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The Church-Turing Thesis

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## 1. The Thesis and its History

❶For the axiom CT in constructive mathematics, see Church's thesis constructive mathematics. Since, as an informal notion, the concept of effective calculability does not have a formal definition, the thesis, although it has near-universal acceptance, cannot be formally proven.
## 2. Misunderstandings of the Thesis

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A variation of the Church—Turing thesis addresses whether an arbitrary but "reasonable" model of computation can be efficiently simulated. This is called the feasibility thesis , [50] also known as the classical complexity-theoretic Church—Turing thesis or the extended Church—Turing thesis , which is not due to Church or Turing, but rather was realized gradually in the development of complexity theory.

This thesis was originally called computational complexity-theoretic Church—Turing thesis by Ethan Bernstein and Umesh Vazirani The complexity-theoretic Church—Turing thesis, then, posits that all 'reasonable' models of computation yield the same class of problems that can be computed in polynomial time. Assuming the conjecture that probabilistic polynomial time BPP equals deterministic polynomial time P , the word 'probabilistic' is optional in the complexity-theoretic Church—Turing thesis.

A similar thesis, called the invariance thesis , was introduced by Cees F. Slot and Peter van Emde Boas. In other words, there would be efficient quantum algorithms that perform tasks that do not have efficient probabilistic algorithms.

This would not however invalidate the original Church—Turing thesis, since a quantum computer can always be simulated by a Turing machine, but it would invalidate the classical complexity-theoretic Church—Turing thesis for efficiency reasons.

Consequently, the quantum complexity-theoretic Church—Turing thesis states: Eugene Eberbach and Peter Wegner claim that the Church—Turing thesis is sometimes interpreted too broadly, stating "the broader assertion that algorithms precisely capture what can be computed is invalid". Philosophers have interpreted the Church—Turing thesis as having implications for the philosophy of mind. Jack Copeland states that it is an open empirical question whether there are actual deterministic physical processes that, in the long run, elude simulation by a Turing machine; furthermore, he states that it is an open empirical question whether any such processes are involved in the working of the human brain.

When applied to physics, the thesis has several possible meanings:. There are many other technical possibilities which fall outside or between these three categories, but these serve to illustrate the range of the concept. One can formally define functions that are not computable.

A well-known example of such a function is the Busy Beaver function. This function takes an input n and returns the largest number of symbols that a Turing machine with n states can print before halting, when run with no input. Finding an upper bound on the busy beaver function is equivalent to solving the halting problem , a problem known to be unsolvable by Turing machines.

Since the busy beaver function cannot be computed by Turing machines, the Church—Turing thesis states that this function cannot be effectively computed by any method. Several computational models allow for the computation of Church-Turing non-computable functions. These are known as hypercomputers. Mark Burgin argues that super-recursive algorithms such as inductive Turing machines disprove the Church—Turing thesis. This interpretation of the Church—Turing thesis differs from the interpretation commonly accepted in computability theory, discussed above.

The argument that super-recursive algorithms are indeed algorithms in the sense of the Church—Turing thesis has not found broad acceptance within the computability research community.

From Wikipedia, the free encyclopedia. For the axiom CT in constructive mathematics, see Church's thesis constructive mathematics. History of the Church—Turing thesis. This section relies largely or entirely upon a single source. Relevant discussion may be found on the talk page. Please help improve this article by introducing citations to additional sources. November Learn how and when to remove this template message. Merriam Webster's New Collegiate Dictionary 9th ed.

Merriam-Webster's Online Dictionary 11th ed. What is effectively calculable is computable. He calls this "Church's Thesis". Church uses the words "effective calculability" on page ff. Church in Davis Archived from the original PDF on Editor's footnote to Post Finite Combinatory Process. With respect to his proposed Gandy machine he later adds LC. Also a review of this collection: Smith, Peter July 11, Archived PDF from the original on July 27, Retrieved July 27, Bulletin of Symbolic Logic.

Archived from the original PDF on November 24, An introduction to quantum computing. Handbook of Theoretical Computer Science A. On tape versus core: Jack November 10, In Zalta, Edward N. Stanford Encyclopedia of Philosophy.

