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Church’s Thesis for Turing Machine

In 1936, A method named as lambda-calculus was created by Alonzo Church in which the Church numerals are well defined, i.e. the encoding of natural numbers. Also in 1936, Turing machines (earlier called theoretical model for machines) was created by Alan Turing, that is used for manipulating the symbols of string with the help of tape.

Church Turing Thesis :

Turing machine is defined as an abstract representation of a computing device such as hardware in computers. Alan Turing proposed Logical Computing Machines (LCMs), i.e. Turing’s expressions for Turing Machines. This was done to define algorithms properly. So, Church made a mechanical method named as ‘M’ for manipulation of strings by using logic and mathematics. This method M must pass the following statements:

• Number of instructions in M must be finite.
• Output should be produced after performing finite number of steps.
• It should not be imaginary, i.e. can be made in real life.
• It should not require any complex understanding.

Using these statements Church proposed a hypothesis called

Church’s Turing thesis

that can be stated as: “The assumption that the intuitive notion of computable functions can be identified with partial recursive functions.”

Or in simple words we can say that “Every computation that can be carried out in the real world can be effectively performed by a Turing Machine.”

In 1930, this statement was first formulated by Alonzo Church and is usually referred to as Church’s thesis, or the Church-Turing thesis. However, this hypothesis cannot be proved. The recursive functions can be computable after taking following assumptions:

• Each and every function must be computable.
• Let ‘F’ be the computable function and after performing some elementary operations to ‘F’, it will transform a new function ‘G’ then this function ‘G’ automatically becomes the computable function.
• If any functions that follow above two assumptions must be states as computable function.

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Church Turing Thesis in Theory of Computation

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In this article, we have explain the meaning and importance of Church Turing Thesis in Theory of Computation along with its applications and limitations.

Table of contents:

• Introduction to Turing Church Thesis

Applications of Church Turing Thesis

Limitations of church turing thesis.

Prerequisites: Algorithms, Turing Machine

Let us get started with Church Turing Thesis in Theory of Computation.

Definition of Church Turing Thesis

Church Turing Thesis states that:

A computation process that can be represented by an algorithm can be converted to a Turing Machine.

In simple words, any thing that can be done by an Algorithm can be done by a Turing Machine as well. So, all algorithms can be implemented in a Turing Machine.

Specific Computation Models are equivalent which means any one model can be coverted to another model. These Computation Models include:

• One tape Turing Machine
• K tape Turing Machine where K >= 1
• Non Deterministic Turing Machine
• Programs in Programming Languages such as Java, C++, Lisp and others.

So, a program in C++ can be converted to a K tape Turing Machine and vice versa.

The applications of Church Turing Thesis are as follows:

• Church Turing Thesis is used to define an Algorithm (traditionally)
• Used in solving 10th Problem by Hilbert.
• Used in defining modern computing devices including Molecular and Quantum Computers.

10th Problem by Hilbert

It has been used to solve the 10th Problem by Hilbert in 8th August 1900 at the Second International Congress of Mathematicians in Paris. These problems were listed as critical problems that should be solved for progress in Mathematics.

The 10th Problem by Hilbert was:

Does there exists a process with finite number of steps that can determine if a given polynomial with integer coefficients has integral roots?

Another way to look at the problem is to find if there is an Algorithm to find if there exists an integral root for a given polynomial or not.

For example: This is a Polynomial:

35x 10 y 2 z 9 + 11x 6 z 7 + 103xyz + 17y 31 z 3 = 0.

Is there an algorithm that can find if there exist a solution in integers?

Note the solution is not needed. Only we need to find if such a solution exists or not.

In 1970, it was proved that no such algorithm exists. This was done by Matiyasevich.

Algorithm = Church Turing Thesis

To solve the 10th Hilbert Problem, one needs to understand what is meant by an algorithm. In fact, there have been different definitions and all have proved to be equivalent. Some definitions were:

• 1936: Algorithm = Turing Machine
• 1936: Algorithm = Lambda Calculus
• 1970+: Algorithm = Implementation in Programming Languages like C and Lisp
• Final: Algorithm = process converted to Turing Machine.

Finally, it was agreed that an Algorithm is based on Church Turing Thesis which said:

"Any computational process can be considered as an Algorithm if it can be converted to a Turing Machine." Note: This does not hold true as of now.

