Edwin J Beggs ´ Fe ´lix Costa Jose John V Tucker
Physical Oracles: The Turing Machine and the Wheatstone Bridge
Abstract. Earlier, we have studied computations possible by physical systems and by algorithms combined with physical systems. In particular, we have analysed the idea of using an experiment as an oracle to an abstract computational device, such as the Turing machine. The theory of composite machines of this kind can be used to understand (a) a Turing machine receiving extra computational power from a physical process, or (b) an experimenter modelled as a Turing machine performing a test of a known physical theory T . Our earlier work was based upon experiments in Newtonian mechanics. Here we extend the scope of the theory of experimental oracles beyond Newtonian mechanics to electrical theory. First, we specify an experiment that measures resistance using a Wheatstone bridge and start to classify the computational power of this experimental oracle using non-uniform complexity classes. Secondly, we show that modelling an experimenter and experimental procedure algorithmically imposes a limit on our ability to measure resistance by the Wheatstone bridge. The connection between the algorithm and physical test is mediated by a protocol controlling each query, especially the physical time taken by the experimenter. In our studies we find that physical experiments have an exponential time protocol; this we formulate as a general conjecture. Our theory proposes that measurability in Physics is subject to laws which are co-lateral effects of the limits of computability and computational complexity. Keywords: Turing machine, physical oracle, experimental procedure, theory of measurement, Wheatstone bridge, physically measurable numbers
1.
Introduction and overview
We have developed a methodology, mathematical theory, and several applications, exploring physical aspects of computability and complexity. In a sequence of papers [11, 4, 3, 8], we have studied computations possible by physical systems and by algorithms combined with physical systems. In particular, we have started to develop a theory of Turing Machines with a physical experiment as an oracle. The theory of physical oracles has been shaped by the study of experiments in Newtonian mechanics. Typically, in
Presented by Diederik Aerts, Sonja Smets, and Jean Paul Van Bendegem; Received February 1, 2010
Studia Logica (2010) 94: 1–22
c Springer
2010
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[3], we analysed in detail an experiment, called the scatter machine experiment SME, to measure distance and we showed that this oracle boosts the computational power of Turing machines beyond the Church-Turing barrier: for example, with the experimental oracle SME, Turing machines can compute in polynomial time non-uniform complexity classes such as P/poly and BP P//log∗, depending upon assumptions about precision. In [8] we asked the new question: 1. If a physical experiment were to be completely controlled by an algorithm, what effect would the algorithm have on the physical measurements made possible by the experiment? To answer, we introduced a method that uses Turing machines to govern the conduct of physical experiments. Specifically, in [8], we specified a new experiment, called the collider machine experiment CME, that measures inertial mass and we extended our existing theory of physical oracles to analyse the protocols between this experiment and machine. We modelled an experimenter following an experimental procedure by means of an algorithm and we uncovered a new form of uncertainty principle that imposes a limit on our knowledge of the values of the physical quantities involved in physical experiments in Newtonian mechanics. We also found that the computational power of this experimental oracle was classified by different non-uniform complexity classes, namely P/log∗ and BP P//log∗, depending upon assumptions about precision. In this paper we expand the scope of the theory of physical oracles, beyond Newtonian mechanics, by analysing a classical experiment in electricity: the measurement of resistance using a Wheatstone bridge. The Wheatstone bridge experiment (WBE for short) has several different features: in addition to the laws of circuits, some theory of heat is used to specify the experiment. Given our earlier work [8], obvious questions are: 2. What is the computational power of the WBE as an oracle? 3. What are the scope and limits of measurement by the WBE when it is governed by an algorithm? Indeed, given our research programme, some bigger questions are: 4. What differences are there between computations and measurements in the electrical world of WBE and the mechanical world of CME and SME? 5. Should we expect each experiment in each physical theory to be destined to reveal special properties? We will answer Questions 1 and 2 directly with theorems. For example, Theorem 1.1. Turing machines equipped with the Wheatstone bridge experiment WBE as an oracle can compute the complexity class P/log∗ in
Physical oracles . . .
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polynomial time, assuming the resistances of the test bridges can be set with infinite precision. Theorem 1.2. There are uncountably many resistances % such that for every experimental procedure governing the WBE it is only possible to determine finitely many digits of %, even allowing arbitrary long run times for the procedure. Theorems 1.1 and 1.2 will be later discussed as Theorems 5.3 and 5.7, respectively. We will answer the last three questions with observations, arguments and conjectures: our view is that there are many common properties and that unified mathematical theories of computation and measurement that apply to most experiments in most theories is possible. In Section 2, we recall the principles shaping the theory of physical oracles from [3, 8]. In Section 3, we discuss some operational aspects of measurement from the point of view of physicists Geroch and Hartle [13] and offer a possible formalization of their informal conception of measurable number. In Section 4, we develop the experimental procedures to perform the measurements of an electrical resistance. These algorithms correspond to gedankenexperimente in Physics. In Section 5, we reflect on the idea of time of a measurement process and the fact that the computational perspective implies that not all quantities are measurable.
