination: contradiction. P Lemma 5.25. Fix linear functions f1 , . . . , fn : Qm → Q, fi (x1 , . . . , xm ) = j aij xj with A = (aij ) ∈ Qn×m . a) If f1 , . . . , fn (i.e. the rows ai• of A, i = 1, . . . , n) are linearly dependent over Q, then so are the real (!) numbers f1 (~x), . . . , fn (~x) for any choice of ~x ∈ Rm . b) If both f1 , . . . , fn and x1 , . . . , xm ∈ R are linearly independent over Q, then so are the real numbers f1 (~x), . . . , fn (~x). c) Suppose rankQ (A) = n < m. Then the set of vectors (x1 , . . . , xm ) ∈ Rm which are linearly dependent over Q but mapped to linearly independent numbers f1 (~x), . . . , fn (~x) is dense.

114


d) The same holds for finitely many matrices A1 , . . . , Ak ∈ Qn×m : If they all have rankQ (A) = n < m, then the following set is dense in Rm : ~x ∈ Rm linearly dependent over Q ∧ Aj · ~x linearly independent over Q ∀j = 1, . . . , k Of course Q ⊆ R may be replaced by any other field extension F ⊆ E in Claims a) and b). Claim c) trivially fails for rankQ (A) < n or for n = m; and so does d) for the case of countably many matrices. Proof.P a) By prerequisite there exists non-zero ~z ∈ Qn such that ~z† · A = 0 ∈ Rm ; hence x) = ~z† · (A · ~x) = 0 ∈ R. i zi fi (~ P b) Let ~z ∈ Qn be such that R 3 0 = i zi fi (~x) = ~z† · A · ~x. Then Qm 3 ~z† · A = 0 because the xj are linearly independent over Q; hence ~z = 0 by linear independence of f1 , . . . , fn . c) Observe that f1 (~x), . . . , fn (~x) are linearly dependent over Q iff ~x is perpendicular (‘⊥’) to some ~z = A† · ~y with non-zero ~y ∈ Qn . Since rank(A† ) = n by prerequisite, such ~z must itself be non-zero: f1 (~x), . . . , fn (~x) linearly dependent over Q

⇔

∃~z ∈ rangeQ (A† ) \ 0 : ~x⊥~z .

(5.2)

Also by prerequisite, rangeQ (A† ) is strictly a subspace of Qm ; hence Qm \ rangeQ (A† ) is dense. Any ~z ∈ Qm \ rangeQ (A† ) gives rise to a (real!) hyperplane ⊥~z := {~x ∈ Rm : ~x⊥~z} of vectors linearly dependent over Q. By the choice of ~z 6∈ rangeQ (A† ), ⊥~z is neither included in, nor coincides with, the real hyperplane ⊥~y for any ~y ∈ rangeQ (A† ). (Observe that ~z 6= 0!) Thus, (⊥~z)∩(⊥~y ) constitutes a strict, nowhere dense subspace of ⊥~z. Removing from ⊥~z the countably many subspaces (⊥~z) ∩ (⊥~y ) with ~y ∈ rangeQ (A† ) \ 0 leaves, by Baire’s Category Theorem, a dense subset of ⊥~z: consisting according to Equation (5.2) only of vectors ~x linearly dependent over Q such that f1 (~x), . . . , fn (~x) are linearly independent over Q. Since ~z was arbitrary from the dense set Qm \ rangeQ (A† ), this proves the claim. S d) follows similarly to c) by considering ~z ∈ Qm \ j rangeQ (A†j ).

5.4

Arithmetical Hierarchy

The diagonalization argument establishing the undecidability of the real Halting Problem H =: ∅ 0 (Fact 2.17d) relativizes and yields by iteration a strict hierarchy of problems ∅ (d)

:=

(d−1) hM? , ~xi M∅ terminates on input ~x

R∗

⊆

(5.3)

similar to Equation (3.2). In the discrete case, Theorem 3.9 established an equivalent characterization in syntactical terms. However, any problem L ⊆ N of the form (3.3) is BCSS–decidable by Example 2.25, and replacing X = N with X = R in (3.3) does not render it undecidable either because of Fact 2.13. Instead, Cucker in [Cuck92], motivated by Corollary 2.27a), arrives at the following Definition 5.26. For d, k ∈ N and a field F ⊆ R, consider the class d,F (2R ) of real languages of the form ~x ∈ Rk ∃n1 ∈ N ∀n2 ∈ N ∃n3 ∈ N . . . θd nd ∈ N : ~x ∈ Bhn1 ,...,nd i (5.4) k

where Bn ⊆ Rk is a sequence of sets (not necessarily basic-) semi-algebraic over F ; recall DefiniS S ∗ k ∗ ∗ tion 2.10. Furthermore let d,F (2R ) := k d,F (2R ) and d (2R ) := F d,F (2R ) where the last union is understood to range over all finite real field extensions: F = Q(c1 , . . . , cj ), j ∈ N, c1 , . . . , cj ∈ R. Similarly for classes d and d . . .

5.4. ARITHMETICAL HIERARCHY

115

Hence 1 (2R ) are the BCSS semi-decidable problems. Even just this bottom class (d = 1) lies skew to Σd and Σd , as we illustrate by extending [Zhon98, Section 3]: ∗

k

Lemma 5.27. Every set U ∈ Σ1 (2R ) is BCSS semi-decidable, but not every set in Π1 (2R ) is. k k Conversely, every BCSS semi-decidable L ⊆ Rk belongs to Σ2 (2R ) but not necessarily to Σ2 (2R ) k or to Π2 (2R ). S Proof. Every open set U ⊆ Rk is a countable union of certain open rational balls U = n B(~cn , rn ); (recall Section 2.2.2). By encoding the sequence of centers ~cn and radii rn in a real constant, a BCSS machine may effectively search for some n such that a given input belongs to B(~cn , rn ) and thus semi-decide U . The Cantor set belongs to Π1 (2R ) (Example 2.50) but is not BCSS semidecidable (Example 2.18a). Conversely, a basic semi-algebraic subset of Rk is by Definition 2.10 a finite intersection of preimages qi−1 [(0, ∞)] and p−1 j [{0}] of open or closed sets, respectively, under (continuous) polynok

k

k

mials and thus (Section 2.2.2) belongs to Σ1 (2R ) ∪ Π1 (2R ) ⊆ Σ2 (2R ). According to Corollary 2.27a), a semi-decidable set is a countable union of basic semi-algebraic ones and therefore k still an element of Σ2 (2R ). The set Q of rational numbers is BCSS semi-decidable (Example 2.21) but not in Π2 (2R ) (Example P2.8). The complement R \ {c} of every singleton is trivially BCSS-decidable but, for c := n∈∅00 2−n ∈ Σ2 (R) \ Π2 (R), not in Σ2 (2R ) according to Fact 4.15a). In contrast, the following refinement of [Cuck92, Theorems 2.11 and 2.13] holds: Theorem 5.28. For L ⊆ R∗ , d ∈ N, and c1 , . . . , cj , the implication chain “a⇒b⇒c⇒d” holds where a) to d) refer to the following claims: a) L is BCSS semi-decidable by a ∅ (d−1) –oracle machine with constants c1 , . . . , cj . b) L ∈d,F (2R ) where F = Q(c1 , . . . , cj ). ∗

c) There is some cj+1 ∈ R and a many-one reduction f : R∗ → R∗ from L to ∅ (d) , computable by a BCSS-machine with constants c1 , . . . , cj , cj+1 . d) L is BCSS semi-decidable by a ∅ (d−1) –oracle machine with constants c1 , . . . , cj , cj+1 from c). Proof. c⇒d) ∅ (d) is semi-decidable relative to ∅ (d−1) by the universal BCSS-machine without constants. b⇒c) holds by induction: For d = 1, encode the rational coefficients of the descriptions of the (polynomials whose solutions constitute the) sequence of sets (Bn,k ) as elements in F = Q(c1 , . . . , cj ) in the real number cj ; compare Lemma 2.26. Then devise a BCSS machine M which, using this encoded information, generates and searches the sequence (Bn,k ) for a member to contain a given input ~x ∈ Rk and which terminates when one is found. The mapping ~x 7→ hM, ~xi is thus a reduction from L to ∅ 0 computable by a machine with constants c1 , . . . , cj , cj+1 . The induction step now proceeds similarly to that of Theorem 3.9 with K replaced by

K := ~x, n1 m ∈ N, ∀n2 ∈ N ∃n3 ∈ N . . . θd nd ∈ N : ~x ∈ Bhn1 ,...,nd i,m . a⇒b) also proceeds inductively, the case d = 1 having been treated in Corollary 2.27a). For the inductive step d ≥ 2, unroll a computation of a ∅ (d−1) –oracle machine on input ~x ∈ Rk . as described above. Let gu ∈ F [X1 , . . . , Xk ] denote the quolynomial (vector) associated ∗ with tree node u. By induction hypothesis, it holds ∅ (d−1) ∈d−1,Q (2R ): a universal BCSS-machine without constants can semi-decide ∅ (d−1) relative to ∅ (d−2) . According to Lemma 5.29a-c) below, the inputs ~x leading to positive/negative answers to the single oracle

116

CHAPTER 5. BCSS HYPERCOMPUTATION query “gu (~x) ∈ ∅ (d−1) ?” thus belong to sets in d−1,F and d−1,F , respectively; that is, in d,F . And the sign conditions “gu (~x) : 0” give rise to sets in 1,F ⊆d,F , too. A leaf set Xv of the above ∅ (d−1) –oracle machine, thus being a finite intersection of sets in d,F , therefore still lies in d,F by virtue of Lemma 5.29d); and so does the set L accepted by the machine, that is the countable union over these Xv : Lemma 5.29e).

We point out some interesting differences between (the last step of) the proof of Theorem 5.28 and that of its discrete counterpart: on the one hand the quantifiers over N in Equation (5.4 would not merge (or commute) with one ranging over a finite collection O± of putative real oracle queries; on the other hand, the (merely) decidable set R in Equation (3.3) is in (5.4) replaced by the much more explicit countably semi-algebraic sequence (Bn,m ). Thus, instead of Lemma 3.11 one now has the following Lemma 5.29 (Closure Properties). Fix d ∈ N and field F ⊆ R. a)

d,F (2R ) ⊆d+1,F (2R ). ∗

∗

If E ⊆ R is a field extension of F , then

d,F (2R ) ⊆d,E (2R ). ∗

∗

b) For k ∈ N and L ⊆ Rk , L ∈d,F (2R ) ⇔ Rk \ L ∈d,F (2R ). ∗ ∗ For L ⊆ R∗ , L ∈d,F (2R ) ⇔ R∗ \ L ∈d,F (2R ). k

k

c) For f :⊆ R` → Rk a vector of quolynomials over F and L ∈d,F (2R ), it holds f −1 [L] ∈d,F ` (2R ). k

d) Let L1 , L2 ∈d,F (2R ). Then L1 ∩ L2 ∈d,F (2R ). S ∗ ∗ e) Ln ∈d,F (2R ) for n ∈ N implies that also n Ln ∈d,F (2R ). ∗

Dual claims hold for

5.4.1

∗

-classes.

Completeness and Other Applications of the Normal Form Theorem

Neglecting the exact choice and number of real constants, one arrives at a formulation even more similar to Post’s Theorem 3.9: Corollary 5.30. A language L ⊆ R∗ is semi-decidable by an ∅ (d−1) –oracle BCSS machine iff there exists a BCSS-decidable P ⊆ R∗ such that ~x ∈ R∗ ∃n1 ∈ N ∀n2 ∈ N ∃n3 ∈ N . . . θd nd ∈ N : (~x; n1 , . . . , nd ) ∈ P . (5.5) L = Proof. In view of Theorem 5.28, let L ∈d and Bm,k ⊆ Rk be basic semi-algebraic over F = Q(c1 , . . . , cj ) ⊆ R. Consider [ P := Bhn1 ,...,nd i,k × {(n1 , . . . , nd )} ⊆ R∗ . n1 ,...,nd ,k

This language is decidable by a BCSS machine M with constants c1 , . . . , cj : Given (~x, n1 , . . . , nd ) ∈ Rk × Nd (for inputs not of this form, M may reject right away) simply check whether “~x ∈ Bhn1 ,...,nd i,k ” holds. Moreover we have, for any choice of (n1 , . . . , nd−1 ), θd nd ∈ N : ~x ∈ Bhn1 ,...,nd−1 ,nd i,k

⇔

θd nd ∈ N : (~x; n1 , . . . , nd−1 , nd ) ∈ P

and thus the claimed form (5.5). ∗ Conversely let L be of this by a machine with constants c1 , . . . , cj . S form with P ⊆ R decidable Then, by Lemma 2.26, P = m,k Bm,k with Bm,k ⊆ Rk basic semi-algebraic over F = Q(c1 , . . . , cj ). We shall suppose w.l.o.g. that d is odd, i.e. θd = ∃; otherwise above representation applyk the to the complement of P. Define B0hn1 ,...,nd−1 ,nd ,mi,k := ~x ∈ R (~x, n1 , . . . , nd−1 , nd ) ∈

5.4. ARITHMETICAL HIERARCHY

117

Bm,k+d (which is basic semi-algebraic over F ) and observe that for ~x ∈ Rk and (n1 , . . . , nd−1 ) ∈ Nd−1 it holds ∃nd : (~x, n1 , . . . , nd ) ∈ P

⇔

∃nd , m : ~x ∈ Bhn1 ,...,nd−1 ,nd ,mi,k .

