The Church-Turing Thesis in Quantum and Classical Computing∗

The Church-Turing thesis in quantum and classical computing∗

Petrus Potgieter† 31 October 2002

Abstract The Church-Turing thesis is examined in historical context and a survey is made of current claims to have surpassed its restrictions using quantum computing — thereby calling into question the basis of the accepted solutions to the Entscheidungsproblem and Hilbert’s tenth problem.

1 Formal computability

The need for a formal definition of algorithm or computation became clear to the scientific community of the early twentieth century due mainly to two problems of Hilbert: the Entscheidungsproblem—can an algorithm be found that, given a statement • in first-order logic (for example Peano arithmetic), determines whether it is true in all models of a theory; and Hilbert’s tenth problem—does there exist an algorithm which, given a Diophan- • tine equation, determines whether is has any integer solutions? The Entscheidungsproblem, or decision problem, can be traced back to Leibniz and was successfully and independently solved in the mid-1930s by Alonzo Church [Church, 1936] and Alan Turing [Turing, 1935]. Church defined an algorithm to be identical to that of a function computable by his Lambda Calculus. Turing, in turn, first identified an algorithm to be identical with a recursive function and later with a function computed by what is now known as a Turing machine1. It was later shown that the class of the recursive functions, Turing machines, and the Lambda Calculus define the same class of functions. The remarkable equivalence of these three definitions of disparate origin quite strongly supported the idea of this as an adequate definition of computability and the Entscheidungsproblem is generally considered to be solved.

∗Talk at Budepest University of Economic Sciences and Public Administration, 31 October 2002. †Department of Quantitative Management, University of South Africa, PO Box 392, Unisarand 0003, fax: +27-12-429-4898, e-mail: [email protected]. 1 The curious reader will ﬁnd Turing machine simulators on the Internet, i.a. at http://www.nmia.com/ ~soki/turing/.

1 Using the same definition of computability Yuri Matiyasevich ([Mati seviq, 1970] or [Matijasevich, 1970]) showed in 1970 (refining work by Julia Robinson, Martin Davis and Hilary Putnam—see [Matiyasevich, 1992] for a discussion of Robinson’s contri- bution) that, within the Churh-Turing framework of computation, no algorithm for determining whether a given Diophantine equation admits integer solutions. The negative answers to the Entscheidungsproblem and to Hilbert’s tenth problem are related to the Halting Problem for a Turing machine: given a program or input for a Turing machine, is it possible to determine algorithmically (i.e. using another Turing machine) whether the first Turing machine ever halts on the given program or input. The answer to this question is NO. The funcion fT which assigns to each x the value 1 if a fixed Turing machine T halts on input x and 0 otherwise is the classical example of a non- computable function for a sufficiently powerful T, specifically when T is a Universal Turing Machine (UTM) which is capable of simulating all other Turing machines.

2 The Church-Turing theses

There are many versions and interpretations of what is loosely known as the Church- Turing thesis, and it would be more accurate to speak of Church-Turing theses. The equivalence of the deﬁnitions of computability given by Church and by Turing gave rise to the Church-Turing Thesis (CTT) which, as formulated by Turing, says that Every ‘function which would naturally be regarded as computable’ can by computed by a Turing machine. (CTT) Now, certainly any recursive function or function deﬁne by lambda calculus or by any of a number of other computational schemes, including Markov algorithms, can provably be computed by a Turing machine. The problem is that the vagueness of the concept ‘function which would naturally be regarded as computable’ certainly ensures that the statement of the CTT above is not something that can ever be proven because the ‘natural’ notion of computable may change. The following quo- tations (taken directly from [Copeland, 1997] of Alan Turing will shed some light on his views of the universal machine and of computability.

A man provided with paper, pencil, and rubber, and subject to strict discipline, is in eﬀect a universal machine. (1948)

The idea behind digital computers may be explained by saying that these machines are intended to carry out any operations which could be done by a human computer. (1950)

The class of problems capable of solution by the machine [the Automatic Computing Engine (ACE)] can be defined fairly specifically. They are [a subset of] those problems which can be solved by human clerical labour, working to fixed rules, and without understanding. (1946)

2 Computers always spend just as long in writing numbers down and deciding what to do next as they do in actual multiplications, and it is just the same with ACE ... [T]he ACE will do the work of about 10,000 computers ... Computers will still be employed on small calculations ... (1947)

An absolutely crucial point here is that Turing was still thinking about a human computer which, in his time, was a calculating clerk. In the sense of Turing we can accept CTT as stated above. A somewhat stronger version of CTT, call it the Physical CTT (PCTT) would be Every function that can be physically computed, can by computed by a Turing machine. (PCTT) Here the notion ‘physically computed’, if including the result of any physical process (or of our model of any physical process), is so vague that the thesis PCTT cannot hold. For example, a Turing machine cannot effectively approximate any of the values of one-dimensional Brownian motion at rational points in time almost surely (w.r.t the Wiener measure), as pointed out by Willem Fouché[Fouché,2000]. Yet another version of the thesis is the following, as stated in [Bernstein and Vazirani, 1997]: Any ‘reasonable’ model of computation can be efficiently2 simulated on a probabilistic Turing machine. (SCTT) We shall call this version the Strong Church-Turing Thesis (SCTT).