2.
Methodological principles for combining experiments and algorithms
In [9, 10] we proposed a methodology for analysing how physical experiments, in isolation, could be employed to compute sets and functions. It was based upon four methodological principles that emphasised the role of precisely defined physical theories. Here we summarise our methodology focussing on two further principles that concern the interaction between experiments and algorithms. 2.1.
Computing with physical systems
To explore what may be computed by experimenting with physical systems we must develop a methodology that is independent of theory of algorithms. For conceptual clarity, and mathematical precision and detail, we proposed, in [9, 10], the following four methodological principles for such an investigation:
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Principle 1. Defining a physical subtheory: Define precisely a subtheory T of a physical theory and examine experimental computation by the systems that are valid models of the subtheory T . Principle 2. Classifying computers in a physical theory: Find systems that are models of T that can through experimental computation implement specific algorithms, calculators, computers, universal computers and hyper-computers. Principle 3. Mapping the border between computer and hypercomputer in physical theory: Analyse what properties of the subtheory T are the source of computable and non-computable behaviour and seek necessary and sufficient conditions for the systems to implement precisely the algorithmically computable functions. Principle 4. Reviewing and refining the physical theory: Determine the physical relevance of the systems of interest by reviewing the truth or valid scope of the subtheory. Criticism of the system might require strengthening the subtheory T in different ways, leading to a portfolio of theories and examples. Our methodology depends upon a careful formulation of a physical theory T to lay bare all the concepts and technicalities to be found in examples and to classify their computational behaviour. From the theory T are derived the central concepts of experimental procedures and equipment. The theory T is everywhere and so we are actually studying T -computability and T -computational complexity. Our approach has been applied in a new discussion of the physical basis of the Church-Turing Thesis in Ziegler [17]. 2.2.
Computing with algorithms and physical systems
First, consider using an experiment as a component to boost the performance of an algorithm or class of algorithms. In this case, computations involve some form of protocol for exchanging data between a physical system and an algorithm. A simple general way to do this is to choose an algorithmic model and incorporate the experiment as an oracle. There are many algorithmic models but the advantage of choosing Turing machines is their rich theory of computational complexity. Suppose we wish to study the complexity of computations by Turing machines with experimental oracles. Given an input string w over the alphabet of the Turing machine, in the course of a finite computation, the machine will generate and receive a finite sequence of oracle queries and answers. Specifically, as the i-th question to the oracle, the machine generates
Physical oracles . . .
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a string that is converted into a rational number ρi and used to set an input parameter pi to the equipment. The experiment is performed and, after some delay, returns as output a rational number measurement, or qualitative observation, yi , which is converted into a string or state for the Turing machine to process. The Turing machine may pause while waiting for the oracle, running a clock to keep track of time passing. In summary: Principle 5. Combining experiments and algorithms: Use a physical system as an oracle in a model of algorithmic computation, such as Turing machines. Determine whether the subtheory T , the experimental computation, and the protocol extends the power and efficiency of the algorithmic model. Secondly, consider the nature of an experimental procedure for making a measurement. The process of calculating initial conditions and perfoming an experiment step by step and interpreting the results can be expressed as a sequence of commands and rules. We imagine a human experimenter following an experimental procedure made up of basic operations defined by a physical theory T . The analysis can be compared with Turing’s analysis of the human computer, which resulted in the Turing machine. If the T commands are coded on the tape then the Turing machine represents an abstract model of the procedure. Of course, today many experiments are fully computer controlled. Principle 6. Algorithms controlling experiments: Use a model of algorithmic computation, such as Turing machines, to control a physical system. Determine whether the subtheory T , the experimental procedure and equipment, and the protocol extends or limits the accuracy and efficiency of the physical experiment to make measurements. In [5, 8] we introduced this principle. Let us note that our experiments are idealised. The ontology of our gedankenexperimente means that there is no fundamental difference between the ideal experiments defined by physical theory and the Turing machine. Like the Turing machine, they unify the essential features of examples, and map the limit of physical reality.
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Reflections on measurement in physics
Let us consider Principle 6 in Section 2. The idea of using algorithms to completely control physical experiments seems to us to be both simple and radical, and the task of making a theory to explore its meaning and consequences an intriguing challenge. One aim would be: To create a theory about the nature of physical measurements. Certainly, the experience of thinking about using physical systems for computation is invaluable; we encounter problems of (i) specifying physical theories; and (ii) modelling experimental procedures, equipment, observers and experimenters. The subject is a potent mixture of the philosophical and technical, offering work for philosophers, logicians, computer scientists and physicists. Theoretical studies of measurement are diverse, from reflections of working physicists of the past four centuries to axiomatic analyses by logicians and philosophers. Let us consider the fascinating paper [13], by the physicists Geroch and Hartle, published in 1986 to celebrate the 75th birthday of the late John Archibald Wheeler. We will outline some of their main ideas and speculations, which we will then challenge and refine using the formal models of our theory. Geroch and Hartle start by considering the concept of measurable number in contrast to the concept of computable number: We propose, in parallel with the notion of a computable number in mathematics, that of a measurable number in a physical theory. The question of whether there exists an algorithm for implementing a theory may then be formulated more precisely as the question of whether the measurable numbers of the theory are computable.