Modern Computing Devices

Traditional Computers which are in use today, are limited by Church Turing Thesis. This is because Church Turing Thesis defines an Algorithm which can be implemented in a real system.

Therefore, the Computing Device you are using is basically a Turing Machine.

The only difference is that Computing Devices are efficient while Turing Machine is inefficient. Theoretically, from a point of view of algorithms, there is no difference.

There are 3 different approaches future computers may take:

• Quantum Computer : Solve Computing Problems using atoms by quantum rules. This is an active area of research.
• Molecular Computer : Solve Computing Problems using Molecules by taking advantage of Physical laws of Moleculars. This includes replicating the idea of DNA.
• Super Recursive Algorithm : This domain has not been realized yet and exists in theory but this is the part where Church Turing Thesis fail. We have covered this in the next section on "Limitations of Church Turing Thesis".

Two different futuristic models of Computer which follows Church Turing Thesis:

• Quantum Computers can be represented as Non Deterministic Turing Machine
• Molecular Computers can be represented by Turing Machine with many tapes and heads

Therefore, Quantum and Molecular Computers are same fundamentally and they are only more efficient than Mechnical Computers.

Super Recursive Algorithms proved Church Turing Thesis wrong. The first Super Recursive algorithm was introduced in 1965 by Mark Gold and Hillary Putnam by using ideas of limit recursive and limit partial recursive functions. It was based on ideas from non standard analysis by Abraham Robinson in 1966 and Inductive Definition of sets by Spector in 1959. This resulted in Inductive Inference by Gasarch and Smith in 1997 and is used in Machine Learning.

Super Recursive Algorithms can solve problems that are unsolvable by Turing Machines. To account for this, a new idea was introduced: Inductive Turing Macine. These were not accepted as Algorithm for a long time as it was refuting Church Turing Thesis and Godel Incompleteness Theorem (as proved in 1987 by Burgin).

The idea of Inductive Turing Machine is as follows:

• Turing Machine has a property that it stops after giving a result.
• Most programs stop after giving a result and this seems to be reasonable as what a program should do once it has found the answer.
• Operating Systems are also programs but it does not give a standard output. It gives some strings to the users during its use but it cannot be considered as a output. The functionality of an Operating System is considered to be the output. It does not stop like standard program. If it stops, it cannot give any output.
• There can be programs which give a result at the moment which is good enough but
• This idea of not stopping after giving a result is the basis.

Inductive Turing Machine is more powerful than Conventional Turing Machine. Inductive Turing Machine can solve the Halting Problem which is known to be unsolvable by Conventional Turing Maching.

There are different types of Inductive Turing Machine:

• Inductive Turing Machine + Structured Memory
• Inductive Turing Machine + Structured Rules (control device)
• Inductive Turing Machine + Structured Head (Operating Device)

Today, Church Turing Thesis is not considered as an Universal Principle. Inductive Turing Machine is the most powerful super recursive algorithm.

This lead to the formulation of "Extended Church Turing Thesis".

There are three open questions:

• How to realize Super Recursive algorithms in technological devices?
• How modern computing devices are related to Super Recursive Algorithm?
• What are the new possibilities with Super Recursive Algorithm?

Think about these research open ended problems in Theory of Computation.

With this article at OpenGenus, you must have the complete idea of Church Turing Thesis in Theory of Computation.

Stanford Encyclopedia of Philosophy

The church-turing thesis, the thesis and its history, misunderstandings of the thesis, some key remarks by turing, bibliography, other internet resources, related entries.

• M is set out in terms of a finite number of exact instructions (each instruction being expressed by means of a finite number of symbols);
• M will, if carried out without error, produce the desired result in a finite number of steps;
• M can (in practice or in principle) be carried out by a human being unaided by any machinery save paper and pencil;
• M demands no insight or ingenuity on the part of the human being carrying it out.

A well-known example of an effective method is the truth table test for tautologousness. In practice, of course, this test is unworkable for formulae containing a large number of propositional variables, but in principle one could apply it successfully to any formula of the propositional calculus, given sufficient time, tenacity, paper, and pencils.