In Theorem 5.28, the implication “b⇒c” means that ∅ (d) is complete for d (2R ); compare Example 3.12a). Another complete problem is given in [Cuck92, Theorem 2.13], a uniform variant of Theorem 5.28: ∗

Scholium 5.31. For F = Q(c1 , . . . , cj ), let Bn,k ⊆ Rk be a sequence of sets semi-algebraic over F . Encode this sequence as the real (j + 1)-tuple consisting of c1 , . . . , cj plus cj ∈ R containing the sequence of the rational coefficients (over F ) of the polynomials describing Bn,k S ; recall Example 2.25 ∗ and Fact 2.32. Write hLi := (c1 , . . . , cj , cj+1 ) for this encoding of L = n,k Bn,k ∈d,F (2R ), compare Equation (5.4) in Definition 5.26. Then the language Sd := hL, ~xi L ∈d , ~x ∈ L is

d –complete, that is it holds Sd ∈d (2R ) and every L ∈d (2R ) satisfies L < Sd . ∗

∗

A counterpart to Example 3.12b), [Cuck92, Theorem 2.15] also establishes a real analogue of FIN to be 2 –complete. By combining Lemma 5.27 and Theorem 5.28, we conclude [Cuck92, Theorem 4.3]: Corollary 5.32. For each d ∈ N, the class d (2R ) of real subsets BCSS semi-decidable relative ∗ to ∅ (d−1) lies strictly between Σd and Σd+1 ; that d (2R ) of sets BCSS decidable relative to (d−1) ∅ between ∆d and ∆d+1 . S In particular, the (finite) Borel Hierarchy of Euclidean subsets d,k Σd (Rk ) coincides with the S ∗ (finite) BCSS Arithmetical Hierarchy d d (2R ). ∗

Proof. By induction on d with inductive start d = 1 treated in Lemma 5.27. Observe that in Example 5.7 the relatively computable real function f with image L lacks injectivity. This can be remedied by proceeding from a total to a partial f . More seriously, the oracle O under consideration belongs to but is not complete for 1 ⊆2 (Lemma 5.27). On the other hand, for injective functions computable relative to a d -complete oracle, the real counterpart to Fact 2.17b) does hold: Proposition 5.33. Let (possibly partial) f :⊆ R∗ → R∗ be injective and computable relative to ∅ (d) by a machine with constants c1 , . . . , cD . Then range(f ) is semi-decidable relative to ∅ (d) by a machine with constants c1 , . . . , cD . Proof. By “a⇒b)” of Theorem 5.28, dom(f ) ∈d+1,F where F := Q(c1 , . . . , cD ); that is [ [ \ [ M dom(f ) = ··· Bhn1 ,...,nd i,k k∈N n1 ∈N n2 ∈N n3 ∈N

nd+1 ∈N

where Bn,k ⊆ Rk is semi-algebraic over F (and denotes either union or intersection depending on d’s parity). Moreover fn,k := f |Bn,k is a quolynomial vector over F ; hence f [Bn,k ] ∈ R`k is semi-algebraic over F , too, by virtue of Quantifier Elimination (Fact 2.13). Therefore [ [\[ M range(f ) = f [dom(f )] = f ··· Bhn1 ,...,nd i,k L

k (*)

=

[ [\[ k

n1 n2 n3

···

M

n1 n2 n3

nd+1

f [Bhn1 ,...,nd i,k ]

nd+1

belongs to d+1,F according to (Lemma 5.29e). Step (*) crucially relies on f being injective on its domain! Finally, Claim “b⇒d)” of Theorem 5.28 asserts range(f ) semi-decidable relative to ∅ (d) by a machine with an additional real constant cD+1 . Closer inspection reveals that both sequences (Bn,k ) and (fn,k ) are uniform, therefore this constant can be disposed of.

118

5.4.2


Nondeterministic and Analytic Hierarchy

By Corollary 2.31, the Halting Problem for a non-deterministic BCSS machine is the same as that ∗ for a deterministic one. Put differently, the nondeterministic class n1,F (2R ) of languages of the form ~x ∈ R∗ ∃~y ∈ R∗ : (~x, ~y ) ∈ L = ~x ∈ Rk ∃k, ` ∈ N ∃~y ∈ R` ∃n ∈ N : (~x, ~y ) ∈ Bhk,ì,n (L ∈1,F , B semi-algebraic over F ) coincides with deterministic 1,F because existential quantifiers commute and the real one “∃~y ” can thus be merged into the B’s by virtue of Tarski’s Fact 2.13. However, when we proceed to oracle computation, we already saw in Example 5.7 that Quantifier Elimination may fail. Indeed, nondeterministic semi-decidability relative to H can be characterized k [Cuck92, Theorem 3.5] by the class n2,F (2R ) of languages of the form ~x ∈ Rk ∃~y ∈ R∗ : (~x, ~y ) ∈ L = ~x ∈ Rk ∃` ∈ N ∃~y ∈ R` ∃n1 ∀n2 ∈ N : (~x, ~y ) ∈ B`,hn1 ,n2 i (L ∈2,F (2R ), B semi-algebraic over F ) where we cannot move real “∃~y ” past “∀n2 ” for further S k k elimination. As a matter of fact, [Cuck92, Theorem 4.4] establishes n2 (2R ) := F n2,F (2R ) as so powerful as to coincide with the class Σ11 of analytic subsets of Rk ; recall Section 2.2.2. k Hence n2 (2R ) equals the class ∆11 of Borel sets and thus strictly contains the finite deterministic Arithmetical BCSS Hierarchy. Moreover, the real counterpart to Totality Tot turns out as n2 – complete [Cuck92, Theorem 3.9]. For the d-th nondeterministic level, [Cuck92] arrives at ∗

Definition 5.34. For F = Q(c1 , . . . , cj ) ⊆ R let nd,F denote the class of languages of the form ~x ∈ Rk ∃k ∈ N ∃n1 ∈ N ∃~y1 ∈ Rn1 ∀n2 ∈ N ∀~y2 ∈ Rn2 ∃n3 ∈ N ∃~y3 ∈ Rn3 (5.6) ... θd−1 nd−1 ∈ N θd−1 ~yd−1 ∈ Rnd−1 θd nd ∈ N : (~x, ~y1 , . . . , ~yd−1 ) ∈ Bk,hn1 ,...,nd i with B semi-algebraic over F .

5.5

Real Computational Universality

The preceding sections have revealed BCSS computability theory to be quite similar to recursion theory for discrete Turing machines. Nevertheless, the current state of the first is still far from achieving the depth and breadth to which the latter has been developed [Odif89, Soar87]. This is due to fundamental differences between the two realms, which require many proofs in the real setting—although for claims parallel to the discrete setting—to proceed considerably differently. For instance, Section 5.3 established explicitly an undecidable real problem strictly easier than the real Halting problem. As a matter of fact, most classical proofs for the undecidability of an r.e. problem P proceed by reduction from H—and thus implicitly establish P as Turing complete in the sense that P supports universal computation. The undecidability of a real problem P in contrast quite often exploits algebraic properties of BCSS computation; recall, for instance, Example 2.18). It thus does not (and, at least for P = Q, provably cannot) imply P to be computationally universal. Therefore, where the challenge in the discrete case consisted in finding an undecidable problem not reducible from H, we are in the real case confronted with Question 5.35. Are there ‘natural’ problems P BCSS-complete in the sense that both P < H and H < P hold?

5.5. REAL COMPUTATIONAL UNIVERSALITY

119

One such problem, S1 , has been identified in Scholium 5.31. However, its definition exhibits relations to semi-algebraic geometry and BCSS computability, hence completeness may not come as too great a surprise. This differs significantly from the two milestones in discrete uncomputability mentioned in Section 1.1.2: Both the Word Problem for finitely presented groups and Hilbert’s Tenth Problem arise naturally within pure mathematics; and the importance of their undecidability consists to a considerable part in the (unexpected) relation of these problems to computer science! Now an (at least straightforward) translation of Hilbert’s Tenth Problem to the reals fails BCSS–completeness, recall Lemma 2.30. So we turn in the following subsections to the

5.5.1

Word Problem for Groups

Groups occur ubiquitously in mathematics, and having calculations with and in them handled by computers constitutes an important tool both in their theoretical investigation and in practical applications, as revealed by the flourishing field of Computational Group Theory [FiKa91, FiKa95, HEoB05]. Unfortunately even the simplest question, namely the equality ‘a = b’ of two elements a, b ∈ G is generally undecidable for groups G that are reasonably presentable to a digital computer, that is, in a finite way—the celebrated result obtained independently by Novikov [Novi59] and Boone [Boon58] in the 1950ies. To a BCSS machine, on the other hand, it is trivially decidable (Example 2.25). However, whenever we deal with computational questions involving groups of real or complex numbers, the Turing model seems inappropriate anyway. As an example, take the unit circle in R2 equipped with complex multiplication. There is a clear mathematical intuition of how to compute in this group; such computations can be formalized in the BCSS model. We thus aim at a continuous counterpart to the discrete class of finitely presented groups for which the word problem is universal for the BCSS model.

5.5.2

Classical Setting

We briefly recall the classical word problem: Definition 5.36. a) Let X be a set. The free group generated by X, denoted by F = (hXi, ◦) or more briefly hXi, is the set (X ∪ X −1 )∗ of all finite sequences w ¯ = x11 · · · xnn with n ∈ N, xi ∈ X, i ∈ {−1, +1}, equipped with concatenation ◦ as the group operation subject to the rules x ◦ x−1 = 1 = x−1 ◦ x ∀x ∈ X (5.7) where x1 := x and where 1 denotes the empty word, that is, the unit element. b) For a group H and W ⊆ H, denote by hW iH := w11 · · · wnn : n ∈ N, wi ∈ W, i = ±1 the subgroup of H generated by W . The normal subgroup of H generated by W is hW iHn := h{h · w · h−1 : h ∈ H, w ∈ W }iH . For h ∈ H, we write h/W for its W –coset {h · w : w ∈ hW iHn } of all g ∈ H with g ≡W h. c) Fix sets X and R ⊆ hXi and consider the quotient group G := hXi/hRin , denoted by hX|Ri, of all R–cosets of hXi. If both X and R are finite, the tuple (X, R) will be called a finite presentation of G; if X is finite and R recursively enumerable (by a Turing machine, that is in the discrete sense; equivalently: semi-decidable), it is a recursive21 presentation; if X is finite and R arbitrary, G is finitely generated. 21 This

notion seems misleading as R is in general not recursive; nevertheless it has become established in literature.

120


Intuitively, R induces further rules “w ¯ = 1, w ¯ ∈ R” in addition to Equation (5.7); put differently, distinct words u ¯, v¯ ∈ hXi might satisfy u ¯ = v¯ in G, that is, by virtue of R. Observe that the ¯ = (w11 · · · wnn ) ∈ R can also be applied as rule “w11 · · · wnn = 1” induced by an element w − “w11 · · · wkk = wn−n · · · wk+1k+1 ”. Definition 5.36 (continued). d) The word problem for hX|Ri is the task of deciding, given w ¯ ∈ hXi, whether w ¯ = 1 holds in hX|Ri. The famous work of Novikov and, independently, Boone establishes the Word Problem for finitely presented groups to be Turing-complete: Fact 5.37. a) For any finitely presented group hX|Ri, its associated word problem is semi-decidable (by a Turing machine). b) There exists a finitely presented group hX|Ri whose associated word problem is many-one reducible by a Turing machine from the discrete Halting Problem H. Of course, a) is immediate. For the highly nontrivial Claim b), see for instance one of [Boon58, Novi59, LySs77, Rotm95].

is a recursively presented Example 5.38. H := {a, b, c, d} {a−i bai = c−i dci : i ∈ H} group whose word problem is reducible from H; compare the proof of [LySs77, Theorem §IV.7.2]. In order to establish Fact 5.37b), we need a finitely presented group. This step is provided by the remarkable Fact 5.39 (Higman Embedding Theorem). Every recursively presented group can be embedded in a finitely generated one. Proof. See, e.g., [LySs77, Section §IV.7] or [Rotm95, Theorem 12.18]. Fact 5.39 ensures the word problem from Example 5.38 to be in turn reducible to that of the finitely presented group H is embedded in, because any such embedding is automatically effective: Observation 5.40. Let G = hXi/hRin and H = hY i/hSin denote finitely generated groups and ψ : G → H a homomorphism. Then ψ is (Turing-) computable in the sense that there exists a computable homomorphism ψ 0 : hXi → hY i such that ψ 0 (¯ x) ∈ hSin whenever x ¯ ∈ hRin ; that is, ψ 0 maps R-cosets to S-cosets and makes the following diagram commute: hXi   y

−−−− → 0 ψ

hY i   y

(5.8)

ψ

hXi/hRin −−−−→ hY i/hSin Indeed, due the homomorphism property, ψ is uniquely determined by its values on the finitely many generators xi ∈ X of G, that is, by ψ(xi ) = w ¯i /hSin where w ¯i ∈ hY i. Setting (and storing in a Turing Machine) ψ 0 (xi ) := w ¯i yields the claim.

5.5.3

Presenting Real Groups

Regarding the BCSS-machine as a natural extension of the Turing machine from the discrete to the reals, the following is equally natural a generalization of Definition 5.36c+d): Definition 5.41. Let X ⊆ R∗ and R ⊆ hXi ⊆22 R∗ . The tuple (X, R) is called a presentation of the real group G = hX|Ri. This presentation is algebraically generated if X is BCSS-decidable 22 Most formally, R is a set of vectors of vectors of varying lengths. However, by suitably encoding delimiters, we shall regard R as effectively embedded in single vectors of varying lengths.