Then they add some considerations on numbers being measurable and/or computable: We argue that the measurable numbers are in fact computable in the familiar theories of physics, but there is no reason why this need be the case in order that a theory have predictive power. Indeed, in some recent formulations of quantum gravity as a sum over histories, there are candidates for numbers that are measurable but not computable.
They introduce the notion of observer and physicist for the purpose of measuring physical variables: Regard number w as measurable if there exists a finite set of instructions for performing an experiment such that a technician, given an abundance of unprepared raw materials and an allowed error ε, is able by following those instructions to perform the experiment, yielding ultimately a rational number within ε of w.
Physical oracles . . .
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The set of instructions which Geroch and Hartle refer to, together with some memory to take account of intermediate calculations, can be interpreted as a Turing machine. The Turing machine represents the physicist, or the observer, or the experimenter or the technician. Thus, we propose the assumption: Postulate 1. The physicist is modelled by a Turing machine. The measuring process is controlled by an algorithm that runs on the machine, generating the atomic instructions to be performed at each step of the experimental procedure. This postulate says that the experimenter cannot escape the logic of following rules, which is formalised by computability theory. The logic of experimental procedures can be captured by a Turing machine. The nature of the rules or instructions making up experimental procedures is determined by some physical theory T . A point not considered in [13] is that not all measurements are possible. Assuming the physicist to be a Turing machine, then the limits of computation of the Turing machine imply limits on measurements; and, therefore, on the nature of physical experiments. The accuracy ε is to be understood as arbitrarily small. As we will see, our work makes the concept of measurable as precise as the concept of computable, which was not the intention of [13]: “Measurable” is analogous to, although of course much less precise than, “computable”. The technician is analogous to the computer, the instructions to the computer program, the “abundance of unprepared raw materials” to the infinite number of memory locations, initially blank. Indeed, one can think of the measurable numbers as those that are “computable” using an analog, rather than digital, computer.
Geroch and Hartle stress the need for a theory to specify a gedankenexperiment as follows: The notion “measurable” involves a mix of natural phenomena and the theory by which we describe those phenomena. Imagine that one had access to experiments in the physical world, but lacked any physical theory whatsoever. Then no number w could be shown to be measurable, for, to demonstrate experimentally that a given instruction set shows w measurable would require repeating the experiment an infinite number of times, for a succession of εs approaching zero. One could not even demonstrate that a given instruction set shows measurability of any number at all, for it could turn out that, as ε is made smaller, the resulting sequence of experimentally determined rationals simply fails to converge. It is only a theory that can guarantee otherwise.
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The role of theories in determining observables has long been recognised, of course. Now, how does the Turing machine communicate with Nature? We propose that this interaction is captured by the concept of an evolving physical experiment acting as an oracle. Postulate 2. The measurement process is taken to be an oracle to a Turing machine. The interaction is achieved through a protocol. After each consultation, the oracle may provide one bit of the measurement. This bit also provides information necessary to the machine to proceed with the desired computation and experimental procedure. These considerations are enough for them to “prove” theorems such as: Every computable number is measurable. Then the authors ask the following question: We now ask whether, conversely, every measurable number is computable — or, in more detail, whether current physical theories are such that their measurable numbers are computable. This question must be asked with care.
Actually, the question received a very careful answer in our [11]: the experiment SME demonstrates that there are numbers that are measurable in Newtonian dynamics but not computable. Much more can be said in answer to this question. According to a our framework, everything depends upon the physical theory chosen. For Newtonian mechanics we have shown that for some experiments quantities are always measurable (see [11]) whilst for others there are quantities that are not always measurable (see [5, 8]). Our technical results can be used to show that the task of measuring quantities in physics, let us say in polynomial time, can be classified by well known complexity classes. Principle 6 and the postulates leads to a deeper understanding of experimenters and experiments which impose a theoretical and absolute limit on the measurability of a physical quantity. Indeed, we are exploring the conjecture that Any quantity in physics may not be always measurable. The ideas of protocol and precision discussed earlier play a influential role in the model. In [15, 1], inter alia, we find early explicit statements on measurability, accuracy, and time complexity of physical quantities. For instance, Bachelard starts digressing about this fact in his Philosophy of No ([1]) and, in [2], he states that improving accuracy of an experiment increases exponentially the time needed.
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Physical oracles . . .
In the oracle model, we are not interested in the time taken to build the experimental apparatus from the unprepared raw materials a la Geroch and Hartle, nor in the time taken to tune the instruments to some precision. We are interested in the run time of the physical experiment. In [6], we developed a computational model of the axiomatic measurement theory due to Campbell, Hempel, Carnap and Suppes (see, e.g., [14]). In this theory, a measurement is a function over a set of physical objects, allowing different objects to have the same value assigned by the measurement function, and the function satisfies certain laws. We allow, at least in theory, for different objects with the same attribute.