3 Quantum computing

In 1982, Richard Feynman proposed the idea of using quantum mechanical phenomena in order to perform computations in a talk at MIT [Feynman, 1982]. He identified the immense compactness of representing information in qubits due to the superposition principle and the potential of quantum systems to be used in computation. In 1985, David Deutsch [Deutsch, 1985] a universal quantum computer (UQC) for this model, similar to the universal Turing machine (UTM), which can simulate any other quantum computer. Deutsch also showed that his UQC can compute any function computable by a Turing machine. Furthermore, any UQC can be simulated (but not effectively) by a probabilistic Turing machine (PTM) and hence on a deterministic Turing machine ([Simon, 1994] or [Bernstein and Vazirani, 1997], for example). Different models of quantum computation exist [Benioff, 1998] but we will take the model of Deutsch as canonical.

3.1 Successes of quantum computing Quantum computing oﬀers enormous possibilities for speeding up calculations and the following (see [Bennett and DiVincenzo, 1995] for a more complete, though dated, survey) milestones are worth noting. 2 An eﬃcient simulation is one whose running time is bounded by some polynomial in the running time of the simulated machine.

3 Shor’s 1994 algorithm for integer factorisation [Shor, 2000] which was imple- • mented on a 7-qubit quantum computer in 2001 by IBM, factoring the number 15 correctly. The successful construction of a quantum computer able to handle larger numbers will be devastating to the ﬁeld of public-key cryptography. Lou Grover’s 1996 algorithm [Grover, 1996] to speed up database search. • Perhaps more importantly, quantum mechanical eﬀects can be used for secure secret key exchange in cryptography and this has been done in practice over medium distance.

4 Quantum processes

A quantum computer as defined in [Deutsch, 1985] cannot compute any function that cannot also be computed by an ordinary Turing machine. However, since the quantum computer is much faster than a classical computer for several problems it would seem that the Strong Churh-Turing Thesis is in fact violated by quantum computers as illustrated in [Bernstein and Vazirani, 1997], unless P = NP ! However the classical Churh-Turing Thesis (CTT) is not affected by this argument since it places no restriction on the time required to simulate the computation on an ordinary Turing machine. In a recent publication, Calude and Pavlov [Calude and Pavlov, 2002] have described a quantum process that ‘solves’ the halting problem correctly with probability tending to 0 constructively as the time for computation tends to infinity. Their (theoretical) device therefore ‘computes’ a function which cannot be computed by a Turing machine3. However, as clearly pointed out by the authors, their device operates in an infinite dimensional Hilbert space in contrast to the canonical quantum computer of Deutsch. Although it is without doubt very interesting that a physical process can be conceived that solves the halting problem, in our view the Church-Turing thesis is not necessarily overturned by this discovery. It was pointed out in 1982 [Pour-El and Richards, 1982] already by Pour-El and Richards that computable boundary conditions for the three- dimensional wave equation exist that give rise to non-computable solutions. More recently, Fouchéhas pointed out [Fouché,2000] that for a generic one-dimensional Wiener process (one from a certain measure one set) all the values assumed by the process at recursive real numbers are non-recursive. The idea that a physical process can produce non-computable output is therefore not new at all (also see [Svozil, 1998]).

5 Other attempts to usurp CTT

Hermann Weyl in 1949 [Weyl, 1949] raised the posssibility of a feasible device performing an infinite number of steps in finite time, where we should understand the ‘steps’ to be in some sense identical except for the time taken for their execution. It can be easily seen that allowing arbitrary speed-up of the computing device, one can solve the halting problem for Turing machines in finite time, but not by a Turing machine. Call 3 A popular discussion, copied from New Scientist magazine may be found at http://www.cs.auckland. ac.nz/~cristian/smashandgrab.htm.

4 the accelerated machine a Zeno machine (ZM). Unfortunately (see [Svozil, 1998]) there exists no ZM that can solve the halting problem for ZMs. Nevertheless, an appropri- ately constructed ZM would be able to compute a function that cannot be computed by a Turing machine.

5.1 General relativity In so-called Malament-Hogarth space-times, it is possible for an observer in finite time to observe arbitrarily long calculation due to relativistic effects. This has been describe in some detail by Etesi and Németi([Etesi, 2002] and [Etesi and Németi,2002]). It has been argued in [McLaughlin, ] that the existence of Malament-Hogarth space-times is not consistent with Nelson’s internal set theory in non-standard analysis.

5.2 Biomolecular computing Adleman in the 1990s demonstrated the solution of the Hamiltonian path problem for a seven-node graph using DNA computing (see [Reif, 1998] or [Reif, 2002]). The main advantage of DNA computing appears to be the physical compactness and cost of the data storage and the main problem is that of autonomous computing, i.e. of performing more than a small number of steps without intervention from outside the computing system (by the lab assistant, for example).

5.3 Kalmár LászlóKalmárgave a counter-argument to CTT [Kalmár,1959] based on a function of Kleene,

min y > 0 ψ(x, y) = 0 , if min ψ(x, y) 0, f(x) = ( y 0 { | } , otherwise; ≤ for an appropriate recursive (Turing machine computable) function ψ. Identification of Turing machine computable with effectively computable results in an apparent con- tradiction. Kalmáralso postulated that effective computability is a pre-mathematical notion and can never be fully captured by mathematics.

6 Conclusion

Clearly a computation should be in principle be a physically realisable or (at least) theorisable process (where we customarily assume unbounded time and space resources), bu the central polemic seems to be whether any physically realisable or theorisable process should be considered a computation. Our answer is cautiously in the negative. The solution to Hilbert’s tenth problem shows that the class of recursively enumerable sets (sets that constitute the range of a function computed by a Turing machine) is exactly equivalent to the class of Diophantine sets. The Diophantine sets form such a natural class that we do not feel ready to expand the deﬁnition of computability beyond CTT. However, a radical revision in our view of space and time (see [Longo, 2002] for

5 a very non-technical overview) might in the future necessity a radical revision of our ideas about mechanical computability.

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