121

and X ⊆ RN for some N ∈ N. G is termed algebraically enumerated if R is in addition BCSS semi-decidable; if R is even BCSS-decidable, call G algebraically presented. The word problem for the presented real group G = hX|Ri is the task of BCSS-deciding, given w ¯ ∈ hXi, whether w ¯ = 1 holds in G. Turing finitely generated recursively presented finitely presented

BCSS algebraically generated algebraically enumerated algebraically presented

Figure 5.1: Correspondence between Classical Discrete and New Real Group Presentations

Remark 5.42. a) Although X inherits from R algebraic structure such as addition + and multiplication ×, the Definition 5.36a) of the free group G = (hXi, ◦) considers X a plain set only. In particular, (group-) inversion in G must not be confused with (multiplicative) inversion: 5 ◦ 15 6= 1 = 5 ◦ 5−1 for X = R. This difference may be stressed notationally by writing ‘abstract’ generators xa¯ indexed with real vectors a ¯; here x−1 5 6= x1/5 holds more intuitively. b) Isomorphic (that is, essentially identical) groups hX|Ri ∼ = hX 0 |R0 i may have different pre0 0 sentations (X, R) and (X , R ); see Example 5.47 below. Even when R = R0 , X need not be unique! Nevertheless, we adopt from literature such as [LySs77] the convention23 of speaking of “the group hX|Ri”, meaning a group with presentation (X, R). This, however, requires some care, for instance when w ¯ is considered (as in Definition 5.36d) both an element of hXi and of hX|Ri! For that reason we prefer to write hW iH rather than for instance Gp(W ), to indicate the group in which we consider a subgroup to be generated. For a BCSS-machine to read or write a word w ¯ ∈ hXi = (X ∪ X −1 )∗ of course means to input or N output a vector (w1 , 1 , . . . , wn , n ) ∈ (R × N)n . In this sense, the Rules (5.7) implicit in the free group are obviously decidable and may w.l.o.g. be included in R.

5.5.4

Examples of Presented Real Groups

The purpose of this section is to convince the reader of the reasonableness of Definition 5.41 by showing that it captures many classical groups from pure mathematics. We begin with a trivial Example 5.43. Every finite or recursive presentation is an algebraic presentation. Its word problem is BCSS-decidable. As long as X in Definition 5.36c) is at most countable, so will any group hX|Ri be. Only proceeding to real groups as in Definition 5.41 can include many interesting uncountable groups in mathematics. Example 5.44. Let S denote the unit circle in C with complex multiplication. The following is an algebraic presentation hX|R1 ∪ R2 i of S: • X := xr,s : (r, s) ∈ R2 \ {0} , • R1 := xr,s ◦ x−1 a,b : (r, s), (a, b) 6= 0 ∧ rb = sa ∧ ar > 0 , • R2 := xr,s ◦ xa,b ◦ x−1 u,v : (r, s), (a, b), (u, v) 6= 0 ∧ ∧ r2 + s2 = 1 ∧ a2 + b2 = 1 ∧ u = ra − sb ∧ v = rb + sa . 23 This can be justified with respect to the solvability of the word problem in the case of finite presentations and nonuniformly by virtue of Tietze’s Theorem [LySs77, Propositions §II.2.1 and §II.2.2].

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Intuitively, R1 yields the identification of (generators whose indices represent) points lying on the same half line through the origin. In particular, every xr,s is ‘equal’ (by virtue of R1 ) to some xa,b of ‘length’ a2 + b2 = 1. R2 in turn identifies xr,s ◦ xa,b with xu,v whenever, over the complex numbers, it holds (r + is) · (a + ib) = u + iv. Clearly, the presentation of a group need not be unique; e.g. we also have S ∼ = hY |R2 i where Y = {xr,s : r2 + s2 = 1}. Here is a further algebraic presentation of the same group: Example 5.45. Let X := {xt : t ∈ R}, R := {xt = xt+1 , xt xs = xt+s : t, s ∈ R}. Then hX|Ri is a 1D (!) algebraic presentation of the group [0, 1), + isomorphic to (S, ×) via t 7→ exp(2πit + ic) for any c ∈ R. Yet none of these isomorphisms is BCSS-computable! Next consider the group SL2 (R) of real 2 × 2 matrices A with det(A) = 1. A straightforward algebraic presentation of it is given as hX|Ri where X := {x(a,b,c,d) : ad − bc = 1} and R := {x(a,b,c,d) x(q,r,s,t) = x(u,v,w,z) : u = aq + bs ∧ v = ar + bt ∧ w = cq + ds ∧ z = cr + dt}. Here as well as in the above examples, any group element w ¯ ∈ hXi is equivalent (w.r.t. R) to an appropriate single generator x ∈ X. This is different for the following alternative, far less obvious algebraic presentation: Example 5.46 (Weil Presentation of SL2 (R)). For each b ∈ R, write 1 b 0 1 U (b) := , V := , S(a) := V · U ( a1 ) · V · U (a) · V · U ( a1 ) ∈ SL2 (R) . 0 1 −1 0 Let X = {xU (b) : b ∈ R} ∪ {XV }. Also let R denote the union of the following four families of relations (which are easy but tedious to state formally as subsets of hXi): SL1: “ U (·) is an additive homomorphism”; SL2: “ S(·) is a multiplicative homomorphism”; SL3: “ V 2 = S(−1)”; SL4: “ S(a) · U (b) · S(1/a) = U (ba2 ) ∀a, b”. According to [Lan85], hX|Ri is isomorphic to SL2 (R) under the natural homomorphism. In all the above cases, the word problem — in Example 5.44 basically the question of whether (r, s) = (1, 0) and in Example 5.45 whether t = 0 — is decidable. We next illustrate that, in the real case, different presentations of the same group may affect the solvability of the word problem. Example 5.47. The following are presentations hX|Ri of (Q, +): a) X = xr : r ∈ Q , R = xr xs = xr+s : r, s ∈ Q . b) X = {xp,q : p, q ∈ Z, q 6= 0}, R = xp,q xa,b = x(pb+aq,qb) : p, q, a, b ∈ Z ∪ xp,q = x(np,nq) : p, q, n ∈ Z, n 6= 0 . c) Let (bi )i∈I denote an algebraic basis24 of the P Q–vector space R; w.l.o.g. 0 ∈ I and b0 = 1. Consider the linear projection P : R → Q, i ri bi 7→ r0 with ri ∈ Q. X = xt : t ∈ R , R = xt xs = xt+s : t, s ∈ R ∪ xt = xP (t) : t ∈ R . Case b) yields an algebraic presentation, a) is not even algebraically generated, but c) is. The word problem is decidable for a): e.g. by effective embedding P in (R, +); and so is it for b), although not for c): xt = 1 ⇔ P (t) = 0 but both P −1 (0) = { j∈J bj qj : 0 6∈ J ⊆ I finite, qj ∈ Q} and its complement are totally disconnected and uncountable, hence BCSS-undecidable. 24 That is, as opposed to a Banach space basis, every vector admits of a representation as a linear combination of finitely many of these (here uncountably many) base elements.


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Example 5.48. (Undecidable) real membership “t ∈ Q” is reducible to the word problem of an algebraically presented real group: Consider X = {xr : r ∈ R}, R = xnr = xr , xr+k = xk , xr xs = xs xr : r, s ∈ R, n ∈ N, k ∈ Z . Then xr = x0 ⇔ r ∈ Q; also, R ⊆ R2 ∪ R4 is decidable because Z ⊆ R is. Note that the group with the undecidable word problem given in Example 5.48 is in fact a commutative one! However, this does not establish the BCSS-hardness of the real word problem because Q is provably easier than the BCSS Halting Problem H (Section 5.3). In contrast, without the restriction to algebraically presented groups (and thus parallel to Example 5.38), it is easy to find a real group with BCSS-hard word problem: Example 5.49. Let X := {xr : r ∈ R} ] {t} ∼ = R ] {∞} and, for (r1 , . . . , rd ) ∈ R∗ , write −1 −1 w ¯(r1 ,...,rd ) := xrd · · · xr1 · t · xr1 · · · xrd · t. It is tedious but possible to confirm that hXi × R∗ 3

g¯, ~r

7→ g¯−1 · w ¯~r · g¯ ∈ hXi

is injective. Moreover the words g¯−1 · w ¯~r · g¯ are Nielsen-reduced (cf. Fact 5.54b) below) and thus generate a free normal subgroup in hXi; recall Definition 5.36b). Therefore, for r.e. relations R := {w ¯~r : ~r ∈ H} and in G := hX|Ri, it holds w ¯~r = 1 if and only25 if ~r ∈ H: a reduction of the real Halting Problem to the Word Problem of G. Notice that G’s relations R are just semi -decidable. The construction of an algebraically presented group with BCSS-complete word problem in Section 5.5.9 is the main contribution of the present chapter.

5.5.5

Reducibility to the Real Halting Problem

We first show that, parallel to Fact 5.37a), the word problem for any algebraically enumerated real group is not harder than the BCSS Halting Problem. Theorem 5.50. Let G = hX|Ri denote an algebraically enumerated real group. Then the associated word problem is BCSS semi-decidable. The proof is based on the following Lemma 5.51. For Y ⊆ R∗ , it holds: If Y is (semi-)decidable, then so is hY i. Proof. Given a string w ¯ = (y1 , . . . , yk ) ∈ Rk , consider all 2k−1 partitions of w ¯ into non-empty subwords. For each subword, decide or semi-decide whether it belongs to Y ∪ Y −1 . Accept iff, for at least one partition, all its subwords succeed. Proof (Theorem 5.50). By Definition 5.36b+c), w ¯≡1⇔w ¯ ∈ hRin , that is, if and only if ∃n ∈ N ∃¯ x1 , . . . , x ¯n ∈ hXi ∃¯ r1 , . . . , r¯n ∈ hRi :

w ¯=x ¯1 r¯1 x ¯−1 ·x ¯2 r¯2 x ¯−1 ¯n r¯n x ¯−1 . n 1 2 ··· x

(5.9)

Since both X and R were required to be semi-decidable, the same holds for hXi and hRi. Since semi-decidability is equivalent to enumerability (Fact 2.17b), this carries over to (5.9). Indeed, let f, g :⊆ R∗ → R∗ be BCSS-computable with hXi = range(f ) and hRi = range(g); then it is easy to construct (but tedious to formalize) from f and g a BCSS-computable function on R∗ ranging over all n ∈ N, all w ¯ ∈ hXi, all x ¯1 , . . . , x ¯n ∈ hXi, and all r¯1 , . . . , r¯n ∈ hRi. Compose its output with the decidable test “w ¯=x ¯1 r¯1 x ¯−1 · · · x ¯n r¯n x ¯−1 ¯ This constitutes n ?” and, if successful, return w. 1 ∗ a function on R with range exactly hW in . 25 This

−1 crucial direction would fail for instance for w ¯(r1 ,...,rd ) := x−1 rd · · · xr1 · t · xr1 · xrd !

124


5.5.6

Basics from Group Theory and Their Presentations

This subsection briefly recalls some constructions from group theory and their properties which will be used intensively later on. For a more detailed exposition as well as proofs of the cited results, we refer to the two textbooks [LySs77, Rotm95]. Our notational emphasis for each construction and claim lies on the particular group presentation under consideration—for two reasons: First and as opposed to the discrete case23 , different presentations of the same group may severely affect its effectivity properties (Example 5.47). Second, sometimes there does not seem to be a ‘natural’ choice for a presentation (Remark 5.53, Footnote 26). No assumptions (e.g. effectivity) are made here concerning the set of generators or concerning the relations presenting a group. To start with and just for the record, let us briefly extend the standard notions of a subgroup and a homomorphism to the setting of presented groups: Definition 5.52. A subgroup U of the presented group G = hX|Ri is a tuple (V, S) with V ⊆ hXi and S = R ∩ hV i. This will be denoted by U = hV |RV i or, more relaxed, U = hV |Ri. A realization of a homomorphism ψ : G → H between presented groups G = hX|Ri and H = hY |Si is a mapping ψ 0 : X → hY i whose unique extension to a homomorphism on hXi maps R-cosets to S-cosets, that is, makes Equation (5.8) commute. A realization of an isomorphism φ is a realization of φ as a homomorphism. In the above notation, hψ 0 (X) Si is a presentation of the subgroup ψ(G) of H. For an embedding ψ, G is classically isomorphic to ψ(G); Lemma 5.65 below contains a computable variation of this fact. Remark 5.53. The intersection A ∩ B of two subgroups A, B of G is again a subgroup of G. For presented sub-groups A = hU |Ri and B = hV |Ri of G = hX|Ri, however, hU ∩ V |Ri is in general not a presentation of A ∩ B. Roughly speaking, a group is free if it is generic in the sense that it satisfies no relations other than the ‘basic’ Rules (5.7). Fact 5.54 (Nielsen). Let U ⊆ hXi. a) Suppose U is finite. Then there exists a finite V ⊆ hXi such that hU ihXi = hV ihXi and V is Nielsen reduced in the sense that it satisfies for all u, v, w ∈ V ∪ V −1 : (N0) u 6= 1. (N1) If uv 6= 1, then |uv| ≥ max{|u|, |v|}. (N2) If uv 6= 1 6= vw, then |uvw| > |u| − |v| + |w| where |u| denotes the length of u ∈ hXi. b) Suppose U is Nielsen-reduced. Then u1 , . . . , un ∈ U ∪U −1 with ui ui+1 6= 1 implies |u1 · · · un | ≥ n. In particular the subgroup hU ihXi of hXi is again free and isomorphic to hU i. c) Every subgroup of hXi is free. Proof. Cf. [LySs77, Section I.2]. Definition 5.55 (Free Product). Consider two presented groups G = hX|Ri and H = hY |Si with disjoint generators X ∩ Y = ∅ — e.g. by proceeding to X 0 := X × {1}, Y 0 := Y × {2}, R0 := R × {1}, S 0 := S × {2}. The free product of G and H is the presented group

G ∗ H := X ∪Y R∪S . Similarly for the free product

*G

i∈I

i

with Gi = hXi |Ri i, i ranging over arbitrary index set I.