Classical and Contemporary Readings. The Blackwell guide to the philosophy of computing and information. The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics. Monographs in Computer Science. Barwise, Jon ; Keisler, H. Bernstein, E; Vazirani, U. Blass, Andreas ; Gurevich, Yuri October Monographs in computer science.

Church, Alonzo April American Journal of Mathematics. Date and year link Church, Alonzo June Journal of Symbolic Logic. Date and year link Church, Alonzo March The Calculi of Lambda-Conversion. Perspectives East and West. Davis, Martin , ed. Handbook of Philosophical Logic.

The universal Turing Machine: Kleene and Rosser lecture note-takers ; Institute for Advanced Study lecture sponsor. Ergenbnisse eines mathematishen Kolloquiums in German. Cited by Kleene Gurevich, Yuri June Gurevich, Yuri July Church, Turing, Tarski, and Others".

Kleene, Stephen Cole American Mathematical Society Transactions. Reprinted in The Undecidable , p. Kleene refined his definition of "general recursion" and proceeded in his chapter " Algorithmic theories" to posit "Thesis I" p.

The Art of Computer Programming. Kugel, Peter November Communications of the ACM. Elements of the Theory of Computation. Manna, Zohar []. Mathematical Theory of Computation. American Mathematical Society Translations. Church's Thesis After 70 Years. Rejecting the conventional view, Kripke suggests that, on the contrary, the Church-Turing thesis is susceptible to mathematical proof. Furthermore he canvasses the idea that Turing himself sketched an argument that serves to prove the thesis.

Put somewhat crudely, the latter theorem states that every valid deduction couched in the language of first-order predicate calculus with identity is provable in the calculus. The first step of the Kripke argument is his claim that error-free, human computation is itself a form of deduction:. One is given a set of instructions, and the steps in the computation are supposed to follow—follow deductively—from the instructions as given.

So a computation is just another mathematical deduction, albeit one of a very specialized form. The execution of this two-line program can be represented as a deduction:. In the case of Turing-machine programs, Turing developed a detailed logical notation for expressing all such deductions Turing In fact, the successful execution of any string of instructions can be represented deductively in this fashion—Kripke has not drawn attention to a feature special to computation.

The instructions do not need to be ones that a computer can carry out. Nachum Dershowitz and Yuri Gurevich and independently Wilfried Sieg have also argued that the Church-Turing thesis is susceptible to mathematical proof. In their Dershowitz and Gurevich offer. Dershowitz and Gurevich According to Turing, his thesis is not susceptible to mathematical proof. He did not consider either argument I or argument II to be a mathematical demonstration of his thesis: The statement is … one which one does not attempt to prove.

Propaganda is more appropriate to it than proof, for its status is something between a theorem and a definition. Are rhubarb and tomatoes vegetables or fruits?

Is coal vegetable or mineral? What about coal gas, marrow, fossilised trees, streptococci, viruses? Has the lettuce I ate at lunch yet become animal? Turing in Copeland b: This myth has passed into the philosophy of mind, theoretical psychology, cognitive science, computer science, Artificial Intelligence, Artificial Life, and elsewhere—generally to pernicious effect.

Turing showed that his very simple machine … can specify the steps required for the solution of any problem that can be solved by instructions, explicitly stated rules, or procedures. Richard Gregory writing in his Turing had proven—and this is probably his greatest contribution—that his Universal Turing machine can compute any function that any computer, with any architecture, can compute That is, it can display any systematic pattern of responses to the environment whatsoever.

These various quotations are typical of writing on the foundations of computer science and computational theories of mind. In reality Turing proved that his universal machine can compute any function that any Turing machine can compute; and he put forward, and advanced philosophical arguments in support of, the thesis that effective methods are to be identified with methods that the universal Turing machine is able to carry out.

The Church-Turing thesis is a thesis about the extent of effective methods, and therein lies its mathematical importance. Putting this another way, the thesis concerns what a human being can achieve when working by rote, with paper and pencil ignoring contingencies such as boredom, death, or insufficiency of paper.