Statements that there is an effective method for achieving such-and-such a result are commonly expressed by saying that there is an effective method for obtaining the values of such-and-such a mathematical function. For example, that there is an effective method for determining whether or not any given formula of the propositional calculus is a tautology -- e.g. the truth table method -- is expressed in function-speak by saying that there is an effective method for obtaining the values of a function, call it T, whose domain is the set of formulae of the propositional calculus and whose value for any given formula x, written T(x), is 1 or 0 according to whether x is, or is not, a tautology.

The notion of an effective method is an informal one, and attempts to characterise effectiveness, such as the above, lack rigour, for the key requirement that the method demand no insight or ingenuity is left unexplicated. One of Turing's achievements in his paper of 1936 was to present a formally exact predicate with which the informal predicate ‘can be calculated by means of an effective method’ may be replaced. Church did the same (1936a). The replacement predicates that Turing and Church proposed were, on the face of it, very different from one another, but they turned out to be equivalent, in the sense that each picks out the same set of mathematical functions. The Church-Turing thesis is the assertion that this set contains every function whose values can be obtained by a method satisfying the above conditions for effectiveness. (Clearly, if there were functions of which the informal predicate, but not the formal predicate, were true, then the latter would be less general than the former and so could not reasonably be employed to replace it.) When the thesis is expressed in terms of the formal concept proposed by Turing, it is appropriate to refer to the thesis also as ‘Turing's thesis’; and mutatis mutandis in the case of Church.

The formal concept proposed by Turing is that of computability by Turing machine . He argued for the claim (Turing's thesis) that whenever there is an effective method for obtaining the values of a mathematical function, the function can be computed by a Turing machine. The converse claim is easily established, for a Turing machine program is itself a specification of an effective method: without exercising any ingenuity or insight, a human being can work through the instructions in the program and carry out the required operations. If Turing's thesis is correct, then talk about the existence and non-existence of effective methods can be replaced throughout mathematics and logic by talk about the existence or non-existence of Turing machine programs.

Turing stated his thesis in numerous places, with varying degrees of rigour. The following formulation is one of the most accessible.

Turing's thesis : LCMs [logical computing machines: Turing's expression for Turing machines] can do anything that could be described as "rule of thumb" or "purely mechanical". (Turing 1948:7.)

This is sufficiently well established that it is now agreed amongst logicians that "calculable by means of an LCM" is the correct accurate rendering of such phrases. (1948: 7.)

By the Entscheidungsproblem of a system of symbolic logic is here understood the problem to find an effective method by which, given any expression Q in the notation of the system, it can be determined whether or not Q is provable in the system. (Church 1936b: 41.)

Church had arrived at the same negative result a few months earlier, employing the concept of lambda-definability in place of computability by Turing machine. (A function of positive integers is said to be lambda-definable if the values of the function can be calculated by a process of repeated substitution.) Church and Turing discovered the result quite independently of one another. Turing's method of obtaining it is rather more satisfying than Church's, as Church himself acknowledged in a review of Turing's work:

computability by a Turing machine ... has the advantage of making the identification with effectiveness in the ordinary (not explicitly defined) sense evident immediately. (1937a: 43.)

Church used the (informal) expression ‘effectively calculable’ to indicate that there is an effective method for calculating the values of the function. He proposed that we

define the notion ... of an effectively calculable function of positive integers by identifying it with the notion of a recursive function of positive integers (or of a lambda-definable function of positive integers). (1936a: 356.)

Post referred to Church's identification of effective calculability with recursiveness as a "working hypothesis", and quite properly criticised Church for masking this hypothesis as a definition.

[T]o mask this identification under a definition ... blinds us to the need of its continual verification. (Post 1936: 105.)

Church's thesis : A function of positive integers is effectively calculable only if recursive.

The term ‘Church-Turing thesis’ seems to have been first introduced by Kleene, with a small flourish of bias in favour of Church:

So Turing's and Church's theses are equivalent. We shall usually refer to them both as Church's thesis , or in connection with that one of its ... versions which deals with ‘Turing machines’ as the Church-Turing thesis . (Kleene 1967: 232.)