125

In many situations one wants to identify certain elements of a free product of groups. These are provided by two basic constructions: amalgamation and Higman-Neumann-Neumann (HNN for short) extension, see [HNN49, LySs77, Rotm95]. The intuition behind the latter is nicely illustrated, e.g., in [Rotm95, Figure 11.9]. Definition 5.56 (Amalgamation). Let G = hX|Ri, H = hY |Si with X ∩Y = ∅. Let A = hV |Ri and B = hW |Si be respective subgroups and φ0 : hV i → hW i the realization of an isomorphism φ : A → B. The free product of G and H amalgamating the subgroups A and B via φ is the presented group

hG ∗ H | φ(a) = a∀a ∈ Ai := X ∪ Y | R ∪ S ∪ {φ0 (¯ v )¯ v −1 : v¯ ∈ V } . (5.10) Definition 5.57 (HNN Extension). Let G = hX|Ri, A = hV |Ri, B = hW |Ri subgroups of G, and φ0 a realization of an isomorphism between A and B. The Higman-Neumann-Neumann (HNN) extension of G relative to A, B and φ is the presented group

hG; t | ta = φ(a)t∀a ∈ Ai := X ∪ {t} | R ∪ {φ0 (¯ v )t¯ v −1 t−1 : v¯ ∈ V } . G is the base of the HNN extension, t 6∈ X is a new generator called the stable letter, and A and B are the associated subgroups of the extension. Similarly, hG; (ti )i∈I | ti a = φi (a)ti ∀a ∈ Ai ∀i ∈ Ii denotes the HNN extension with respect to a family of isomorphisms φi : Ai → Bi and subgroups Ai , Bi ⊆ G, i ∈ I. Both HNN extensions and free products with amalgamation admit of simple and intuitive characterizations for a word to be equivalent to 1 in the resulting group. These results are connected to some very famous names in group theory. Proofs can be found, e.g., in [LySs77, Chapter IV] or [Rotm95, Chapter 11]. Fact 5.58 (Higman-Neumann-Neumann). Let G∗ := hG; t|ta = φ(a)t∀a ∈ Ai be an HNN extension of G. Then identity g 7→ g is an embedding of G in G∗ . Fact 5.59 (Britton’s Lemma). Let G∗ := hG; t|ta = φ(a)t∀a ∈ Ai be an HNN extension of G. Consider a sequence (g0 , t1 , g1 , . . . , tn , gn ) with n ∈ N, gi ∈ G, i ∈ {−1, 1}. If it contains no consecutive subsequence (t−1 , gi , t) with gi ∈ A or (t, gj , t−1 ) with gj ∈ B, then it holds g0 · t1 · g1 · · · tn · gn 6= 1 in G∗ . Fact 5.60 (Normal Form). Let P := hG ∗ H|φ(a) = a∀Ai denote a free product with amalgamation. Consider c1 , . . . , cn ∈ G ∗ H, n ∈ N, such that – Each ci is either in G or in H; – Consecutive ci , ci+1 come from different factors; – If n > 1, then no ci is in A or B; – If n = 1, then c1 6= 1. Then c1 · · · cn 6= 1 in P .

5.5.7

First Effectivity Considerations

Regarding finitely generated groups, the cardinalities of the sets of generators (that is their ranks) behave additively under free products [LySs77, Corollary §IV.1.9]. Consequently, they can straightforwardly be bounded under both HNN extensions and free products with amalgamation. Similarly, for real groups, we have easy control over the dimension N of the set of generators according to Definition 5.41:

126


Observation 5.61. For groups Gi = hXi |Ri i with Xi ⊆ RN for all i ∈ I ⊆ R, the free product [

[ Gi = (X × {i}) (R × {i})

*

i∈I

i∈I

i∈I

is of dimension at most N + 1. In the countable case I ⊆ N, the dimension can even be made to not grow at all: by means of a bi-computable bijection R × N → R like (x, n) 7→ hbxc, ni + (x − bxc). Similar properties hold for free products with amalgamation and for HNN extensions. Moreover, free products, HNN extensions, and amalgamations of algebraically generated/enumerated/ presented groups are, under reasonable presumptions, again algebraically generated/enumerated/ presented: Lemma 5.62. a) Let Gi = hXi |Ri i for all i ∈ I ⊆ N. If I is finite and each Gi algebraically generated/enumerated/presented, then so is i∈I Gi . The same applies to I = N, provided that Gi is algebraically generated/enumerated/presented uniformly in i.

*

b) Let G = hX|Ri and consider the HNN extension G∗ := hG; (ti )i∈I | ti a = φi (a)ti ∀a ∈ Ai ∀i ∈ Ii with respect to a family of isomorphisms φi : Ai → Bi between subgroups Ai = hVi |Ri, Bi = hWi |Ri for Vi , Wi ⊆ hXi, i ∈ I. Suppose that I is finite, each Gi is algebraically enumerated/presented, Vi ⊆ R∗ is semi/decidable, and finally each φi is effective as a homomorphism; then G∗ is algebraically enumerated/presented as well. The same applies to I = N, provided that the Vi are uniformly semi-/decidable and effectivity of the φi holds uniformly. c) Let G = hX|Ri and H = hY |Si; let A = hV |Ri ⊆ G and B = hW |Si ⊆ H be subgroups with V ⊆ hXi, W ⊆ hY i, V ⊆ R∗ semi-/decidable, and φ : A → B an isomorphism and effective homomorphism. Then their free product with amalgamation (5.10) is algebraically enumerated/presented whenever G and H are. Remark 5.63. Uniform (semi-)decidability of a family Vi ⊆ R∗ of course means that every Vi is (semi-)decidable not only by a BCSS-machine Mi , but all Vi by one common machine M; similarly for uniform computability of a family of mappings. By virtue of (the proof of ) [Cuck92, Theorem 2.4], a both necessary and sufficient condition for such uniformity is that the real constants employed by the Mi can be chosen to all belong to one common finite field extension Q(c1 , . . . , ck ) over the rationals. Recall (Observation 5.40) that a homomorphism between finitely generated groups is automatically effective and, if injective, has decidable range and effective inverse. For real groups however, in order to make sense of the prerequisites in Lemma 5.62b+c), we explicitly have to specify the following Definition 5.64. An homomorphism ψ : hX|Ri → hY |Si of presented real groups is called an effective homomorphism if it admits of a BCSS-computable realization ψ 0 : X → hY i in the sense of Definition 5.52. For ψ to be called an effective embedding, it must not only be an effective homomorphism and injective; but ψ 0 is also required to be injective and have decidable image ψ 0 (X) plus a BCSScomputable inverse χ0 : ψ 0 (X) ⊆ hY i → X. Effective embeddings arise in Lemmas 5.65 and 5.68c+e+f). For an injective effective homomorphism φ as in Lemma 5.62c), on the other hand, a realization need not be injective; for instance, φ0 might map two equivalent (w.r.t. the relations R) yet distinct elements to the same image word. S Proof (Lemma 5.62). a) If Xi is decidable for each i ∈ I, I finite, then so is i∈I (Xi × {i}); similarly for S semi-decidable/decidable Ri . Uniform (semi-)decidability of each Xi means exactly that i∈N (Xi × {i}) is (semi-)decidable.


127

b) The set of generators of the HNN extension is decidable as in a). The additional relations {φ0i (¯ v )ti v¯−1 t−1 : v¯ ∈ Vi } are semi-/decidable since, due to the presumption, Vi is and i φ0i : hVi i → hWi i is computable. Uniformity enters as in a). c) Similarly. Lemma 5.65. Let ψ : G = hX|Ri → hY |Si = K denote an effective embedding. a) There is an effective embedding χ : ψ(G) → G (i.e. we have an effective isomorphism). b) If V ⊆ hXi is decidable, then the restriction ψ|H to H = hV |Ri ⊆ G is again an effective embedding. c) If G is algebraically generated and K algebraically presented then ψ(G) is algebraically presented as well. Proof. a) Let ψ 0 : X → hY i denote the effective realization of ψ with inverse χ0 according to Definition 5.64. The unique extension of ψ 0 to a homomorphism has image ψ 0 (hXi) = hψ 0 (X)i. In a similar way as with Lemma 5.51, we can decide, given w ¯ ∈ hY i, whether w ¯ ∈ ψ 0 (hXi). Moreover, if so, we obtain a partition w ¯ = (¯ v1 , . . . , v¯` ) with v¯i ∈ ψ 0 (X). Then calculating xi := χ0 (¯ vi ) ∈ X yields a computable extension of χ0 to a homomorphism 0 on ψ (hXi) which satisfies injectivity, has decidable image and ψ 0 as inverse. Moreover χ0 maps S-cosets to R-cosets: Take v¯1 , v¯2 ∈ ψ 0 (hXi) with v¯1 /S = v¯2 /S; then u ¯i := χ0 (¯ vi ) have 0 0 v¯i = ψ (¯ ui ) and thus, since ψ makes Equation (5.8) commute according to the presumption, v¯1 /S = ψ(¯ u1 /R) = ψ(¯ u2 /R) = v¯2 /S; now injectivity of ψ implies u ¯1 /R = u ¯2 /R. b) The range ψ 0 (V ) of the restriction ψ 0 |V coincides with χ0−1 (V ) ∩ hψ 0 (X)i. The first term is decidable since χ0 is computable and V decidable; the second term is decidable by Definition 5.64 and Lemma 5.51. c) becomes clear by staring at ψ(G) = hψ 0 (X)|Si.

5.5.8

Benign Embeddings

The requirement in Lemma 5.62b+c) that the subgroup(s) A be recursively enumerable or even decidable, is of course central but unfortunately violated in many cases. For instance, a subgroup of a finitely presented group in general need not even be finitely generated: Consider for instance the commutator [G, G] := h{uvu−1 v −1 : u, v ∈ G}i of the free group G = h{a, b}i and compare the Remark on p.177 of [LySs77]. Similarly, the algebraically presented real group (R, +) has a subgroup (Example 5.47a) which is not algebraically generated. Nevertheless, both can obviously be effectively embedded into a, respectively, finitely presented and an algebraically presented group. This suggests the notion of benign subgroups, in the classical case (below, Item a) introduced in [Higm61]. Recall that there, the effectivity of an embedding drops off automatically. Definition 5.66. a) Let X be finite, V ⊆ hXi. The subgroup A = hV |Ri of G = hX|Ri is (classically) benign in G if the HNN extension hX; t | ta = at∀a ∈ Ai can be embedded in some finitely presented group K = hY |Si. b) Let X ⊆ R∗ , V ⊆ hXi. The subgroup A = hV |Ri of G = hX|Ri is effectively benign in G if the HNN extension hG; t | ta = at∀a ∈ Ai admits of effective embedding in some algebraically presented group K = hY |Si. c) Let I ⊆ N. A family (Ai )i∈I of subgroups of G is uniformly effectively benign in G if, in the sense of Remark 5.63, there are groups Ki uniformly algebraically presented and uniformly effective embeddings φi : hG; ti |ti ai = ai ti ∀ai ∈ Ai i → Ki . The benefit of benignity is revealed in the following

128


Remark 5.67. In the notation of Definition 5.66b), if A is effectively benign in G then the word problem for A is reducible to that for K: Fact 5.58. Moreover in this case, the membership problem for A in G—that is the question of whether a given x ¯ ∈ hXi is equivalent (w.r.t. R) to an element of A—is also reducible to the word problem for K: According to Fact 5.59, a := x ¯/R satisfies t · a · t−1 · a−1 = 1 ⇔ a ∈ A. We now collect some fundamental properties that will be used frequently later on. They extend corresponding results from the finite framework. Specifically, Lemma 5.68b) generalizes [LySs77, Lemma §IV.7.7(i)] and Claims d+e) generalize [LySs77, Lemma §IV.7.7(ii)]. Lemma 5.68. a) Let A = hV |Ri ⊆ H = hW |Ri ⊆ G = hX|Ri denote a chain of (sub-) groups with V ⊆ hW i and W ⊆ hXi. If W is decidable and A effectively benign in G, then it is also effectively benign in H. b) If G = hX|Ri is algebraically presented and subgroup A = hV |Ri has decidable generators V ⊆ hXi, then A is effectively benign in G. c) If A is effectively benign in G and φ : G → H an effective embedding, then φ(A) is effectively benign in φ(G). d) Let A and B be effectively benign in algebraically presented G. Then A ∩ B admits of a presentation effectively benign in G. e) Let A, B, G as in d); then hA ∪ BiG admits of a presentation26 effectively benign in G. S f ) Let (Ai )i∈I be uniformly effectively benign in G (Definition 5.66c). Then h i∈I Ai i admits of a presentation effectively benign in G. The above claims hold uniformly in that the effective embeddings do not introduce new real constants. Proof. a) Let ψ be an effectively realizable embedding of the HNN extension hX; t|ta = φ(a)t∀a ∈ Ai in some algebraically presented K = hY |Si. Since W ∪ {t} is decidable, Lemma 5.65b) ensures the restriction of ψ to yield an effective embedding of the HNN extension hW ; t|ta = φ(a)t∀a ∈ Ai in K. b) The identity being an effectively realizable embedding (X is decidable, now apply Lemma 5.65b), it suffices to observe that the HNN extension K