Essentially, then, the Church-Turing thesis says that no human computer, or machine that mimics a human computer, can out-compute the universal Turing machine. This loosening of established terminology is unfortunate, since it can easily lead to misunderstandings and confusion.

Some examples from the literature of this loosening are:. If there is a well defined procedure for manipulating symbols, then a Turing machine can be designed to do the procedure.

Geroch and Hartle The behaviour of any discrete physical system evolving according to local mechanical laws is recursive. I can now state the physical version of the Church-Turing principle: Turing and Church were talking about effective methods, not finitely realizable physical systems.

A machine m will be said to be able to generate a certain function e. Mutatis mutandis for functions that, like addition, demand more than one argument. All functions that can be generated by machines working in accordance with a finite program of instructions are computable by effective methods. It is worth noting the existence in the literature of another practice with the potential to mislead the unwary. Although, unlike the terminological practices complained about above, this one is in itself perfectly acceptable.

Thus a function is said to be computable if and only if there is an effective method for obtaining its values. All functions that can be generated by machines working in accordance with a finite program of instructions are computable. Boolos and Jeffrey However, to a casual reader of the technical literature, this statement and others like it may appear to say more than they in fact do. That a function is uncomputable , in this sense, by any past, present, or future real machine, does not entail that the function in question cannot be generated by some real machine past, present, or future.

No possible computing machine can generate a function that the universal Turing machine cannot. But the question of the truth or falsity of the maximality thesis itself remains open.

Although the terminological decision, if accepted, does prevent one from describing any machine putatively falsifying the maximality thesis as computing the function that it generates. For example, statements like the following are to be found:. The stronger-weaker terminology is intended to reflect the fact that the stronger form entails the weaker, but not vice versa.

The stronger form of the maximality thesis is known to be false. Although a single example suffices to show that the thesis is false, two examples are given here.

An ETM is exactly like a standard Turing machine except that, whereas a standard Turing machine stores only a single discrete symbol on each non-blank square of its tape e. The method of storing real numbers on the tape is left unspecified in this purely logical model.

As previously explained, Turing established the existence of real numbers that cannot be computed by standard Turing machines Turing Abramson also proved that ETMs are able to generate functions not capable of being computed by any standard Turing machine.

Therefore, ETMs form counterexamples to the stronger form of the maximality thesis. Accelerating Turing machines ATMs are exactly like standard Turing machines except that their speed of operation accelerates as the computation proceeds Stewart ; Copeland a,b, a; Copeland and Shagrir This enables ATMs to generate functions that cannot be computed by any standard Turing machine.

One example of such a function is the halting function h. The ATM then proceeds to simulate the actions of the n th Turing machine. The weaker form of the maximality thesis would be falsified by the actual existence of a physical hypercomputer. Speculation stretches back over at least five decades that there may be real physical processes—and so, potentially, real machine-operations—whose behaviour conforms to functions not computable by any standard Turing machine. At the close of the 20 th century Copeland and Sylvan gave an evangelical survey of the emerging field in their To summarize the situation with respect to the weaker form of the maximality thesis: At the present time, it remains unknown whether hypercomputation is permitted or excluded by the contingencies of the actual universe.

It is, therefore, an open empirical question whether or not the weaker form of the maximality thesis is true. As previously mentioned, this convergence of analyses is generally considered very strong evidence for the Church-Turing thesis, because of the diversity of the analyses.

However, this convergence is sometimes taken to be evidence for the maximality thesis. Allen Newell, for example, cites the convergence as showing that. Yet the analyses Newell is discussing are of the concept of an effective method, not of the concept of a machine-generatable function.

The equivalence of the analyses bears only on the question of the extent of what is humanly computable, not on the question of whether the functions generatable by machines could extend beyond the functions generatable by human computers even human computers who work forever and have access to unlimited quantities of paper and pencils. The error of confusing the Church-Turing thesis properly so called with one or another form of the maximality thesis has led to some remarkable claims in the foundations of psychology.

