Is the Halting problem effectively solvable non-algorithmically? Bhupinder Singh Anand Draft of April 16, 2012. An earlier version of this manuscript is arXived here.∗
Abstract We consider some consequences of the belief that there are classically two equivalent ways to look at the mathematical notion of proof: logical, as a finite sequence of sentences strictly obeying some axioms and inference rules; and computational, as a specific type of computation. We show that even a ‘weaker’ Arithmetical Provability thesis implies that the Church and Turing Theses are false when expressed as identities, and that the Halting Problem is effectively solvable, albeit non-algorithmically.
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Introduction
Classical theory holds that1 : (a) Every Turing-computable function F is partial recursive2 , and, if F is total3 , then F is recursive4 . (b) Every partial recursive function is Turing-computable5 . ∗ Subject
class: LO; MSC: 03B10 take Elliott Mendelson [Me64], George Boolos et al [BBJ03], and Hartley Rogers [Ro87], as representative, in the areas that they cover, of standard expositions of classical first order logic and of effective computability (in particular, of standard Peano Arithmetic and of classical Turing-computability). 2 Classically (cf. [Me64], p.120-121, p.214), a partial function F of n arguments is called partial recursive if, and only if, F can be obtained from the initial functions (zero function), projection functions, and successor function (of classical recursive function theory) by means of substitution, recursion and the classical, unrestricted, µ-operator. F is said to come from G by means of the unrestricted µ-operator, where G(x1 , . . . , xn , y) is recursive, if, and only if, F (x1 , . . . , xn ) = µy(G(x1 , . . . , xn , y) = 0), where µy(G(x1 , . . . , xn , y) = 0) is the least number k (if such exists) such that, if 0 ≤ i ≤ k, G(x1 , . . . , xn , i) exists and is not 0, and G(x1 , . . . , xn , k) = 0. We note that, classically, F may not be defined for certain n-tuples; in particular, for those n-tuples (x1 , . . . , xn ) for which there is no y such that G(x1 , . . . , xn , y) = 0. 3 We define a number-theoretic function, or relation, as total if, and only if, it is effectively computable, or effectively decidable, respectively, for any given set of natural number values assigned to its free variables. We define a number-theoretic function, or relation, as partial otherwise. We define a partial number theoretic function, or relation, as effectively computable, or decidable, respectively, if, and only if, it is effectively computable, or decidable, respectively, for any given set of values assigned to its free variables for which it is defined. 4 [Me64], p.233, Corollary 5.13. 5 [Me64], p.237, Corollary 5.15. 1 We
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From this, it concludes that the following, essentially unverifiable6 but refutable7 , theses (informally referred to as CT) are equivalent8 : Church’s Thesis: A number-theoretic function is effectively computable if, and only if, it is recursive9 . Turing’s Thesis: A number-theoretic function is effectively computable if, and only if, it is Turing-computable10 . In this paper we offer a plausible thesis under which the Church and Turing Theses do not hold when expressed as identities as above, and consider some consequences.
1.1
CT does not hold strictly, and needs further qualification
We note first that, even classically, the above equivalence does not hold strictly, and needs further qualification. The following argument highlights this, where F is any number-theoretic function: (i) Assume Church’s Thesis. Then: If F is Turing-computable then, by (a), it is partial recursive. If F is total, then it is both recursive11 and, by our assumption, effectively computable. If F is effectively computable then, by our assumption, it is recursive. Hence, by definition, it is partial recursive and, by (b), Turing-computable. (ii) Assume Turing’s Thesis. Then: If F is recursive, it is partial recursive and, by (b), Turing-computable. Hence, by our assumption, F is effectively computable. If F is effectively computable then, by our assumption, it is Turingcomputable. Hence, by (a), it is partial recursive and, if F is total, then it is recursive. The question arises: Can we assume that every partial recursive function is effectively decidable as total or not? Now it follows from Alan Turing’s reasoning12 that such an assumption is, in fact, inconsistent with classical theory. 6 The two theses are essentially unverifiable in classical theory since the notion of ‘effective computability’ is intuitive, and not defined formally. 7 Demonstration of a number-theoretic function that is effectively computable, but not recursive, would falsify Church’s Thesis; similarly, demonstration of a number-theoretic function that is effectively computable, but not Turing-computable, would falsify Turing’s Thesis. 8 [Me64], p.237 9 [Me64], p.227. 10 [BBJ03], p.33. 11 [Me64], p.227. 12 [Tu36].