:=

hG; t | at = ta∀a ∈ Ai

=

hX; t | R ∪ {¯ v t = t¯ v ∀¯ v ∈ V }i

is algebraically presented itself. Indeed, X, R, and the additional relations parametrized by V are decidable by presumption. c) The presented HNN extension under consideration, hφ0 (X); s | φ0 (¯ v )s = sφ0 (¯ v )∀¯ v ∈Vi ,

(5.11)

is the image under φ of hG; t|at = ta∀a ∈ Ai by extending φ0 (t) := s. Due to the presumption, the latter HNN extension embeds in some (finite-dim.) algebraically presented K via some effective ψ. According to Lemma 5.65a), φ admits an effective inverse. Hence the composition ψ ◦ φ−1 constitutes the desired effective embedding of (5.11) in K. d) By assumption there exist two algebraically presented groups K = hY |Si and L = hZ|T i together with realizations φ0 : X ∪ {r} → hY i, ψ 0 : X ∪ {r} → hZi of effective embeddings φ : GA := hG; r|ar = ra∀a ∈ Ai = hX; r | R ∪ { v¯r = r¯ v :¯ v ∈ V }i → K = hY |Si ψ : GB := hG; r|br = rb∀b ∈ Bi = hX; r | R ∪ {wr ¯ = rw: ¯w ¯ ∈ W }i → L = hZ|T i . 26 Possibly

different from hV ∪ W |Ri


129

We shall realize an embedding χ of the HNN extension GC := hG; r|cr = rc∀c ∈ Ci in an algebraically presented group for the presentation27 C := {w ¯ ∈ hW i : w/R ¯ ∈ A} R for A ∩ B. Observe that φ(G) = hφ0 (X)|Si and ψ(G) = hψ 0 (X)|T i are subgroups of K and L, respectively, and isomorphic due to Fact 5.58 with isomorphism27 φ ◦ ψ −1 : ψ(G) → φ(G) realized by φ0 ◦ ψ 0−1 according to Lemma 5.65. Definition 5.56 is thus applicable and we are entitled to consider the free group with amalgamation

P := K ∗ L φ ψ −1 (`) = `∀` ∈ ψ(G) (5.12)

0 0−1 0 z ) = z¯ : z¯ ∈ ψ (X)} . = Y ∪ Z S ∪ T ∪ {φ ψ (¯ P is algebraically presented because of Lemma 5.62c). Moreover, φ(G) = ψ(G) in P according to (5.12). Also, s := φ0 (r) commutes exactly with φ(A) and t := ψ 0 (r) exactly with ψ(B), so s · t commutes exactly with φ(A) ∩ ψ(B). Therefore, χ0 : X ∪ {r} → hY ∪ Zi, x 7→ ψ 0 (x), r 7→ s · t respects cosets in the sense of Equation (5.8) and thus realizes an embedding χ : hG; r|cr = rc∀c ∈ Ci → P as desired. e) With notations as in d), it holds ψ(hA ∪ BiG ) = φ(hA ∪ BiG )

=

hφ(A) ∪ φ(B) P = hφ(A) ∪ ψ(B) P

=

hφ(r · G · r−1 ) ∪ ψ(r · G · r−1 )iP ∩ φ(G) ;

the first line because φ and ψ are injective homomorphisms coinciding on G; the second because A and only A commutes with r in GA due to Britton’s Lemma (Fact 5.59), similarly for B in GB . Now φ(G) is algebraically presented due to Lemma 5.65c) and thus effectively benign in P by Claim b). Similarly, hφ(r · G · r−1 ) ∪ ψ(r · G · r−1 )iP has decidable generators and is thus effectively benign in P as well. Claim d) now ensures the effective benignity of φ(hA ∪ Bi) in P ; and therefore also in φ(G) ⊆ P according to Claim a) combined with Lemma 5.65c). Claim c) combined with Lemma 5.65a) finally yields the effective benignity of hA ∪ Bi in G. f) Let (φ0i )i∈I denote the uniformly computable realizations of embeddings φi : Gi := hG; r|ar = ra∀a ∈ Ai i → Ki . Fix j ∈ I. Similar to Equation 5.12) and the proof of e), we have

D[ E φj Ai G = φi r · G · r−1 ∩ φj G ,

*

i∈I

P :=

i∈I

D

*

i∈I

P

E Ki φi φ−1 (`) = ` ∀` ∈ φ (G) ∀i ∈ I j j

S where (by uniformity, see Lemma 5.62) P and φj (G) are algebraically presented, and i∈I φi (r· G · r−1 ) P has decidable generators of bounded dimension, compare Observation 5.61.

5.5.9

Reduction from the Real Halting Problem

With the preparations and tools from the previous sections, it is now easy to proceed to the main result of this chapter: the construction of an algebraically presented group to whose word problem the real Halting Problem can be reduced to. Let H ⊆ R∗ denote the real Halting Problem, semi-decided by some (constant-free) universal BCSS Machine M. Denote by n 7→ γn an effective enumeration of all computational paths of M, An ⊆ H ∩ Rd(n) the set of inputs accepted at path γn . Definition 5.69. Let X := {xr : r ∈ R} ] {kn : n ∈ N} ] {s}, G := hXi, and U := hk¯n−1 · w~r · kn : n ∈ N, ~r ∈ An i

where

−1 w ¯r1 ,...,rd := x−1 rd · · · xr1 · s · xr1 · · · xrd .

Finally let V := hU ; (kn )i ∩ hs; xr : r ∈ Ri. 27 Notice

the arbitrarily broken symmetry between the groups/embeddings (A, φ) and (B, ψ) involved.

130 Lemma 5.70.

CHAPTER 5. BCSS HYPERCOMPUTATION a) U is decidable.

b) U and V are effectively benign in G. c) The words kn−1 ·w~r ·kn are Nielsen-reduced (and thus freely generate U by virtue of Fact 5.54b). d) It holds V = hw ¯~r : ∃n ∈ N : ~r ∈ An i. S Proof. a) Observe that n∈N ({n}×An ) ⊆ R∗ is decidable: From kn , compute the description γn of the computational path of M corresponding to (and thus a semi-algebraic description of) An . Knowing n, it is simply a matter of evaluating the path (i.e., the system of polynomial inequalities) in order to decide membership of a given ~r to An . b) Apply Lemma 5.68b+d). c) Compare [LySs77, p.223]. ¯~r ·kn )·kn−1 belongs to hU ; (kn )i and to hs; xr : r ∈ Ri; d) For n ∈ N and ~r ∈ An , w ¯~r = kn ·(kn−1 · w hence the inclusion “V ⊇ . . .” follows. Conversely by c), a word in hU ; (kn )i devoid of kn can arise only from kn−1 · w ¯~r · kn ∈ hU i after elimination of kn−1 and kn ; thus “V ⊆ . . .” holds as well. Corollary 5.71. Let K denote the algebraically presented group which the HNN extension hG; t|tv = vt∀v ∈ V i effectively embeds in (Lemma 5.70b) by some effective embedding ψ. Then, according S to Fact 5.59 and Lemma 5.70d), v¯ := ψ t · w ¯~r · t−1 · w ¯~r−1 equals 1 in K iff ~r ∈ H = n An . The simplicity of this construction of an SAP group with BSS-complete word problem may not be as surprising when viewed in the discrete setting, yielding a decidably yet infinitely presented group with Turing-complete word problem: Example 5.72. Let H|n := {e ∈ N : TM no.e terminates after at most n steps}. Let G := hx, y, si, U := hy n · xe · s · x−e · y −n : n ∈ N, e ∈ H|n i, V := hU ; yi ∩ hx, si. Then the HNN extension of G relative to V is embeddable in a group K = hY |Si with Turingdecidable (although not necessarily finite) Y, S.

Chapter 6

Concluding Comparison Real Hypercomputation combines real computability with (discrete) hypercomputation in order to study models of real number computation beyond the Church–Turing Hypothesis. A particular focus lies on (extensions of) the two major relevant notions: Recursive Analysis (Chapter 4) and BCSS machines (Chapter 5). Discrete hypercomputers can usually be characterized in terms of undecidable oracles; their iteration gives rise to the Arithmetical Hierarchy of subsets of integers (Section 3.1.2). Oracleaccess extends real models, too, and gives rise to a taxonomy of certain subsets of reals according to their degree of undecidability: the Arithmetical Hierarchy for BCSS machines (Section 5.4) and, in the case of Recursive Analysis, the effective Borel Hierarchy (Section 4.1). Here, one also has an Arithmetical Hierarchy of single reals (Section 4.2).

6.1

Discrete versus Real Hypercomputation

In the discrete realm, every integer function becomes computable relative to some appropriate oracle. This is different in the real case: a discontinuous function as simple as the sign remains uncomputable in Recursive Analysis relative to any oracle (Corollary 4.21); and so does square root and exponentiation in the BCSS model (Theorem 5.11). This leads to a second difference between the theories of discrete and of real hypercomputers: the latter naturally gives rise to non-oracle extensions, as for instance • in the BCSS case: enhancing the set of operational primitives +, −, ×, ÷, < (Section 5.1); • for Recursive Analysis: weakened notions of function evaluation (Section 4.3.1) or, equivalently, effective measurability rather than effective continuity (Section 4.3.4). Another non-oracle extension brings us to the third major difference between real and discrete hypercomputation: for the latter, nondeterminism can be (reasonably required to be bounded and thus) simulated deterministically. Over the reals, on the other hand, nondeterminism is (inherently unbounded and) of surprising computational power strictly exceeding the (finite) Arithmetical Hierarchy; recall Sections 4.5 and 5.4.2. Finally the fourth difference: Not surprising in view of its origin (Section 1.1), arguments in discrete Computability Theory can typically be described as ‘logical ’ (classic example: diagonalization) and combinatorial. On the other hand, proofs in computability, and even more in hypercomputation, on reals (have to) additionally take into account topological and algebraic properties of R (recall Section 2.2), leading to a much more diverse and rich choice of methods and tools—which brings us back to the fifth difference: Establishing a discrete problem P as undecidable generally proceeds by reduction from the Halting problem H whose undecidability in turn follows by diagonalization. In particular, such a proof implies P to at least as hard as H; and it requires considerable efford to construct a (say, 131

132

CHAPTER 6. CONCLUDING COMPARISON

semi-decidable yet) undecidable problem strictly easier than H (Section 3.1.4). Over the reals, on the other hand, the methods and tools available in addition to diagonalization show problems undecidable without implying reducibility from the Halting problem (Section 5.3); and indeed in the BCSS setting it becomes a nontrivial task to devise a ‘natural’ problem equivalent to H.

6.2

Superrecursive Analysis versus BCSS Hypercomputers

The two major models of real computability focus on complementary aspects of practical computation on R (Section 2.8) and, thus not surprisingly, are incomparable; recall Observation 2.64. One might think that Part b) of the latter can be removed by turning to BCSS hypercomputers in the sense of Section 5.1; however, this is not the case because it holds Fact 6.1. To every finite (or even countably infinite but uniform) family of real operations (fn ), computable in the sense of Recursive Analysis, there exists a (ρ → ρ)–computable real function which is uncomputable to a BCSS machine over (R, +, −, ×, ÷, 0” performed by M are semi-decidable (Observation 2.48) ~ ) is (ρk → ρ)–computable relative to O; and equality “fu (~y ) = 0” because fu ∈ Q(c1 , . . . , cD ; Y can even be decided according to b+c). e) The set (of encodings as in (b) of elements ~y ∈) E k is recursively enumerable relative to O. We acknowledge that (the proofs of) Claims b+c+d) resemble and motivated (those of) Theorem 2.55.

6.3.2

Continuous BCSS-Computable Functions

We now extend Fact 6.5 from the semi-decidability of Euclidean subsets to real function computability. The former requires restriction to open sets, the latter to continuous functions. Theorem 6.8. Let f :⊆ Rk → R be continuous and computable by a BCSS machine M with real constants c1 , . . . , cD . Then f is (ρk → ρ)–computable relative to hc1 , . . . , cD i0 . Recall that Definition 2.42 does not require (ρk → ρ)–computation of f to diverge on inputs ~x 6∈ dom(f ); so while dom(f ) under the above hypothesis automatically satisfies BCSS semidecidability, additional presumptions about that of its complement (like in Fact 6.5) are dispensable here. 28 Miller

pointed out [Mill04] that continuous real functions need not have Turing degrees.

6.4. ROBUST QUASI-STRONGLY δ–Q–ANALYTIC FUNCTIONS

135

Proof. A Turing machine can easily unroll the computations of M into a tree as described in Section 2.3.2 and enumerate its leaves v together with symbolic descriptions (in the form of systems of polynomial (in)equalities over F := Q(c1 , . . . , cD )) of the corresponding sets Xv and the quolynomials gv computed therein. Given (~qn )n with |~x − ~qn | ≤ 2−n and give m ∈ N, we want to find some n ∈ N and (to output) some y ∈ Q such that it holds: for all leaves v and for all ~z ∈ Xv ∩ B(~qn , 2−n ) algebraic over F :

gv (~z) ∈ B(y, 2−m ).