For example, one frequently encounters the view that psychology must be capable of being expressed ultimately in terms of the Turing machine e. To one who makes this error, conceptual space will seem to contain no room for mechanical models of the mind that are not equivalent to Turing machines. Yet it is certainly possible that psychology will find the need to employ models of human cognition transcending Turing machines.

A similar confusion is found in Artificial Life. Christopher Langton, the leading pioneer of A-Life, said the following when writing about foundational matters:.

Turing proved that no such machine can be specified. However, Turing certainly did not prove that no such machine can be specified. It is also worth mentioning that, although the Halting Problem is very commonly attributed to Turing as Langton does here , Turing did not in fact formulate it. Another example is the simulation thesis. For example, the entry on Turing in the Blackwell Companion to the Philosophy of Mind contains the following claims:. Sam Guttenplan writing in his Can the operations of the brain be simulated on a digital computer?

Is there some description of the brain such that under that description you could do a computational simulation of the operations of the brain. Any process that can be given a mathematical description or that is scientifically describable or scientifically explicable can be simulated by a Turing machine.

Paul and Patricia Churchland and Philip Johnson-Laird also assert versions of the simulation thesis, with a wave towards Church and Turing by way of justification:.

Assuming, with some safety, that what the mind-brain does is computable, then it can in principle be simulated by a computer. Churchland and Churchland If you assume that [consciousness] is scientifically explicable … [and] [g]ranted that the [Church-Turing] thesis is correct, then the final dichotomy rests on … functionalism.

If you believe [functionalism] to be false … then … you hold that consciousness could be modelled in a computer program in the same way that, say, the weather can be modelled … If you accept functionalism, however, then you should believe that consciousness is a computational process. But Turing had no result entailing what the Churchlands say. In fact, he had a result entailing that there are patterns of responses that no standard Turing machine is able to generate.

One example of such a pattern is provided by the function h , described earlier. In reality the Church-Turing thesis does not entail that the brain or the mind, or consciousness can be modelled by a Turing machine program, not even in conjunction with the belief that the brain or mind, or consciousness is scientifically explicable, or rule-governed, or scientifically describable, or characterizable as a set of steps Copeland c.

The simulation thesis is much stronger than the Church-Turing thesis: This is equally so if the simulation thesis is taken narrowly, as concerning processes that conform to the physics of the real world. If, on the other hand, the thesis is taken as ranging over all processes, including merely possible or notional processes, then the thesis is known to be false, for exactly the same reasons that the stronger form of the maximality thesis is false.

Any device or organ whose internal processes can be described completely by means of what Church called effectively calculable functions can be simulated exactly by a Turing machine providing that the input into the device or organ is itself computable by Turing machine.

But any device or organ whose mathematical description involves functions that are not effectively calculable cannot be so simulated. As Turing showed, there are uncountably many such functions.

It is an open question whether a completed neuroscience will need to employ functions that are not effectively calculable. We may compare a man in the process of computing a … number to a machine.

Computability and Complexity the Church-Turing Thesis: types of evidence • large sets of Turing-Computable functions many examples no counter-examples • equivalent to other formalisms for algorithms Church’s l calculus and others • intuitive - any detailed algorithm for manual calculation can be implemented by a Turing Machine.

Quantum Computation and Extended Church-Turing Thesis Extended Church-Turing Thesis The extended Church-Turing thesis is a foundational principle in computer science.

There are various equivalent formulations of the Church-Turing thesis. A common one is that every effective computation can be carried out by a Turing machine. The Church-Turing thesis is often misunderstood, particularly in recent writing in the philosophy of mind. The Church-Turing thesis in a quantum world Ashley Montanaro Centre for Quantum Information and Foundations, Department of Applied Mathematics and Theoretical Physics.

In computability theory, the Church–Turing thesis (also known as computability thesis,[1] the Turing–Church thesis,[2] the Church–Turing conjecture, Church's thesis, Church's conjecture, and Turing's thesis) is a hypothesis about the nature of computable functions. This PDF version matches the latest version of this entry. To view the PDF, you must Log In or Become a Member. You can also read more about the Friends of the SEP Society.