4.
Wheatstone bridge
We consider (a variant of) the Wheatstone bridge to measure an electrical resistance. 4.1.
Theory
An electric network in equilibrium (or steady state) is characterized by the laws of the theory of electrical circuits of Kirchhoff-Helmholtz, which blend two different aspects of a network: the topological side and the physical side. We follow the axiomatic formulation due to Bunge [12]. We consider electric networks in equilibrium without self-induction in all branches of the network and without mutual inductance between any pair of branches. We will consider only ohmic resistances and capacitances. Let n name an edge of a circuit then Rn is the ohmic resistance, Cn the capacitance, en (t) is the impressed voltage, Vn (t) is the electrical potential, and in (t) represents the intensity of the electrical current at time t in edge n: 1. At every vertex of the circuit the sum of the current intensities along branches represented by the edges meeting at the given vertex equals zero. 2. For every loop in the circuit, the sum of the potentials at the vertexes of that loop vanishes. 3. For an arbitrary edge n between two vertices A and B of the circuit,
Rn in (t) +
1 Cn
Z dt in (t) + en (t) = Vn,A (t) − Vn,B (t) .
(I)
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4.2.
Equipment
In the following arrangement two copies of the unknown resistance % and two copies of the resistance ρ used for the test are needed.1
gu % i -
au
i∗
@ @ ρ i0 ? @ @ @
r
@ @
bu
i -
@ @
i − i@ ∗@ R@
@ ρ @ @
% @u
h Figure 1. A Wheatstone bridge made simple.
The two equations for the voltage are (see Figure 1) VA − VB = %i∗ + ρ(i∗ − i0 ) ,
(II)
VA − VB = ρ(i − i∗ ) + %(i − i∗ + i0 ) ,
(III)
and the equation for the left loop is %i∗ + ri0 + ρ(i∗ − i) = 0 .
(IV)
From these three equations we find the two currents i∗ and i0 , given by: i0 = i
%−ρ , % + 2r + ρ
(V)
1 To be completely general with the Wheastone bridge experiment we ought to consider four resistances (the four edges of the square of Figure 1). But then the calculations are more cumbersome, involving more complex algebraic expressions, and no result in the paper would change.
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i∗ = i
ρ+r . % + 2r + ρ
(VI)
From Equation V, we can see that when the resistance ρ approaches the value of the unknown resistance %, the current in the branch of the resistance r (GH) goes to zero. Moreover, • if, in the experimental procedure, ρ > %, then the current will flow from H to G; • if, in the experimental procedure, ρ < %, then the current will flow from G to H. In principle, this idea can be used in a binary search procedure targeted at % to find its successive dyadic approximates ρ. To have the value of the current i, which depends on ρ, we have to make a few more calculations, namely involving the resistance equivalent to the whole network of resistances: R =
VA − VB %r + 2%ρ + ρr = . i % + 2r + ρ
(VII)
The equivalent resistance, as ρ → %, approches a non-zero well defined value Rlim = %
(VIII)
We conclude that, during the bisection procedure, the value of the current i will be close to the value i =
VA − VB %
(IX)
which is a well defined value, since the diference VA − VB does not change during the procedure. It means that the current i is in a well defined interval in the overall experiment. The same applies to all parameters VA − VB , %, r: they will keep their values during the procedure. Only the current i0 will be sensitive to the convergence process as ρ → %.
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4.3.
Bisection algorithm
Now we can describe the experimental procedure in full detail. Let T : N → N be the time given for the experiment to take place as a total function of the size of the sequence of bits setting the value of the test resistance. The function T can be seen as a schedule, i.e., in each experiment, in order to read the |ρ|-th bit of the resistance %, T (n) gives the amount of time steps that the experimenter accepts to wait until restarting with new experimental initial conditions. The function T can either be a computable function or a non-computable function of its argument. Bisection(T ) — the bisection algorithm: a procedure to read the first n bits of a unknown resistance % 1. input n — required precision coded by the number of places to the right of the left leading 0; 2. ρ1 := 0, ρ2 := 1, ρ := 0 — initial values with no physical significance; note |ρ1 | = 0, |ρ2 | = 1, and |ρ| = 0; 3. place two copies of the resistance % in their places on the bridge; it is supposed that the resistance % is in the desired range; 4. while |ρ| ≤ n do (a) ρ :=
ρ1 +ρ2 2
;
(b) place two copies of the resistance ρ in their places on the bridge; (c) close the circuit (having a fixed emf E) to let some current i to flow; (d) if a current is detected from G to H in time T (|ρ|) then ρ1 := ρ; append 1; — it is known that % ∈ (ρ, ρ1 ); (e) if a current is detected from H to G in time T (|ρ|) then ρ2 := ρ; append 0; — it is known that % ∈ (ρ1 , ρ); (f) if no current is detected in time T (|ρ|) then return time out; 5. end while; 6. output dyadic rational denoted by ρ.