Much evidence has been amassed for the ‘working hypothesis’ proposed by Church and Turing in 1936. One of the fullest surveys is to be found in chapters 12 and 13 of Kleene (1952). In summary: (1) Every effectively calculable function that has been investigated in this respect has turned out to be computable by Turing machine. (2) All known methods or operations for obtaining new effectively calculable functions from given effectively calculable functions are paralleled by methods for constructing new Turing machines from given Turing machines. (3) All attempts to give an exact analysis of the intuitive notion of an effectively calculable function have turned out to be equivalent in the sense that each analysis offered has been proved to pick out the same class of functions, namely those that are computable by Turing machine. Because of the diversity of the various analyses the latter is generally considered strong evidence. For example, apart from the analyses already mentioned in terms of lambda-definability and recursiveness, there are analyses in terms of register machines (Shepherdson and Sturgis 1963), Post's canonical and normal systems (Post 1943, 1946), combinatory definability (Schönfinkel 1924, Curry 1929, 1930, 1932), Markov algorithms (Markov 1960), and Gödel's notion of reckonability (Gödel 1936, Kleene 1952).

While there have from time to time been attempts to call the Church-Turing thesis into question (for example by Kalmar (1959); Mendelson (1963) replies), the summary of the situation that Turing gave in 1948 is no less true today: "it is now agreed amongst logicians that ‘calculable by means of an LCM’ is the correct accurate rendering" (of the informal notion in question).

A myth seems to have arisen concerning Turing's paper of 1936, namely that he there gave a treatment of the limits of mechanism and established a fundamental result to the effect that the universal Turing machine can simulate the behaviour of any machine. The myth has passed into the philosophy of mind, generally to pernicious effect. For example, the Oxford Companion to the Mind states: "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" (Gregory 1987: 784). Dennett maintains that "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" (1991: 215); also that every "task for which there is a clear recipe composed of simple steps can be performed by a very simple computer, a universal Turing machine, the universal recipe-follower (1978:. xviii). Paul and Patricia Churchland assert that Turing's "results entail something remarkable, namely that a standard digital computer, given only the right program, a large enough memory and sufficient time, can compute any rule-governed input-output function. That is, it can display any systematic pattern of responses to the environment whatsoever" (1990: 26). These various quotations are typical of current writing on the foundations of the computational theory of mind. It seems on the surface unlikely that these authors mean to restrict the general notions of ‘explicitly stated rule’, ‘instruction’, ‘clear recipe composed of simple steps', ‘computer with any architecture’, ‘rule-governed function’ and ‘systematic pattern’ so as to apply only to things that can be obeyed, simulated, calculated, or produced by a machine that implements ‘effective’ methods in Turing's original sense. But unless these notions are restricted in this way from the start, we should reject such claims.

Turing did not show that his machines can solve any problem that can be solved "by instructions, explicitly stated rules, or procedures", nor did he prove that the universal Turing machine "can compute any function that any computer, with any architecture, can compute". He 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 here called Turing's thesis. But a thesis concerning the extent of effective methods -- which is to say, concerning the extent of procedures of a certain sort that a human being unaided by machinery is capable of carrying out -- carries no implication concerning the extent of the procedures that machines are capable of carrying out, even machines acting in accordance with ‘explicitly stated rules’. For among a machine's repertoire of atomic operations there may be those that no human being unaided by machinery can perform.

The further proposition, very different from Turing's own thesis, that a Turing machine can compute whatever can be computed by any machine working on finite data in accordance with a finite program of instructions, is sometimes also referred to as (a version of) the Church-Turing thesis or Church's thesis. For example, Smolensky says:

connectionist models ... may possibly even challenge the strong construal of Church's Thesis as the claim that the class of well-defined computations is exhausted by those of Turing machines. (Smolensky 1988: 3.)

This loosening of established terminology is unfortunate, for neither Church nor Turing endorsed, or even formulated, this further proposition. There are numerous examples of this extended usage in the literature. The following are typical.

That there exists a most general formulation of machine and that it leads to a unique set of input-output functions has come to be called Church's thesis . (Newell 1980: 150.) [T]he work of Church and Turing fundamentally connects computers and Turing machines. The limits of Turing machines, according to the Church-Turing thesis, also describe the theoretical limits of all computers. (McArthur 1991: 401.) [I]t is difficult to see how any language that could actually be run on a physical computer could do more than Fortran can do. The idea that there is no such language is called Church's thesis. (Geroch and Hartle 1986: 539.)

Also (more distant still from anything that Church or Turing actually wrote):

I can now state the physical version of the Church-Turing principle: "Every finitely realizable physical system can be perfectly simulated by a universal model computing machine operating by finite means." This formulation is both better defined and more physical than Turing's own way of expressing it. (Deutsch 1985: 99.)