(6.1)

Now oracle access to O := hc1 , . . . , cD i permits enumeration, in the sense of Claim 6.7b), of all ~z ∈ E k ∩ Xv ∩ B(~qn , 2−n ) and ρ–evaluation of gv on them. Therefore “gv (~z) ∈ B(y, 2−m )” is co-r.e. in ~z relative to O (a minor generalization of Observation 2.48, see for instance [16, Lemma 4.1c]); and Π1 [O] satisfies invariance under (relatively) effective universal quantification with respect to v and to ~z, compare Lemma 3.11b). Therefore, relative to the jump O0 , a Turing machine can decide (6.1) and effectively search for n and y satisfying this equation. By continuity of f and by density of E k ∩ S in S := Xv ∩ B(~qn , 2−n ) (Claim 6.7a), any such (n, y) found will satisfy gv (~z) ∈ B(y, 2−m ) for all ~z ∈ B(~qn , 2−n ) and thus be a valid 2−m – approximation to f (~x). Conversely, since f is continuous, there exists for any ~x ∈ dom(f ) and m ∈ N some n ∈ N and y ∈ Q such that, for all ~z ∈ B(~qn , 2−n )∩dom(f ), it holds f (~z) ∈ B(y, 2−m ); and in particular for all restrictions of f |Xv = gv to Xv ∩ B(~qn , 2−n ): hence the above search will eventually succeed. Granting the additional jump in Theorem 6.8 seems reasonable in view of Example 6.4, since the domains of BCSS-computable functions are exactly the BCSS semi-decidable sets. In fact, one jump is in general necessary even for total functions: Example 6.9. There exists a continuous function f : R → R computable by a constant-free BCSS machine which is not (ρ → ρ)–computable. Proof. Let h : N → N denote a recursive non-repeating enumeration of all terminating Turing machines, i.e. such that H = h[N] ∈ Σ1 \ ∆1 . Let ϕ : R → R denote a computable piecewise linear ‘hat’ function vanishing outside [0, 1] and having height maxx ϕ(x) = ϕ( 21 ) = 1 and consider the non-overlapping superposition of shifts of such hats, the m-th one scaled by am := 2−h(m) ; refer to Equation (4.10) and Figure 4.9 on page 85. Again, f is a continuous function on R vanishing for x ≤ 0 and for x ≥ 1; but, since (am )m converges by construction to 0 only noneffectively, f does not admit effective evaluation at x = 0—formally: it has no recursive modulus of continuity. In contrast, a constant-free BCSS machine can compute x 7→ f (x): first use the equality test to detect the cases x ≥ 1 and x ≤ 0 and handle them by output of f (x) = 0; for 0 < x < 1 determine the unique m ∈ N such that x ∈ (2−m , 2−m+1 ], calculate h(m) by Turing machine simulation, and finally output 2−h(m) · ϕ(2m x − 1).

6.4

Q–Analytic Functions Robust Quasi-Strongly δ –Q

constitute a structurally interesting class because one the one hand, it contains discontinuous functions, while on the other hand it satisfies closure under composition (cmp. Lemma 6.11 below); also it strictly includes both all functions computable according to Recursive Analysis as well as all (even discontinuous) BCSS–computable29 ones; see Fact 6.12 below. This model of real hypercomputation can therefore be regarded as a synthesis of those from Sections 2.3 and 2.4.2. On the other hand, the expressive power of TTE permits this model to be incorporated in the form of another real number representation extending Example 2.58b+d), namely by replacing in Example 2.58b) the condition “|qn −x| ≤ 2−n ” with the asymptotic relaxation “|qn −x| ≤ O(2−n )”: Definition 6.10 (Asymptotic Cauchy Representation). A ρb–name for x ∈ R is some (qn )n ⊆ Q such that ∃N, C ∈ N ∀n ≥ N : |qn − x| ≤ C · 2−n . (6.2) 29 Provided

only computable real constants are employed, compare Section 2.3.4

136


We first show that this definition is uniformly equivalent to the one fixing C to 1; that is a ρb–name encodes x ∈ R in the form of a sequence of rational approximations which converge fast but with the exception of some initial segment of finite (yet unknown) length. Lemma 6.11. a) Given a rational sequence (qn )n satisfying Equation (6.2), one can compute a similar one satisfying Equation (6.2) with C = 1. b) It holds ρ ¬ ρb–computable.

ρb

¬

ρCn ; non-uniformly, a real number x is ρ–computable iff it is

c) A function f :⊆ R → R is (ρ → ρb)–computable δ–Q–analytic machine.

iff

it is computable by a quasi-strongly

d) (ρ → ρb)–computability is equivalent to (b ρ → ρb)–computability. In particular, the class of (ρ → ρb)–computable functions is closed under composition. Proof. a) Output every second element of the input, that is the sequence qñ := q2n . Indeed, ˜ := it follows from Equation (6.2) that |q2n − x| ≤ C · 2−2n ≤ 2−n holds for all n ≥ N max{N, dlog2 Ce}. b) is immediate. c) Observe that the robustness of the program π required in [ChHo99, top of p.157] amounts to the argument x ∈ R of f being accessible by rational approximations qn ∈ Q of error |qn − x| ≤ 2−n , that is, in terms of a ρ–name. The output y = f (x) on the other hand proceeds by way of two sequences (pm )m , (m )m ⊆ Q such that m → 0 and |pm − y| ≤ m hold for all sufficiently large m. By effectively proceeding to an appropriate subsequence, we can w.l.o.g. suppose m = 2−m , hence (pm ) is ρb–name of y. d) According to a), every (b ρ → ρb)–computable function is also (ρ → ρb)–computable. For the converse implication, take the Type-2 Machine M converting ρ–names for x ∈ R to ρb–names for y = f (x). Let (qn ) satisfy Equation (6.2) for some unknown N ∈ N. Now simulate M on (qn )n≥0 , implicitly supposing that it is a valid ρ–name, i.e., that N = 0. Simultaneously check the consistency of Condition (6.2), that is, verify |qn −qk | ≤ 2−n+1 ∀k ≥ n ≥ N . If (or, rather, when) the latter fails for some (k0 , n0 ), M has output only finitely (say M0 ∈ N) many pm ∈ Q. In that case, restart M on (qn )n≥1 presuming N = 1 while, again, checking whether this presumption is consistent with (6.2); but this time throw away the first M0 elements of the sequence printed by M. Continue analogously for N = 2, 3, . . .. We claim that this yields the output of a ρb–name for y. Since (qn ) is a valid ρb–name, a feasible N will eventually be found. Before that happens, the several partial runs of M have produced only finitely (say M ∈ N) many rational numbers pm ; and after that, the final simulation generates according to the presumption a valid ρb–name for y. The first M entries in this sequence (pm )m may have been exchanged by outputs of previous simulation trials; however, according to Definition 6.10, the representation ρb is immune to such finite modifications.

6.4.1

Analytic Machines vs. BCSS-Computation or Recursive Analysis

Fact 6.12.

a) Let f :⊆ Rk → R be (ρk → ρ)–computable; then it is also (b ρk → ρb)–computable.

b) Let f :⊆ Rk → R be computable by a BCSS machine with constants from ∆1 (R) only; then it is also (b ρk → ρb)–computable. Proof. a) Since ρ ρb (Lemma 6.11b), (ρ → ρ)–computability implies (ρ → ρb)–computability which in turn ensures (b ρ → ρb)–computability (Lemma 6.11d).

6.4. ROBUST QUASI-STRONGLY δ–Q–ANALYTIC FUNCTIONS

137

b) has been established in [ChHo99, Theorem 3]. Instead of repeating the (very nice) proof, we illustrate the concept of conservative branching it employs with the example below. Example 6.13. Heaviside’s Function h : R → {0, 1} is (b ρ → ρb)–computable: Given x ∈ R by means of (qn ) ⊆ Q with (6.2) and unknown N ∈ N, let pn := 0 if qn ≤ 2−n and pn := 1 otherwise. Indeed if x ≤ 0 then, for all n ≥ N , qn ≤ 2−n and thus pn = 0 = f (x). If on the other hand x > 0, x > 2−M for some M ∈ N; then for all n ≥ max{M + 1, N }, qn > 2−n so pn = 1 = f (x).

6.4.2

Analytic and Nondeterministic Machines

Quasi-strongly δ–Q–analytic machines (and thus also both major models of real number computation) in turn can be simulated by a nondeterministic Type-2 machine: Corollary 6.14. Let f :⊆ R → R be (b ρ → ρb)–computable relative to some oracle O ∈ ∆11 . Then f is nondeterministically Type-2 computable. Proof. The nondeterministic simulation can answer queries to O due to Theorem 4.72. As ρ ≡n ρb ≡n ρ0 by Corollary 4.77a) and Lemma 6.11b), the claim follows. We generalize this result from the real case to arbitrary represented spaces below:

6.4.3

Finitely Revising Type-2 Computation

In order to characterize quasi-strong δ–Q–analytic computability in terms of TTE, Definition 6.10 has weakened real representation ρ to ρb. We now introduce a generic way of turning a representation α into α b having similar properties as ρb. Recall from Section 4.4.2 the ‘jump’ (meta-)mapping α 7→ α0 = α ◦ ı0 on representations: it generalized the classical Definition 3.6 of revising computation from finite to the Type-2 setting: in both cases the requirement was that the sequence (¯ τ )m of ‘preliminary’ outputs converges to the ‘true’ one σ ¯ —with respect to the discrete topology for the case of finite strings, with respect to Cantor topology for infinite ones. As already indicated in Section 4.4.1, Definition 3.6 a slightly different notion may make sense as well: Definition 6.15. Let the finitely revising representation bı :⊆ {0, 1}ω → {0, 1}ω encode (via a computable standard pairing h · i : {0, 1}ω×ω → {0, 1}ω of ) an infinite string σ ¯ ∈ {0, 1}ω as a sequence of infinite strings (¯ τ )m stabilizing (uniformly rather than character-wise) to σ ¯ , i.e. such that ∃M ∈ N : ∀m ≥ M : τ¯m = σ ¯ . (6.3) Indeed, this can also be regarded as a generalization of Definition 3.6 because a sequence of finite strings converges iff it stabilizes. For a representation α :⊆ {0, 1}ω → A, call α ◦ bı its finitely revising jump. Here come two counterparts to Lemmas 4.51 and 4.53: Lemma 6.16. It holds a) ı bı ı0 ; an infinite string σ ¯ is (ı-) computable iff it is bı–computable. b) bı ◦ bı ≡ bı c) ı0 ◦ bı ≡ ı0 ≡ bı ◦ ı0 QN QN d) n=1 bı ≡ ı for N ∈ N. n=1 ı ◦ b 0 Q Q Q ı ¬ n∈N bı ¬ e) n∈N ı ◦ b m∈N ı ◦ ı . Q Q f ) There is a bı –computable sequence in {0, 1}ω which is not ı–computable; n∈N n∈N ı ◦ b 0 Q Q ω and there is a ı ◦ ı –computable sequence in {0, 1} lacking b ı –computability. m∈N n∈N

138


Claim b) means that, as opposed to the jump operator α 7→ α0 = α ◦ ı0 from Section 4.4.2, iterated application of α 7→ α b := α ◦ bı does not yield anything new. Proof.

a) immediate.

b) Reducibility “bı bı ◦ bı” follows from a). For the converse reduction, let (¯ υk )k denote a sequence in {0, 1}ω with υ¯k = h(¯ τm )m i for all k ≥ K ∈ N and τ¯m = σ ¯ for all m ≥ M ∈ N. ¯ for all n ≥ N := hK, M i ∈ N. It Then τ˜hk,mi := h · i−1 (¯ υk ) m ∈ {0, 1}ω satisfies τñ = σ is easy to computationally extract τñ for all n where it is defined and to otherwise define it arbitrarily. c) Similarly, consider (¯ υk )k with limk υ¯k = h(¯ τm )m i and limm τ¯m = σ ¯ . Then, by Claim 4.49, ~τk,m := h · i−1 (¯ υk ) m is (a finite string, effectively arbitrarily extendable to an infinite one) converging to τ¯m as k → ∞ for each m ∈ N—yet not necessarily uniformly in m. However if either υ¯k = υ¯K for all k ≥ K ∈ N, or if τ¯m = τ¯M = σ ¯ for all m ≥ M ∈ N, then limm ~τm,m = σ ¯ does hold. e) Let (the encoding of) (¯ υk )k be an bı–name of (¯ τm )m , that is with υ¯k = h(¯ τm )m i for all Q k ≥ K. Then ~τk,m := h · i−1 (¯ υk ) m coincides with τ¯m for all k ≥ K, hence h(~τk,m )k,m is a m (bı)– name of (¯ τm )m . This proves the first claimed reducibility; the second one follows from a) in connection with Lemma 4.53c). Failure of the converse reductions is implied by f). d) One reduction has already been shown in e) above. For the converse, suppose τ¯n,m = σ ¯n for all m ≥ Mn and in particular beyond M := max{M1 , . . . , MN }. Then h(¯ τn,m )n=1,...,N i QN stabilizes for m ≥ M and is thus the desired ı–name. n=1 ı ◦ b f) For the Halting Problem H ⊆ N, let τ¯m := 1ω if m ∈ H and τ¯m := 0ω if m 6∈ H. Semidecidability of H yields an algorithm bı–computing τ¯m from given m: For each k ∈ N enumerate the first k elements of H and output τ¯k,m := 1ω if m is among them, otherwise output τ¯k,m := 0ω . Since every m ∈ H occurs in the enumeration at some position K ∈ N, τ¯k,m = τ¯K,m all k ≥ K; whereas in case m Q for Q 6∈ H we have τ¯k,m = τ¯1,m for all k. QThus, (¯ τm )m is ı –computable indeed; were it ı–computable, it would also be nb n ı ◦b nı – computable by a) contradicting undecidability of H. This time let each τ¯m ∈ {0, 1}ω encode H independently of m by setting τm,n := 1 if n ∈ H and τm,n := 0 if n 6∈ H. Each such string is obviously ı0 –computable, Q 0 e.g. by Remark 4.50; and thus the entire (because constant) sequence (¯ τ ) is m m n ı ◦ ı –computable Q (Lemma 4.53c). But its b ı –computability would imply b ı –computability (and thus n∈N ı–computability by a) of, say, τ¯1 : contradiction. Lemma 6.17. Fix representations α :⊆ {0, 1}ω → A and β :⊆ {0, 1}ω → B. a) For any function f :⊆ A → B, (α → β ◦ bı)–computability is equivalent to (α ◦ bı → β ◦ bı)– computability. b) Every (α → β)–computable function f is also (α ◦bı → β ◦bı)–computable; even uniformly in f . c) An (α ◦ bı → β ◦ bı)–computable function need not be (α → β)–continuous. Proof. It suffices to treat the case (A, α) = (B, β) = ({0, 1}ω , ı). a) By Lemma 6.16a), every (ı → bı)–computable function is also (bı → bı)–computable. For the converse implication, take the Type-2 Machine M converting ı–names for x ∈ R to bı–names for y = f (x). Let (¯ σm ) be given with σ ¯m = σ ¯M for all m ≥ M , M ∈ N unknown. Now simulate M on σ ¯1 (implicitly supposing M = 1) and simultaneously check that σ ¯1 = σ ¯m for all m ≥ 1. If (or, rather, when) the latter turns out to fail, restart under the presumption M = 2 and so on. The check will however succeed after finitely many tries (after the ‘true’ M used in the input is reached). We thus obtain a finite sequence of output strings, that is a valid bı–name for f (¯ σ ).