The algorithm is parameterised by the time schedule T given to the experimenter to test for direction of the current flow.
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Physical oracles . . .
4.4.
Timing of experiment
How can we detect the existence and direction of current i0 in the wire GH? Before considering a way to detect the direction of the current flow, let us make a calculation of the time needed to detect a current in the wire. (Do not forget that we are discussing a gedankenexperiment.) Method 1: Heat. We will use a second physical theory with laws of heat. Consider the thermal effects of the existence of a non-zero current in the branch GH. The energy Q dissipated into heat after time t is given by Q = ri20 t. If the threshold (Q > χ) to detect an amount of heat Q is χ, then the time needed is given by
texp ∝
χ(% + 2r + ρ)2 1 , ri2 (% − ρ)2
(X)
which is exponential in the size of (the binary word) ρ during the bisection procedure. We don’t expect that other methods can provide much better lower bounds on the time taken for a detection. Method 2: Dipole. But to detect if the current flows in one direction or another we need for a capacitor to measure the orientation of, let us say, an electrical dipole within its plates. If the dipole is placed in the direction of the plates of the capacitor, then it will turn around according to the electrical field, indicating if the current flows from G to H or from H to G. This process has a twofold exponential delay. First, the capacitor will be charged in exponential time until it indicates the small difference of voltage between nodes G and H; second, the rotation of p the dipole will ultimately take a time inversely proportional to the value of |% − ρ|, which translates in a time exponential in the size of (the binary word) ρ during the bisection procedure. In fact, supposing that the capacitor would be charged instantaneously, then, the deflection α of the dipole would be related with time by the equation:
α =
(VH − VG )e 2 e %−ρ t = ri t2 , 2dm 2dm % + 2r + ρ
(XI)
where d is the distance between the plates of the capacitor, m is the mass
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of the dipole, e is the electrical charge of the dipole, that is, texp ∝
2md(% + 2r + ρ) eri
1/2
1 , (|% − ρ|)1/2
(XII)
that is, time is exponential in the size of the dyadic resistance used for the test in the bisection procedure. Although the two methods give different inverse powers in Equations X and XII, they each have the form: A B ≤ texp ≤ , γ |ρ − %| |ρ − %|γ
(XIII)
where A and B are constants determined by the range of resistences used and the rational γ > 0. The two inequalities in (XIII) serve different purposes: the upper bound on the experimental time allows us to make positive statements of what the WBE can compute, and the lower bound allows us to make statements of what the WBE cannot compute. 4.5.
Precision
Consider now the precision of the experiment. When measuring the output state the situation is simple: either the current i0 flows from H to G or it flows from G to H or, up to time T (|ρ|), no current is detected. As we said, errors in observation do not arise, since it is a gedankenexperiment. There are different postulates for the precision of the experiment, and we list some in order of decreasing strength: Definition 4.1. The WBE is error free if the test resistance can be set exactly to any given dyadic rational number. The WBE is error prone with arbitrary precision if the test resistance can be set only to within a non-zero, but arbitrarily small, dyadic precision. The WBE is error prone with fixed precision if there is a value ε > 0 such that the test resistance can be set up to a given precision ε. 4.6.
Oracle
The Turing machine is connected to the WBE in the same way as it would be connected to an oracle: we replace the query state with a test state (qt ),
Physical oracles . . .
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the yes state with a less than state (ql ), and the no state with a more than state (qm ). The resulting computational device is called the analogue-digital Wheatstone machine, and we refer to the unknown resistance of an analoguedigital Wheatstone machine when meant to discuss the unknown resistance of the corresponding WBE. In order to carry out a Wheatstone machine experiment, the analoguedigital Wheatstone machine will write a word ρ in the query tape and enter the test state. This word will either be “1”, or a binary word beginning with 0. We will use ρ indifferently to denote both a word P ρ1 . . . ρn ∈ {1} ∪ {0s : s ∈ {0, 1}∗ } and the corresponding dyadic rational ni=1 2−i+1 ρi ∈ [0, 1]. In this case, we write |ρ| to denote n, i.e., the size of ρ1 . . . ρn , and say that the analogue-digital Wheatstone machine is aiming at ρ. The Turing machine computation will then be interrupted, and the WBE will attempt to set the test resistance at ρ. After setting the resistance ρ, the WBE will start an electric current i, wait T (|ρ|) time units, and then check if the current did flow. If the current flows in the direction GH, then the Turing machine computation will be resumed in the state ql . If the current flows in the direction HG, then the Turing machine computation will be resumed in the state qm . Definition 4.2. An error free analogue-digital Wheatstone machine is a Turing machine connected to an error free WBE. In a similar way, we define an error prone analogue-digital Wheatstone machine with arbitrary precision, and an error prone analogue-digital Wheatstone machine with fixed precision. If an error prone analogue-digital Wheatstone machine, with unknown resistance % ∈ (0, 1), has test resistance with dyadic rational ρ ∈ [0, 1] Ω, then we are certain that the computation will be resumed in the state ql if ρ < %, and that it will be resumed in the state qm when ρ > %. We define the following decision criteria: Definition 4.3. Let A ⊆ Σ∗ be a set of words over Σ. We say that an error free analogue-digital Wheatstone machine M decides A if there exists a time constructible schedule T to operate the coupled WBE and an oracle % such that, for every input w ∈ Σ∗ , w is accepted if w ∈ A and rejected when w∈ / A. We say that M decides A in polynomial time, if M decides A, and there is a polynomial p such that, for every w ∈ Σ∗ , the number of steps of the computation is bounded by p(|w|). Definition 4.4. Let A ⊆ Σ∗ be a set of words over Σ. We say that an error prone analogue-digital Wheatstone machine M decides A if there exists a