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Reason: In his seminal paper on computable numbers13 , Turing considers the Halting problem, which can be expressed as the query: Halting problem for T 14 : Given a Turing machine T, can one effectively decide, given any instantaneous description alpha, whether or not there is a computation of T beginning with alpha? Turing then shows that the Halting problem is unsolvable by a Turing machine. Since a function is Turing-computable if, and only if, it is partially Markovcomputable15 , it is essentially unverifiable algorithmically whether, or not, a Turing machine that computes a random, n-ary, number-theoretic function will halt classically on every n-ary sequence of natural numbers (for which it is defined) as input, and not go into a non-terminating loop for some natural number input, where: Non-terminating loop: A non-terminating loop is defined as any repetition of the instantaneous tape description16 of a Turing machine during a computation. and where we note that: Effective Looping oracle: Any Turing machine T can be provided with an auxiliary infinite tape17 to effectively recognise a non-terminating looping situation; it simply records18 every instantaneous tape description at the execution of each machine instruction on the auxiliary tape, and compares the current instantaneous tape description with the record. If an instantaneous tape description is repeated, it can be meta-programmed to abort the impending non-terminating loop, and return a meta-symbol indicating self-termination.
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The Provability Thesis for PA
Now, in Theorem VI of his seminal paper19 on undecidable propositions, Kurt G¨ odel constructs a primitive recursive relation that can be shown to be true for all assignments of natural number values to its free variables. Hence, treated 13 [Tu36]. 14 [Me64],
p.256. p.233, Corollary 5.13 & p.237, Corollary 5.15. 16 “An instantaneous tape description describes the condition of the machine and the tape at a given moment. When read from left to right, the tape symbols in the description represent the symbols on the tape at the moment. The internal state qs in the description is the internal state of the machine at the moment, and the tape symbol occurring immediately to the right of qs in the tape description represents the symbol being scanned by the machine at the moment.” ([Me64], p.230, footnote 1). 17 see [Ro87], p.130. 18 It is convenient to visualise the tape of such a Turing machine as that of a two-dimensional virtual-teleprinter, which maintains a copy of every instantaneous tape description in a random-access memory during a computation. 19 [Go31]. 15 [Me64],
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as a Boolean function, it is a total function that is Turing-computable20 . However, none of its representations in first order Peano Arithmetic, PA, are PAprovable.21 More precisely, G¨ odel constructs—in an intuitionistically unobjectionable manner—an arithmetical predicate, say [R(x)]22 , that is unprovable in PA, but which is Tarskian-true under the standard interpretation of PA since, given any natural number n, the formula [R(n)] is PA-provable. Further, in his Theorem VII23 , G¨odel shows that every primitive recursive relation is instantiationally equivalent to an arithmetical relation. The result can be extended to show that every recursive relation, treated as a Boolean function, is representable in PA, and is therefore also instantiationally equivalent to an arithmetical relation24 . Taken together, the above suggest the possibility that: Arithmetical Uncomputability Thesis: There is a total arithmetical function that is not Turing-computable. Unlike the Church and Turing Theses, this thesis is classically irrefutable since, by Turing’s Halting argument, Turing-computability is essentially unverifiable. Moreover, we shall show below that: Corollary: The Arithmetical Uncomputability Thesis implies that the Church and Turing Theses, when expressed as identities, are false. More specifically, we shall replace the Arithmetical Uncomputability Thesis by another, intuitively unobjectionable, Arithmetical Provability Thesis, and use