6.5. OMISSIONS

139

b) The apply operator (F, σ ¯ ) → F (¯ σ ) is (η ωω × bı → bı)–computable: if τ¯m = σ ¯ for all m ≥ M , then also F (¯ τm ) = F (¯ σ ) for all m ≥ M . c) See Example 6.13.

6.4.4

Finitely Revising Real Representation

Similarly to Proposition 4.56, we have Proposition 6.18. The asymptotic Cauchy Representation ρb from Definition 6.10 is uniformly equivalent to ρ ◦ bı. Lemma 6.17a) thus generalizes Lemma 6.11d) and Lemma 6.17b) includes Fact 6.12b). Proof. ρ ◦ bı ρb: Given (¯ τm )m with τ¯m = σ ¯ for all m ≥ M , consider the following decidable property P (m, n): An initial segment of τ¯m (of length `(m, n), say) decodes to a (finite) rational sequence q1 , . . . , qn with |qi − qj | ≤ 2−i+1 for all 1 ≤ i ≤ j ≤ n; moreover, τ¯m , τ¯m+1 , . . . , τ¯m+n−1 all coincide up to the first `(m, n) symbols. Our algorithm starts with m := 1 and outputs, iteratively for n = 1, 2, . . ., qn as defined above as long as P (1, n) holds; otherwise it proceeds to m := 2 and, now continuing with n, n + 1, n + 2, . . ., appends qn to the output—possibly until P (2, n) fails; and so on. In each phase m < M , i.e. when τ¯m0 6= τ¯M for some m ≤ m0 < M , the above coincidence condition will fail for some n and then increase m. This means that after output of finitely many (say N ) elements, one arrives at phase no. M where τ¯m0 = σ ¯ for all m0 ≥ M is a valid ρ–name; therefore P (M, n) is true for all n, that is, the algorithm appends to the output an infinite rational sequence qN , qN +1 , . . . satisfying |qi − qj | ≤ 2−i+1 for all N ≤ i ≤ j. This constitutes (modulo dropping the very first element in order to change error bound 2−i+1 to 2−i ) a ρb–name as desired. ρb ρ ◦ bı: Given a rational sequence (qn )n satisfying Equation (6.2). Let m := 1 and, for n = 1, 2, . . ., output finite initial segments to identical strings τ¯1 , τ¯2 , . . . , τ¯n consisting of encodings of (q2 , q3 , . . . , qn ) as long as all 1 ≤ i ≤ j ≤ n satisfy |qi − qj | ≤ 2−i+1 . In case N = 1, this iteration will continue indefinitely and output a ρ ◦bı–name as desired (notice that, as above, we dropped the very first rational approximation in order to compensate for the factor 2 in 2−i+1 ). Otherwise extend arbitrarily the partial imprints of τ¯1 , τ¯2 , . . . , τ¯n ; simultaneously output, now for n, n+1, n+2, . . ., output initial segments to strings τ¯n , τ¯n+1 , . . . , τm encoding (qn+1 , qn+2 , . . . , qn+m ) as long as |qn+i − qn+j | ≤ 2−i for all 1 ≤ i ≤ j ≤ m, and so on until some n ≥ N is reached from when on condition |qn+i − qn+j | ≤ 2−i is always fulfilled.

6.5

Omissions

Scientific topics which may be regarded as belonging to Real Hypercomputation are numerous. The selection made for the present work is certainly a subjective one. Among the many relevant omissions I plead guilty of are • Higher recursion theory [Sack90] • Infinite-time Turing machines • Analog computers

(recall Section 3.3.1)

(recall Section 2.6.4)

• Refined hierarchies of real numbers [Zhen07] • And certainly many more.

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Own Publications

(preceeding Dissertation)

[1] M. Fischer, T. Lukovszki, M. Ziegler: ”Geometric Searching in Walkthrough Animations with Weak Spanners in Real Time”, pp.163–174 in Proc. 6th Annual European Symposium on Algorithms (ESA’98), Springer LNCS vol.1461. [2] M. Fischer, T. Lukovszki, M. Ziegler: “A Network Based Approach for Realtime Walkthrough of Massive Models”, pp.133–142 in Proc. of the 2nd Workshop on Algorithms Engineering (WAE’98). [3] M. Fischer, T. Lukovszki, M. Ziegler: “Partitioned Neighborhood Spanners of Minimal Outdegree”, in Proc. of the 11th Canadian Conference on Computational Geometry (CCCG’99). [4] A. Czumaj, C. Sohler, M. Ziegler: “Property Testing in Computational Geometry” pp.155–166 in: Proc. 8th Annual European Symposium on Algorithms (ESA’00), Springer LNCS vol.1879. [5] C. Sohler, M. Ziegler: “Computing Cut Numbers”, pp.73–79 in Proc. of the 12th Canadian Conference on Computational Geometry (CCCG’00). [6] V. Brattka, M. Ziegler: “Computing the Dimension of Linear Subspaces”, pp.450–458 in: Proc. of the 27th Annual Conference on Current Trends in Theory and Practice of Informatics (SOFSEM’2000), Springer LNCS vol.1963. [7] V. Brattka, M. Ziegler: “A Computable Spectral Theorem”, pp.378–388 in: Proc. of the 4th Computability and Complexity in Analysis (CCA’2000), Springer LNCS vol.2064. [8] M.R. Emamy-K., M. Ziegler: “New Bounds for Hypercube Slicing Numbers”, pp.155–164 in Proc. of the First International Conference on Discrete Models - Combinatorics, Computation and Geometry, DMTCS vol.AA (2001). [9] V. Brattka, M. Ziegler: “Turing Computability of (Non-)Linear Optimization”, pp.181– 184 in Proc. of the 13th Canadian Conference on Computational Geometry (CCCG’01). [10] M. Ziegler: “Computability on Regular Subsets of Euclidean Space” pp.157–181 in Mathematical Logic Quarterly (MLQ), vol.48 (2002). [11] V. Damerow, L. Finschi, M. Ziegler: “Point Location Algorithms of Minimum Size”, pp.5–9 in Proc. of the 14th Canadian Conference on Computational Geometry (CCCG’02). [12] V. Brattka, M. Ziegler: “Computability of Linear Equations”, S.95-106 in Proc. 2nd IFIP International Conference on Theoretical Computer Science (TCS2002@Montreal), Kluwer. [13] M. Ziegler: “Zur Berechenbarkeit reeller geometrischer Probleme”, Dissertation, HNI Verlagsschriftenreihe vol.115 (2002), ISBN 3-935433-24-7.

152

Own Publications

(succeeding Dissertation)

[14] M. Ziegler: “Fast Relative Approximation of Potential Fields”, pp.140–149 in Proc. 8th Workshop on Algorithms an Data Structures (WADS’03), Springer LNCS vol.2748. 55 [15] M. Ziegler: “Quasi-Optimal Arithmetic for Quaternion Polynomials”, pp.705–715 in Proc. 14th Annual International Symposium on Algorithms and Computation (ISAAC’03), Springer LNCS vol.2906. 55 [16] M. Ziegler: “Computable Operators on Regular Sets”, pp.392–404 in Mathematical Logic Quarterly vol.50 (2004). 135 ¨sken, M. Ziegler: “Fast Multipoint Evaluation of Bivariate Polynomials”, pp.544– [17] M. Nu 555 in Proc. 12th Annual European Symposium on Algorithms (ESA’04), Springer LNCS vol.3221. [18] M. Ziegler, V. Brattka: “Computability in Linear Algebra”, pp.187–211 in Theoretical Computer Science vol.326 (2004). [19] K. Volbert, C. Schindelhauer, M. Ziegler: “Spanners, Weak Spanners, and Power Spanners”; pp.805–821 in Proc. 15th Annual International Symposium on Algorithms and Computation (ISAAC’04), Springer LNCS vol.3341. [20] M. Ziegler: “Computability and Continuity on the Real Arithmetic Hierarchy”, pp.562–571 in Proc. CiE 2005: New Computational Paradigms, Springer LNCS vol.3526. 100 ¨ hler, C. Schindelhauer, M. Ziegler: “On Approximating Real-Word Halting [21] S. Ko Problems”, pp.433–455 in Proc. 15th International Symposium on Fundamentals of Computation Theory (FCT’05), Springer LNCS vol.3623. [22] K. Meer, M. Ziegler: “An Explicit Solution to Post’s Problem over the Reals” (Extended Abstract), pp.456–467 in Proc. 15th International Symposium on Fundamentals of Computation Theory (FCT’05), Springer LNCS vol.3623; full version in the Journal of Complexity vol.24 (2008). 2 [23] M. Ziegler, B. Fuchssteiner: “Nonlinear Reformulation of Heisenberg’s Dynamics”, pp.693–717 in International Journal of Theoretical Physics vol.44 (2005). 53 [24] M. Ziegler: “Computational Power of Infinite Quantum Parallelism” pp.2059–2071 in International Journal of Theoretical Physics vol.44 (2005). 2, 13, 54 [25] M. Ziegler: “Stability versus Speed in a Computable Algebraic Model”, pp.14–26 in Theoretical Computer Science vol.351 (2006). 39 [26] K. Meer, M. Ziegler: “Uncomputability Below the Real Halting Problem”, pp.368–377 in Proc. 2nd Conference on Computability in Europe (CiE’06), Springer LNCS vol.3988. 2, 112 [27] M. Ziegler: “Effectively Open Real Functions”, pp.827–849 in Journal of Complexity vol.22 (2006). 2, 43 153

154

OWN PUBLICATIONS

[28] M. Ziegler: “Real Hypercomputation and Continuity”, pp.177–206 in Theory of Computing Systems vol.41 (2007). 2 [29] M. Ziegler: “Revising Type-2 Computation and Degrees of Discontinuity”, pp.255–274 in Proc. 3rd International Conference on Computability and Complexity in Analysis (CCA’06), Electronic Notes in Theoretical Computer Science vol.167 (2007). 2, 67 [30] K. Meer, M. Ziegler: “Real Computational Universality: The Word Problem for a Class of Groups with Infinite Presentation”, pp.726–737 in Proc. 32nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2007), Springer LNCS vol.4708; journal-version submitted. 2 [31] M.R. Emamy-K., M. Ziegler: “On the Coverings of the d-Cube for d ≤ 6”, to appear in Discrete Applied Mathematics / Discrete Optimization, special issue for Proceedings of the Cologne/Twente Workshop on Graphs and Combinatorial Optimization (CTW’05). 22 [32] M. Ziegler: “(Short) Survey of Real Hypercomputation”, pp.809–824 in Proc. 3rd Conference on Computability in Europe (CiE’07), Springer LNCS vol.4497. [33] S. Le Roux, M. Ziegler: “Singular Coverings and Non-Uniform Notions of Closed Set Computability”, pp.169–185 in Proc. 4th International Conference on Computability and Complexity in Analysis (CCA’07), Electronic Notes in Theoretical Computer Science. 2 ¨rwer-Bru ¨ggemeier, M. Ziegler: “Faster Integer Calculations using Non[34] K. Lu Arithmetic Primitives”, http://arxiv.org/abs/0709.0624 (Sept.2007) [35] M. Ziegler: “(A Meta-Theory of) Physics and Computation”, accepted at Verhandlungen der Deutschen Physikalischen Gesellschaft (DPG, Spring 2008, Freiburg).