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time constructible schedule T to operate the coupled WBE with a given oracle % and a number δ < 21 , such that the error probability of M for any input w is smaller than δ. We say that M decides A in polynomial time, if M decides A, and there is a polynomial p such that, for every input w ∈ Σ∗ , the number of steps in every computation of M on w is bounded by p(|w|). We can end this section with two results that encourage our curiosity in an algorithmic view of physical experiments. Looking at physical experiments in our formal setting, we find that intuitive concepts from abstract computability and complexity theories lift, extending the scope of the theoretical physics. Here we illustrate how the concept of decidability takes form in the analogue-digital WBE framework. Proposition 4.5. That the test resistance coincides with the given unknown resistance cannot be established experimentally in finite time by the WBE under any experimental procedure. Proof. In [8] we prove that the bisection method is an universal measurement method. According with the Equation X, as ρ → %, through the binary search procedure described, the time the experimenter has to wait goes to infinity, texp → +∞. If the two resistances coincide, then the experimenter will never know. As a trivial consequence of this statement we have the following theorem: Proposition 4.6. To know if the unknown resistance is a dyadic rational cannot be established experimentally in finite time by the WBE under any experimental procedure.
5. 5.1.
Reflections on oracles and measurements Computational power of the Wheatstone oracle
In a previous paper (see in [8], but also in [7] for a short account) we discussed the computational power of a Newtonian mechanical device, the collider (CME). By reading information on a physical quantity by an experimental procedure, we can combine the computational power of a Turing machine with an advice function. Consider a Turing machine determining whether to accept or reject words from an alphabet. In addition to its normal processes for input w, it is allowed to access an advice function f , which is an element of an advice class F.
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Definition 5.1. Let B be a class of sets and F a class of functions. The non-uniform class B/F is the class of sets A for which some B ∈ B and some f ∈ F are such that, for every w, w ∈ A if and only if hw, f (|w|)i ∈ B. We will be considering two instances for the class F: poly is the class of functions with size bounded by a polynomial, i.e., poly is the class of functions f : N → Σ∗ such that, for some polynomial p, |f (n)| ∈ O(p(n)); log is the class of functions g : N → Σ∗ such that |g(n)| ∈ O(log(n)). We will also need to treat prefix non-uniform complexity classes. For these classes we may only use prefix functions, i.e., functions f such that f (n) is always a prefix of f (n + 1). The idea behind prefix non-uniform complexity classes is that the advice given for inputs of size n must also be useful to decide smaller inputs. Definition 5.2. Let B be a class of sets and F a class of functions. The prefix non-uniform class B/F∗ is the class of sets A for which some B ∈ B and some prefix function f ∈ F are such that, for every length n and input w with |w| ≤ n, w ∈ A if and only if hw, f (n)i ∈ B. As we shall see in Subsection 5.2, there are some values of resistance which are very difficult to read. However to see what the machine can compute, we code an advice function as an element of a Cantor set, replacing every 0 or 1 by a triple (for example replace every 0 in its binary form by 100 and every 1 by 010 — see [3]). Any element % of this Cantor set has the property that, for all k, n ∈ N, |% − k/2n | > 1/2n+5 . If we substitute this into (XIII), we see that there is an upper bound on the time of any experiment used by the bisection method to determine the first n places of % of, for some constant C, texp ≤ C 2γn .
(XIV)
Theorem 5.3. A Turing machine using the WBE as oracle has computational power at least P/log∗. Proof. For a set in P/log∗ we take an advice function f . This advice function is coded into the vertex position %, which is then guaranteed to be in the Cantor set discussed above. Now for an input word w we need to read the a number of places of % logarithmic in |w|. We do this using the bisection procedure, and by using XIV, this takes an amount of time which is polynomial in |w|. Then, by using this advice, we solve the problem in polynomial time.
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This result was proved for the CME in [8], but there we had γ = 1 in XIII. Here the complexity class lower bound is shown to be independent of γ, and this allows a greater number of physical experiments to be considered. In fact, it is remarkably difficult to find a physical measurement process in which the lower time bound in XIII is not valid for some γ > 0. 5.2.