This formulation may be ‘more physical’ than Turing's own, but it is scarcely ‘better defined’. The notion of an effective method played an important role in early debates about the foundations of mathematics, and it was sufficiently clear to allow Turing, Church, and others to recognize that different formal accounts gave alternative modellings of the notion. Their notion was certainly not that of a ‘finitely realizable physical system’.

Gandy (1980) is one of the few writers to distinguish explicitly between Turing's thesis and the stronger proposition that whatever can be calculated by a machine can be calculated by a Turing machine. Borrowing Gandy's terminology, I will call the stronger proposition ‘Thesis M’. I will use expressions such as ‘the Church-Turing thesis properly so-called’ for the proposition that Church and Turing themselves endorsed.

Thesis M : Whatever can be calculated by a machine (working on finite data in accordance with a finite program of instructions) is Turing-machine-computable.

Thesis M itself admits of two interpretations, according to whether the phrase "can be generated by a machine" is taken in the narrow, this-worldly, sense of "can be generated by a machine that conforms to the physical laws (if not to the resource constraints) of the actual world", or in a wide sense that abstracts from the issue of whether or not the notional machine in question could exist in the actual world. Under the latter interpretation, thesis M is false. It is straightforward to describe notional machines, or ‘hypercomputers’ ( Copeland and Proudfoot (1999a)) that generate functions not Turing-machine-computable (see e.g. Abramson (1971), Copeland (2000), Copeland and Proudfoot (2000), Stewart (1991)). It is an open empirical question whether or not the narrow this-worldly version of thesis M is true. Speculation that there may be physical processes -- and so, potentially, machine-operations -- whose behaviour conforms to functions not computable by Turing machine stretches back over at least five decades; see, for example, da Costa and Doria (1991), (1994), Doyle (1982), Geroch and Hartle (1986), Hogarth (1994), Kreisel (1967), (1974), (1982), Pour-El and Richards (1979), (1981), Scarpellini (1963), Siegelmann and Sontag (1994), and Stannett (1990). (Copeland and Sylvan (1999) is a survey; see also Copeland and Proudfoot (1999b).)

The literature on the computational theory of the mind contains numerous endorsements of propositions equivalent or similar to thesis M that are supported by nothing more than a nod toward Turing or Church (as is illustrated by a number of the quotations given earlier). Perhaps some writers are simply misled by the terminological practice whereby a thesis concerning which there is little real doubt, the Church-Turing thesis properly so called, and a different thesis of unknown truth-value, are referred to indiscriminately as Church's thesis or the Church-Turing thesis -- albeit with accompanying hedges like ‘strong form’ and ‘physical version’. Other writers may maintain thesis M (or some equivalent or near equivalent) on the spurious grounds that the various, and prima facie very different, attempts -- by Turing, Church, Post, Markov, and others -- to characterise in precise terms the informal notion of an effective procedure have turned out to be equivalent to one another. This is evidence concerning the extent of effective procedures, and not evidence concerning the extent of what can be calculated by machine.

The error of confusing the Church-Turing thesis properly so-called with thesis M 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.g. Fodor 1981: 130; Boden 1988: 259). To one who makes the 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 that transcend Turing machines.

Note that in some cases, an author's apparent endorsement of M is merely apparent. In this connection, it is important to remember that in the technical literature the word ‘computable’ is often tied by definition to effective calculability. Thus a function is said to be computable if and only if there is an effective procedure for determining its values. Accordingly, a common formulation of the Church-Turing thesis in the technical literature and in textbooks is:

All computable functions are computable by Turing machine.

Corollaries such as the following are sometimes offered:

certain functions are uncomputable in an absolute sense: uncomputable even by [Turing machine], and, therefore, uncomputable by any past, present, or future real machine. (Boolos and Jeffrey 1980: 55.)

The word ‘mechanical’, too, in technical usage, is tied to effectiveness and, as already remarked, ‘mechanical’ and ‘effective’ are used interchangeably. (Gandy (1988) outlines the history of this usage of the word ‘mechanical’.) Thus statements like the following are to be found in the technical literature:

Turing proposed that a certain class of abstract machines could perform any ‘mechanical’ computing procedure. (Mendelson 1964: 229.)