Index (α → β)–computable, see also representation, 36 (α → β)–continuous, see also representation, 36 , see representation of closed sets ρ, see representation, Cauchy ρ> , see representation, right ρb, see representation, asymptotic Cauchy ρ< , see representation, left ρCn , see representation, naive (ρ → ρ), see also (α → β), 31 supK , 20 T, see transcendental reals θ< , see representation of O Tot, see Problem, Totality

topology, 25 Analytic Machine, 40–41, 135–137 set, 19 Apple, 12, 13 apply operator, 80, 82, 86, 90 Aristotle, 60 arithmetical hierarchy, see Hierarchy, Arithmetical automaton B¨ uchi, 100 cellular, 57 finite, 100 axiom of choice, 16 Zermelo-Fraenkel ∼atic set theory, 16 Baire Category, see Theorem, Baire René–Louis, 19 space, 32, 66 BCSS Machine, 21 linear, 112 nondeterministic, 28, 29, 118 universal, see universal real machine with exponentiation, 105 with oracle, 106 Bell’s Inequality, 53 ´ Bézout, Etienne, 20 Black Hole, 13, 14, 53, 60 Borel σ–algebra, 19, 118 ´ ´ Félix Edouard Justin Emile, 19 Hierarchy, see Hierarchy, Borel Brouwer, Luitzen Egbertus Jan, 26, 60 Cantor Georg Ferdinand Ludwig Philipp, 36 set, 19, 23, 24, 34, 115 space, 19, 31, 36–38 topology, 36, 37, 89, 137 Cantor space, 90 characteristic function, see function, characteristic Church

admissible, see representation, admissible algebraic element, 16 field extension, 16, 28 independence, 18, 107 real, 16, 109 155

156 ∼’s Thesis, 42 Alonzo, 12 Church–Turing Hypothesis, 12, 13, 52–54, 59, 60, 72 extended, 12 harnessable, 13, 60 physical, 12, 13 strong, 12 co–r.e., 9, see also r.e., 34 closed, see ψ> Cohen, Paul Joseph, 16 computability non-uniformly, 35–36 computable BCSS, 22, 27 relatively, 106 α–∼, see representation (α → β)–∼ function, see (α → β) binarily ∼ real number, 31, see also representation, binary real, 31 Cauchy–∼ real number, 30, see also representation, Cauchy, 30, 31, 33, 69 left ∼ real number, see also left representation, 34 relatively, 70 Markov, see Markov computability naively ∼ real number, 30, see also representation, naive, 30, 69 non–uniformly, 31, 35, 36 real function, 33, see also effectively continuous, see also effectively evaluable, see also Weierstraß computable relatively, 72 real number, 31, see also computable, binarily, see also computable, Cauchy, see also computable, left, 36 relatively, 69 real sequence, 35, 38 relatively, 48, see also oracle, 49, 65 revisingly, 48, 89–99 uniformly, 31, 33–36, 58 upper ∼ real number, 34 Weierstraß–∼ real function, see also Weierstraß representation, 31, 33 continuous (α → β)–∼, see (α → β)–continuous effectively, see effectively continuous –δ, 38 function, see function, continuous sequentially, 38 uniformly, see uniform continuity Conway, John Horton, 57, see also Life, Game of Coulomb

INDEX Charles-Augustin de, 55 Law, see law, Coulomb’s cube Klee–Minty, 22 decidable, 9 BCSS–, 22, 28, 106 relatively, 106 semi–, see semi–decidable Dedekind cut, 34, 34, 36, 37 degree algebraic, 16, 26 field extension, 16, 28 polynomial, 25 quolynomial, 25 transcendence, 18, 28, 108 differential equation algebraic, 44 ordinary, 15, 41, 55 partial, 15, 88 Diophantine Equation, 11, see also Hilbert’s Tenth Problem Dirac, Paul, 14 domain invariance of, 26 theory, 42 effectively Borel semi-measurable, 85 Borel-measurable, 85 continuous, 33, 33, 85 evaluable, see also (ρ → ρ), 33 semi-continuous, 85, 86 semi-evaluable, 34, see also (ρ → ρ< ), 85 Einstein, Albert, 13, 14, 53 Electrodynamics, 52 Ellipsoid Method, 11 entanglement, 53 enumerable, see r.e. EPR, see paradox, Einstein–Podolski–Rosen equation Bailey–Borwein–Plouffe, 30 differential, see differential equation Diophantine, see Diophantine Equation Dirac, 14 system of linear ∼s, 22 Euclid, 11, 110 Exact Computation Paradigm, 44, 45 Fermat ∼’s Last Theorem, 10 Pierre de, 10 Feynman, Richard Phillips, 53 function

INDEX algebraic differential, 44 Borel semi-measurable, 85, 98 Borel-measurable, 85, 98 characteristic, 22, 33, 34 padded, 101 computable real, see computable real function continuous, 20, 25, 32, see also Main Theorem, 33, 35, 73, 74, 98 (α → β)–∼, see (α → β)–continuous effectively, see effectively continuous uniformly, see uniform continuity generalized, 88 graph, 22, 23 Heaviside, 32, 39, 43, 73, 79, 88, 99 injective, 25 multi-valued, 39 nondecreasing, 73, 74, 81, 98 open, 43 pairing, see pairing function partial, 10, 22, 26, 36 rational, see quolynomial (α → β)-realization, 36 semi-continuous, 20, 73, 74, 80, 81, 88, 98 surjective, 36 totally discontinuous, 26 Gauß Elimination, 11, 22, 110 Johann Carl Friedrich, see Lemma or Elimination Lemma, 17 Global Positioning System, see GPS Gödel Incompleteness Theorem, 9 index, 10, 43, 44, 88 Kurt, 9, 16 Prize, 11 Gödelization, 10, 23, 29, see also G¨ odel index Goldbach Conjecture, 10 GPS, 53 Grzegorczyk, Andrzej, 33 Halting Problem, 9, 10, 10, 12, 13, 27, 30, 47, 52–54, 56, 58, 60, 138 for C64, 11 for iMac, 11 iterated, 49, 114 numerical, 88 real, 23, 109, 114, 118, 123, 129 linear, 112 nondeterministic, 118 Heaviside, Oliver, see function, Heaviside

157 Hierarchy Analytical, 51 Arithmetical, 49, 67, 71, 89, 97, 115 BCSS, 114, 117 of real numbers, 69, 70, 102 of real representations, 70 relativized, 65 Borel, 19, 51, 67, 115, 117 effective, 65, 66, 101 Chomsky, 58, 100 Hyperarithmetical, 51, 101 Polynomial, 51 Projective, 19 Hilbert ∼’s Program, 9 ∼’s Tenth Problem, 11, 28, 119 David, 9, 60 space, 54 hypercomputer, 47 Hypothesis Church–Turing, see Church–Turing Hypothesis Continuum, 16 Riemann, see Riemann Hypothesis inequality Bell, 53 system of linear inequalities, 22 system of polynomial (in)equalities, see semi-algebraic jump, 49, 52 Kleene, Stephen Cole, 49, 52 law Coulomb’s, 55 Moore’s, 59 Newton’s, see Newton of physics, 14 of thermodynamics, see also thermodynamics Lebesgue Henri Léon, 19 Leibniz, Gottfried Wilhelm, 60 Lemma Gauß, see Gauß Lemma König, 42, 56 Limit, 48, 69 Padding, 10, 48 library Core, 45 iRRAM, 45 LEDA, 45

158 Life, Game of, 57 Limited Principle of Omniscience, 14 Lindemann ∼-Weierstraß Theorem, 18 Ferdinand von, 18 linear system of ∼ equations, 22 system of ∼ inequalities, 22 system of ∼ inequalities, 23 LPO, see Limited Principle of Omniscience machine BCSS, see BCSS Machine analog, 44 analytic, 137, see Analytic Machine hyper, see hypercomputer infinite–time, 56 nondeterministic, 26 BCSS, see BCSS Machine fair, 56 Turing, 28, 56 Type-2, 99–103, 137 random access ∼, see Random Access Machine parallel, 15 revising, 48 Turing, see Turing machine universal, see universal machine Main Theorem of Recursive Analysis, 32, see also function, continuous, 33, 37, 38, 40, 43, 55, 72, 89 Malament–Hogarth Space–Time, 53, see also Relativity, Theory of General, 56 Mandelbrot Set, 23 Markov Andrey Andreyevich Jr., 43 computability, 37, 43 Principle, 44 Maxwell, James Clerk, 52 Mechanics classical, 54 Newtonian, 13, 52, 54 Quantum, see Qantum Mechanics1 Michelson&Morley experiment, 52 name, see representation Neumann, John von, 11 Newton Gravitation, 55 Isaac, see also Mechanics, Newtonian, 55 Iteration, 23 Law of Motion, 15, 55 nondeterminism, see machine, nondeterministic

INDEX ODE, see differential equation, ordinary open function, 43 set, 25, 33 class of ∼s, 18, 37 representation of ∼s, see representation of open sets oracle, 47, see also computable, relatively ordinal, 19 Church–Kleene, 51, 67 recursive, 51, 67 Painlevé, Paul, 55 pairing function integer, 10 mixed, 23 real, 23, 25, 109 paradigm Exact Computation, 44 paradox Einstein–Podolski–Rosen, 53 Twins, 53 parallel cellular automaton, see automaton, cellular computing, 57 infinite ∼ism, 54, 57 quantum ∼ism, see Quantum parallelism random access machine, see Random Access Machine, parallel Podolski, Boris, 53 Poincaré, Jules Henri, 55 Post ∼’s Problem, see Problem, Post’s Emil Leon, 49, 52 Normal Form Theorem, see Theorem, Post PRAM, see Random Access Machine, parallel Principle Limited ∼, of Omniscience, see Limited Principle of Omniscience Markov’s, 44 Prize Gödel, 11 Millennium, 35 Problem diagonal, 10 Finiteness, 48, 50, 117 Halting, see Halting Problem Hilbert’s Tenth, see Hilbert’s Tenth Problem integer factorization, 53 Linear Programming, see linear system of inequalities

INDEX Non-emptiness, 29 Post’s, 52 linear, 112 real, 109 real decision, 22 Totality, 48, 50, 58, 67, 118 Word, see Word Problem quantifier alternating, 43, 49, 51 bounded, 50 constructive interpretation, 42, 43 elimination, 20, 28, 96, 107, 118 polynomial, 51 swapping, 35 Quantum computer, 53, 54 entanglement, see entanglement Field Theory, 13 Gravitation, 12, 60 Leap, 60 Mechanics, 13, 53, 54 parallelism, 60 system, 54 qubit, 53 quolynomial, 23, 24–27, 29, 106–108, 111, 115–117, 134, 135 r.e., 9, see also semi–decidable, 34 BCSS, 22, 123 relativized, 106–108, 117 co–, see co–r.e. open, 34, see also θ< , 65 RAM, see Random Access Machine Random Access Machine, 11, 15 parallel, 15 realization, see function realization recursive, 9, see also decidable, 93 recursively enumerable, see r.e. reducible BCSS, 111 many-one, 47, 49, 115 nondeterministic, see representation reducibility Turing, 47 relative computability, see computable,relatively Relativity, see also Einstein, Albert Theory of Special and General, 13, 14, 53, 54 representation, 36, 37 admissible, 38, 72, 92 equivalence, 36 jump, 92 finitely revising, see revising

159 of {0, 1}ω , 90 of C(R) Weierstraß, 37 of closed sets, 67 of LSC, 80, 85 of MLSC, 81 of open sets, 37, see also r.e. open of Q, 37 of R Cauchy, 37, 70 left, 37 naive, 37, 70 right, 37 on Borel’s Hierarchy, 67 reducibility, 36 nondeterministic, 101, 102 revising, 90 finitely, 137 standard, 93 Riemann Georg Friedrich Bernhard, 10 Hypothesis, 10 Rosen, Nathan, 53 semi–algebraic, 20 basic, 20, 24, 115 countably, 20, 27, 115 function, 26 over a real subfield, 20, 27, 35, 114, 116, 117 semi–decidable, 9, see also r.e. BCSS, 22, 26–29, 115, 123 relativized, 107, 108, 117 BCSS relativized, 106 Recursive Analysis, 34, see also r.e. open semi-continuous, see continuous, semi Shannon, Claude Elwood, 44 Simplex Algorithm, 22 software verification, automated, 10, 48 space Baire, see Baire space Cantor, see Cantor space Hilbert, see Hilbert space vector ∼, 14 Sundman, Karl Frithiof, 55 Suslin Mikhail Yakovievich, 19 Theorem, 19 Tarski, Alfred, 20, 28, 118 tertium non datur, 13 theorem Baire, 19, 25, 114

160 Cook’s, 51 Fermat’s Last, see Fermat’s Last Theorem Gödel’s Incompleteness, 9 Lindemann-Weierstraß, 18 Main ∼, of Recursive Analysis, see Main Theorem of Recursive Analysis Post Normal Form, 49, 116 Rice’s, 10 SMN, 10, 23, 38, 43, 44 UTM, 10, 23, 38, 43, 44 Weierstraß Approximation, 20, 37, 42, 72, 83 effective, 33, 83 Thermodynamics, 13 topology, 18, see also open set algebraic, 25 Cantor, see Cantor topology Euclidean, 18, 38 final, 38 second countable, 18, 92 transcendental, 16, 107, 109 transfinite, 19 TTE, 36, see also representation Turing Alan Mathison, 9, 12, 30, 31, 47, 55 complete, 118 degree, 47, 52 hypothesis, see Church–Turing Hypothesis machine, see anywhere nondeterministic, see machine, nondeterministic probabilistic, 12 reducible, 47 Twins’ Paradox, 53 Type-2 Theory of Effectivity, see TTE uniform computability, see computable, uniformly continuity, 35 convergence, 31, 32 non–∼ computability, see computable, non– uniformly universal computation, 13, 52, 118 Life, 57 mechanical system, 54 real machine, 23, 115, 129 Turing machine, 10, 54, 59 simulation overhead, 59 Weierstraß

INDEX –computable, see computable, Weierstraß Approximation Theorem, see Theorem, Weierstraß Karl Theodor Wilhelm, 20, 42 Lindemann-∼, see Lindemann-Weierstraß Word Problem, 11, 12, 119–130 ZFC, 16 Zuse, Konrad, 12