Notions of measurement and the non-measurability of resistences
To actually use an experiment as an oracle we need to take into account that it may take a very large amount of time to give an answer, or even never give an answer. To get round this we have a schedule T which gives a time limit for the experiments. This would be a function of the inputs to the experiment. Definition 5.4. A resistance % is said to be measurable if there exists a schedule T such that the digits of % can be computed by the Wheatstone bridge experiment repeatedly. Otherwise, the resistance is said to be unmeasurable. The measurement is said to be feasible if T can be chosen to be time constructible. The reader is reminded that a time constructible function of n is given by the computation time of a deterministic Turing machine when given as input a word of length n. We use this because all timing can then be carried out internally by the Turing machine — we do not need external timers to complicate things. So what values of % are not measurable? The worst values of all are the dyadic rationals — rationals whose denominator is a power of 2. If the actual value of % (in binary) is 0·1, there is no experiment obeying the lower time bound of XIII (or anything similar) which will determine the first digit after the point. The problem is, given that any measurement carried out in a given time will only be accurate to a certain error, that will mean that it can’t distinguish between 0·1 and 0·011 . . . 10 (for a large number of ones). These non-measurable dyadic rationals are certainly computable. Conversely there are measurable numbers which are not computable: By using the coding trick mentioned before there are uncountably many feasibly measurable numbers, using an exponential schedule. Irrational numbers which are algebraic (i.e., roots of polynomials with integer coefficients) are both computable and measurable — this being a consequence of a well known result on approximating algebraic numbers by rationals. Finally there are non-computable numbers which are not measurable. This can be shown by
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Physical oracles . . .
constructing examples using the Busy Beaver function (see Tibor Rad´o in [16]). Definition 5.5. Let beaver : N → N be the total function defined in this way: beaver(0) = 0 by convention; beaver(n) is the maximum output for input 0 among all Turing machines with n states that halt on input 0. Proposition 5.6. There are uncountably many x ∈ [0, 1] so that, for any program P with specified waiting times, there is an n so that P cannot determine the first n binary places of x. Proof. Consider the numbers . . 0} 1 |0 .{z . . 0} 1 |0 .{z . . 0} 1 |0 .{z . . 0} 1 . . . x = 0·1 |0 .{z . . 0} 1 |0 .{z a1
a2
a3
a4
(XV)
a5
where each ak denotes the number of repeated zeros. We take a choice of the following for each k in XV: ak =
beaver(k) beaver(k) + 1
Because of the choice, there are uncountably many such x. For details of the proof, see [8]. Notice that all the numbers are non-computable. Theorem 5.7. For uncountably many values of the unknown resistance %, for any experimental procedure it is only possible to determine finitely many binary digits of %, even given arbitrarily long run times of the program. 5.3.
Physical oracles in general
As the list of experiments used as physical oracles grows longer, certain behaviours become clearer. In Equation XIII, the lower bound for the experimental time of the WBE can be considered as less controversial than the upper bound. In our fragment of physical theory, we have a somewhat idealised world, in which we do not have to consider, for example, that the wires are made of atoms. But it is likely that trying to make the theory more ‘realistic’ would increase the experimental time. In other words, the lower bound of XIII is likely to remain, though adding ‘realism’ might cast doubt on the upper bound.
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In another paper [3], on the scatter machine, we have aparently an upper bound which is not of the form in XIII — in fact we could take a constant upper bound. In a paper in preparation, we will argue that introducing more ‘realistic’ assumptions in the theory used there not only removes this constant upper bound, but institutes a lower bound of the form of XIII. We shall now make a conjecture about the behaviour of Turing machines using physical oracles, for which we are gathering more evidence. It is based on the idea that the lower bound to the experimental time similar to that in XIII is a universal feature of ‘realistic’ theories. This lower bound in turn limits the rate that the Turing machine can ‘extract’ information from the physical oracle, and so limits any computational power which may be added to the Turing machine as a result of being connected to the physical oracle. Just what is a ‘realistic’ theory as opposed to another fragment of physical theory which does not obey the lower bound of XIII? We do not have any unifying principle of physics to appeal to here, our argument is based on examples, on the fact that every experiment that we have thought up to get round this (from Optics to Thermodynamics) has obvious shortcomings as a model of reality, and that trying to repair this by incorporating a believeable measurement process leads back to the lower bound. So this may be taken as a challenge: go and find a model without this lower bound which does not have ‘obvious shortcomings’ (with apologies for reintroducing quotation marks). We will now try to make more precise the computational content of the conjecture. Definition 5.8. A number ζ over a physical theory T is said to be measurable if there exists a Turing machine M with physical oracle O(T , ζ) which, running over unbounded time, writes on the query tape an infinite sequence of rational approximations to the binary expansion of ζ and that effectively converges to ζ. Let ζ denote the unknown value to be measured and {ρi }i∈N be the sequence of words queried by the Turing machine. Now we consider the sequence ρi uζ of maximal common initial segments of ρi and ζ. Now, we make a conjecture stating: Conjecture 5.9. For all physical theories T , for all realistic physical measurements of ζT , the natural physical time texp is at least exponential in the size of ρ u ζ, where ρ is a query of the experimenter during the bisection procedure. This Conjecture was tested by designing simple experiments in a variety
Physical oracles . . .