this to construct a total arithmetical relation that, treated as a Boolean function, is not Turing-computable. However we consider first the following, related, thesis: Decidability Thesis: An arithmetical relation, which holds for any given assignments of natural numbers to its free variables, is Turingdecidable as always true if, and only if, it is the standard representation of a PA-provable formula. This too is classically both unverifiable and irrefutable since, again, Turing’s Halting argument implies that we cannot assume that there is always an algorithm that will verify whether an arithmetical relation, which holds for any given assignment of natural numbers to its free variables, is Turing-decidable as always true. The Decidability Thesis is of particular interest, since it echoes some implicitly held beliefs in interpretations of computational theory. For instance, in an arXived paper25 , Cristian Calude et al hold that: 20 Since
every total recursive function is Turing-computable ([Me64], p.237, Corollary 5.15). follows from G¨ odel’s argument that, if r is the G¨ odel-number of the formula [R(x)] in 17 his Arithmetic P, then the P-formula, [R(n)], whose G¨ odel-number is Sb(r, Z(n) ), is provable for any given natural number n ([Go31], p.26). 22 We use square brackets to indicate that the expression inside them refers to a particular syntactical string of symbols that is effectively verifiable as a well-defined formula of a specified formal system. 23 [Go31], p.29. 24 [Me64], pp.131-134, Propositions 3.23 & 3.24. 25 [CCS01], v2. 21 This
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“Classically, there are two equivalent ways to look at the mathematical notion of proof: logical, as a finite sequence of sentences strictly obeying some axioms and inference rules, and computational, as a specific type of computation. Indeed, from a proof given as a sequence of sentences one can easily construct a Turing machine producing that sequence as the result of some finite computation and, conversely, given a machine computing a proof we can just print all sentences produced during the computation and arrange them into a sequence.” In other words, the authors seem to hold that Turing-computability of a ‘proof’, in the case of an arithmetical proposition, is equivalent to provability of its representation in PA. This can be expressed more precisely as the assertion (weaker than the Decidability Thesis) that: Arithmetical Provability Thesis: When computing an arithmetical relation F (x1 , . . . , xn ), treated as a Boolean function26 , a classical Turing machine T halts, and returns a value that interprets as ‘true’ on every n-ary sequence of natural numbers as input—without going into a non-terminating loop—if, and only if, [F (x1 , . . . , xn )] is a PAprovable formula.
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Effective solvability of the Halting problem
The significance of the Arithmetical Provability Thesis is seen in the following argument.27 Theorem 1 The Arithmetical Provability Thesis implies that it is always possible to determine whether a Turing machine will halt or not when computing any partial recursive function F . Proof : We assume that the partial recursive function F is obtained from the recursive function G by means of the unrestricted µ-operator; in other words, that28 : F (x1 , . . . , xn ) = µy(G(x1 , . . . , xn , y) = 0). If [H(x1 , . . . , xn , y)] expresses ¬(G(x1 , . . . , xn , y) = 0) in PA29 , we consider the PA-provability—and Tarskian-truth30 in the standard interpretation M of PA—of the arithmetical formula [H(a1 , . . . , an , y)] for a given sequence of numerals {[a1 ], . . . , [an ]} of PA, as below: 26 In other words the Turing machine computes the characteristic function C(x , . . . , x ) of n 1 F (x1 , . . . , xn ), which is defined as follows ([Me64], p.119):
C(x1 , . . . , xn ) = 0 if F (x1 , . . . , xn ) is true; C(x1 , . . . , xn ) = 1 if F (x1 , . . . , xn ) is false. 27 See also Appendix A 28 See [Me64], p.214 29 By definition the interpretation H ∗ (x , . . . , x , y) of [H(x , . . . , x , y)] in M is instantian n 1 1 tionally equivalent to ¬(G(x1 , . . . , xn , y) = 0) (cf. [Me64], p.117). 30 cf. [Me64], p.49.