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of fields in Physics, namely: measuring mass with the collision experiment or a scale balance; measuring a resistance in electric circuit theory using the Wheatstone bridge apparatus; measuring the latitude of a geographic station using Foucault’s pendulum; measuring the mass of a elementary particle by the sattering method in atomic physics; measuring Brewster’s angle in optics; the measurement of the temperature of a fluid. We have more examples, and, for each quantity, a variety of experiments may be described. So far the conclusion is the same: the time needed to fix the n-th bit of a value is at least exponential in n. Acknowledgements. Edwin Beggs, Jos´e F´elix Costa and John V. Tucker would like to thank EPSRC for their support under grant EP/C525361/1. The research of Jos´e F´elix Costa is also supported by FEDER and FCT Plurianual 2007. References [1] Bachelard, Gaston, La Philosophy du Non. Pour une philosophie du nouvel esprit scientifique, Presses Universitaires de France, 1940. [2] Bachelard, Gaston, The New Scientific Spirit, Beacon Press, 1985. ´ Fe ´lix Costa, Bruno Loff, and John V. Tucker, ‘Compu[3] Beggs, Edwin, Jose tational complexity with experiments as oracles’, Proceedings of the Royal Society, Series A (Mathematical, Physical and Engineering Sciences), 464 (2008), 2098, 2777– 2801. ´ Fe ´lix Costa, Bruno Loff, and John V. Tucker, ‘On the [4] Beggs, Edwin, Jose complexity of measurement in classical physics’, in Manindra Agrawal, Dingzhu Du, Zhenhua Duan, and Angsheng Li, (eds.), Theory and Applications of Models of Computation (TAMC 2008), vol. 4978 of Lecture Notes in Computer Science, Springer, 2008, pp. 20–30. ´ Fe ´lix Costa, and John V. Tucker, ‘Quanta in classical [5] Beggs, Edwin, Jose mechanics: uncertainty in space, time, energy’, (2008). Accepted for presentation in Studia Logica International Conference on Logic and the Foundations of Physics: space, time and quanta (Trends in Logic VI), Belgium, Brussels, 11-12 December 2008. ´ Fe ´lix Costa, and John V. Tucker, ‘Computational Models [6] Beggs, Edwin, Jose of Measurement and Hempels Axiomatization’, in Arturo Carsetti, (ed.), Causality, Meaningful Complexity and Knowledge Construction, vol. 46 of Theory and Decision Library A, Springer, 2009, pp. 155–184. ´ Fe ´lix Costa, and John V. Tucker, ‘Physical experiments [7] Beggs, Edwin, Jose as oracles’, Bulletin of the European Association for Theoretical Computer Science, 97 (2009), 137–151. An invited paper for the “Natural Computing Column”. ´ Fe ´lix Costa, and John V. Tucker, ‘Limits to measurement [8] Beggs, Edwin, Jose in experiments governed by algorithms’, (2010), 33 pages. Submitted.
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[9] Beggs, Edwin, and John V. Tucker, ‘Embedding infinitely parallel computation in Newtonian kinematics’, Applied Mathematics and Computation, 178 (2006), 1, 25–43. [10] Beggs, Edwin, and John V. Tucker, ‘Can Newtonian, bounded in space, time, mass and energy compute all functions?’, Theoretical Computer Science, 371 (2007), 1, 4–19. [11] Beggs, Edwin, and John V. Tucker, ‘Experimental computation of real numbers by Newtonian machines’, Proceedings of the Royal Society, Series A (Mathematical, Physical and Engineering Sciences), 463 (2007), 2082, 1541–1561. [12] Bunge, Mario, Philosophy of Physics, Synthese Library, D. Reidel Publishing Company, 1973. [13] Geroch, Robert, and James B. Hartle, ‘Computability and Physical Theories’, Foundations of Physics, 16 (1986), 6, 533–550. [14] Hempel, Carl G., ‘Fundamentals of concept formation in empirical science’, International Encyclopedia of Unified Science, 2 (1952), 7. [15] Jeans, Sir James, Physics and Philosophy, Dover, 1943, 1981. ´ , Tibor, ‘On non-computable functions’, Bell System Technical Jourmal, 41 [16] Rado (1962), 3, 877–884. [17] Ziegler, Martin, ‘Physically-relativized Church-Turing hypotheses: Physical foundations of computing and complexity theory of computational physics’, Applied Mathematics and Computation, 215 (2009), 4, 1431–1447.
Edwin J Beggs & John V Tucker School of Physical Sciences Swansea University, Singleton Park, Swansea, SA2 8PP Wales, United Kingdom
[email protected],
[email protected] ´ Fe ´lix Costa Jose Department of Mathematics, Instituto Superior T´ecnico Universidade T´ecnica de Lisboa Lisboa, Portugal & Centro de Matem´ atica e Aplica¸co ˜es Fundamentais do Complexo Interdisciplinar Universidade de Lisboa
[email protected]