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(a) Let Q1 be the meta-assertion that the PA-formula [H(a1 , . . . , an , y)] does not interpret as always true in M. Since G(a1 , . . . , an , y) is recursive, it follows that there is some finite k such that any Turing machine T1 (y) that computes G(a1 , . . . , an , y) will halt and return the value 0 for y = k. (b) Let Q2 be the meta-assertion that the PA-formula [H(a1 , . . . , an , y)] interprets as always true in M, but there is no Turing machine that computes the corresponding interpreted arithmetical function H ∗ (a1 , . . . , an , y) as a Boolean function that is always true in M. If we assume the Arithmetical Provability Thesis, then the PAformula [H(a1 , . . . , an , y)] is unprovable in PA, but always true under interpretation in M. Hence—since G(a1 , . . . , an , y) is recursive—any Turing machine T2 (y) that computes the instantiationally equivalent arithmetical function H ∗ (a1 , . . . , an , y) as a Boolean function must halt, since its auxiliary tape will return the symbol for self-termination at the first initiation of a non-terminating loop at some y = k 0 . Comment: In his seminal paper on computable numbers31 , Turing considers the Halting problem, which can be expressed as the query: Halting problem for T 32 : Given a Turing machine T, can one effectively decide, given any instantaneous description alpha, whether or not there is a computation of T beginning with alpha? Turing then shows that the Halting problem is unsolvable by a Turing machine. Since a function is Turing-computable if, and only if, it is partially Markov-computable33 , it is essentially unverifiable algorithmically whether, or not, a Turing machine that computes a random, n-ary, number-theoretic function will halt classically on every n-ary sequence of natural numbers (for which it is defined) as input, and not go into a non-terminating loop for some natural number input, where: Non-terminating loop: A non-terminating loop is defined as any repetition of the instantaneous tape description34 of a Turing machine during a computation. and where we note that: Effective Looping oracle: Any Turing machine T can be provided with an auxiliary infinite tape35 to effectively recognise a non-termin- ating looping situation; it simply records36 every instantaneous tape description at the execution of each machine instruction on the auxiliary tape, 31 [Tu36]. 32 [Me64],
p.256. p.233, Corollary 5.13 & p.237, Corollary 5.15. 34 “An instantaneous tape description describes the condition of the machine and the tape at a given moment. When read from left to right, the tape symbols in the description represent the symbols on the tape at the moment. The internal state qs in the description is the internal state of the machine at the moment, and the tape symbol occurring immediately to the right of qs in the tape description represents the symbol being scanned by the machine at the moment.” ([Me64], p.230, footnote 1). 35 see [Ro87], p.130. 36 It is convenient to visualise the tape of such a Turing machine as that of a two-dimensional virtual-teleprinter, which maintains a copy of every instantaneous tape description in a random-access memory during a computation. 33 [Me64],
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and compares the current instantaneous tape description with the record. If an instantaneous tape description is repeated, it can be meta-programmed to abort the impending non-terminating loop, and return a meta-symbol indicating self-termination.
(c) Finally, let Q3 be the meta-assertion that the PA-formula [H(a1 , . . . , an , y)] interprets as always true in M, and that any Turing machine that computes the interpreted arithmetical function H ∗ (a1 , . . . , an , y) as a Boolean function halts on every natural number input for y, and returns the value 0 (‘true’). Now, if we assume an Arithmetical Provability Thesis, then it follows that [H(a1 , . . . , an , y)] is PA-provable. Let h be the G¨ odel-number of [H(a1 , . . . , an , y)]. We consider, then, G¨ odel’s primitive recursive number-theoretic relation xBy 37 , which holds in M if, and only if, x is the G¨odel-number of a proof sequence in PA for the PA-formula whose G¨odel-number is y. It follows that there is some finite k 00 such that any Turing machine T3 (y), which computes the characteristic function of xBh, will halt and return the value 0 (‘true’) for x = k 00 . Since Q1 , Q2 and Q3 are mutually exclusive and exhaustive it follows that, when run simultaneously over the sequence 1, 2, 3, . . . of values for y, one of the parallel trio (T1 (y) // T2 (y) // T3 (y)) of Turing machines will always halt for some finite value of y. If T1 (y) halts, then a Turing machine will halt when computing the partial recursive function F . If either one of T2 (y) or T3 (y) halts, then a Turing machine will not halt when computing the partial recursive function F .2 Thus, the Halting problem is effectively solvable if we assume an Arithmetical Provability Thesis. Corollary 1 The Arithmetical Provability Thesis implies that the parallel trio of Turing machines (T1 (y) // T2 (y) // T3 (y)) is not a Turing machine. Corollary 2 The Arithmetical Provability Thesis implies that the classical ChurchTuring thesis is false.
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Is there a case for an Arithmetical Provability Thesis?
We consider in a companion paper [An12] the question of whether or not there is a case for introducing non-algorithmic effective methods into classical mathematics and for introducing constructive foundational concepts that validate the Arithmetical Provability Thesis. 37 [Go31],
p.22, definition 45.
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Appendix A
Excepting that it always calculates Lazslo Kalm´ar’s g(n) (defined below) constructively—even in the absence of a uniform procedure—within a fixed postulate system, the reasoning used in the above argument in §3 is, essentially, the same as Kalm´ ar’s argument38 , reproduced below from Selmer Bringsjord’s narrational case against Church’s Thesis39 : “First, he draws our attention to a function g that isn’t Turingcomputable, given that f is40 : g(x) = µy(f (x, y) = 0) = the least y such that f (x, y) = 0 if y exists; and 0 if there is no such y Kalm´ ar proceeds to point out that for any n in N for which a natural number y with f (n, y) = 0 exists, ‘an obvious method for the calculation of the least such y ... can be given,’ namely, calculate in succession the values f (n, 0), f (n, 1), f (n, 2), . . . (which, by hypothesis, is something a computist or TM can do) until we hit a natural number m such that f (n, m) = 0, and set y = m. On the other hand, for any natural number n for which we can prove, not in the frame of some fixed postulate system but by means of arbitrary—of course, correct—arguments that no natural number y with f (n, y) = 0 exists, we have also a method to calculate the value g(n) in a finite number of steps. Kalm´ ar goes on to argue as follows. The definition of g itself implies the tertium non datur, and from it and CT we can infer the existence of a natural number p which is such that (*) there is no natural number y such that f (p, y) = 0; and (**) this cannot be proved by any correct means. Kalm´ ar claims that (*) and (**) are very strange, and that therefore CT is at the very least implausible.”
References [BBJ03]
George S. Boolos, John P. Burgess and Richard C. Jeffrey. 2003. Computability and Logic. Cambridge University Press, Cambridge.
[Bri93]
Selmer Bringsjord. 1993. The Narrational Case Against Church’s Thesis. Easter APA meetings, Atlanta.
[CCS01]
Cristian S. Calude, Elena Calude and Solomon Marcus. 2001. Passages of Proof. Workshop, Annual Conference of the Australasian Association of Philosophy (New Zealand Division), Auckland. Archived at: http://arxiv.org/pdf/math/0305213.pdf
38 [Ka59]. 39 [Bri93] http://www.rpi.edu/ brings/SELPAP/CT/ct/ct.html. 40 Bringsjord
notes that the original proof can be found on page 741 of Kleene [Kl36].
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[Go31]
Kurt G¨ odel. 1931. On formally undecidable propositions of Principia Mathematica and related systems I. Translated by Elliott Mendelson. In M. Davis (ed.). 1965. The Undecidable. Raven Press, New York.
[Ka59]
Laszlo Kalm´ ar. 1959. An Argument Against the Plausibility of Church’s Thesis. In Heyting, A. (ed.) Constructivity in Mathematics. North-Holland, Amsterdam.
[Kl36]
Stephen Cole Kleene. 1936. General Recursive Functions of Natural Numbers. Math. Annalen vol. 112 (1936) pp.727-766.
[Me64]
Elliott Mendelson. 1964. Introduction to Mathematical Logic. Van Norstrand, Princeton.
[Ro87]
Hartley Rogers Jr. 1987. Theory of Recursive Functions and Effective Computability. MIT Press, Cambridge, Massachusetts.
[Tu36]
Alan Turing. 1936. On computable numbers, with an application to the Entscheidungsproblem. In M. Davis (ed.). 1965. The Undecidable. Raven Press, New York. Reprinted from the Proceedings of the London Mathematical Society, ser. 2. vol. 42 (1936-7), pp.230-265; corrections, Ibid, vol 43 (1937) pp. 544-546.
[An12]
Bhupinder Singh Anand. 2012. Some consequences of interpreting the associated logic of the first-order Peano Arithmetic PA finitarily. Unpublished.
Authors postal address: 32 Agarwal House, D Road, Churchgate, Mumbai - 400 020, Maharashtra, India. Email:
[email protected],
[email protected].
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