Thesis M is not the only problematic thesis that is linked to the Church-Turing thesis. An error which, unfortunately, is common in modern writing on computability and the brain is to hold that Turing's results somehow entail that the brain, and indeed any biological or physical system whatever, can be simulated by a Turing machine. For example, the entry on Turing in the recent A Companion to the Philosophy of Mind contains the following claims: "we can depend on there being a Turing machine that captures the functional relations of the brain", for so long as "these relations between input and output are functionally well-behaved enough to be describable by ... mathematical relationships ... we know that some specific version of a Turing machine will be able to mimic them" (Guttenplan 1994: 595). Searle writes in a similar fashion:

Can the operations of the brain be simulated on a digital computer? ... The answer seems to me ... demonstrably ‘Yes’ ... That is, naturally interpreted, the question means: Is there some description of the brain such that under that description you could do a computational simulation of the operations of the brain. But given Church's thesis that anything that can be given a precise enough characterization as a set of steps can be simulated on a digital computer, it follows trivially that the question has an affirmative answer. (Searle 1992: 200.)

If you assume that [consciousness] is scientifically explicable ... [and] [g]ranted that the [Church-Turing] thesis is correct, then ... [i]f you believe [functionalism] to be false ... then ... you [should] hold that consciousness could be modelled in a computer program in the same way that, say, the weather can be modelled ... [and if] you accept functionalism ... you should believe that consciousness is a computational process. (Johnson-Laird 1987: 252.)

Church's Thesis says that whatever is computable is Turing computable. 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 1983: 6.)

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, etc.) is scientifically explicable, or exhibits a systematic pattern of responses to the environment, or is ‘rule-governed’ (etc.). Each of the authors quoted seems to be assuming the truth of a close cousin of thesis M, which I will call

Thesis S: Any process that can be given a mathematical description (or that is scientifically describable or scientifically explicable) can be simulated by a Turing machine.

As with thesis M, neither the Church-Turing thesis properly so-called nor any result proved by Turing or Church entails thesis S. This is so even when the thesis is taken narrowly, as concerning processes that conform to the physics of the real world. (Thesis S taken in the wide sense is known to be false; see the references given earlier re the wide version of thesis M.) Any device or organ whose internal processes can be described completely by means of effectively calculable functions can be simulated exactly by a Turing machine program (provided that the input into the device or organ is itself Turing-machine-computable, which is to say, is either finite or expressible as a computable number, in Turing's sense (which is explained below)); 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. (Examples from logic are Turing's famous halting function (described in the entry on Turing machines) and the function D whose domain is the set of well-formed formulae of the predicate calculus and whose values, D(x), are 1 or 0 according to whether x is, or is not, derivable from the Bernays-Hilbert-Ackermann axioms for predicate logic.) It is an open question whether a completed neuroscience will employ functions that are not effectively calculable.

We may compare a man in the process of computing a ... number to a machine. (Turing 1936: 231.)

Turing's "Machines". These machines are humans who calculate. (Wittgenstein 1980, 1096.)

A man provided with paper, pencil, and rubber, and subject to strict discipline, is in effect a universal machine. (Turing 1948: 9.)

The electronic stored-program digital computers for which the universal Turing machine was a blueprint are, each of them, computationally equivalent to a Turing machine, and so they too are, in a sense, models of human beings engaged in computation. Turing chose to emphasise this when explaining these electronic machines in a manner suitable for an audience of uninitiates:

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. (Turing 1950a: 436).

The class of problems capable of solution by the machine [the 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. (Turing 1946: 38-9.)

Electronic computers are intended to carry out any definite rule of thumb process which could have been done by a human operator working in a disciplined but unintelligent manner. (Turing 1950b: 1.)

It was not some deficiency of imagination that led Turing to model his computing machines on what could be achieved by a human computer. The purpose for which the Turing machine was invented demanded it. The Entscheidungsproblem is the problem of finding a humanly executable procedure of a certain sort, and Turing's aim was precisely to show that there is no such procedure in the case of predicate logic. He proved that no Turing machine can compute the values of the function D that I described earlier, and he argued that his model of human computation is sufficiently general, in the sense that there are no intuitively computable (i.e. effectively calculable) functions that Turing machines are incapable of computing.

The latter claim is, of course, Turing's thesis. Here are two additional formulations of the thesis, from his paper of 1936.

[T]he "computable numbers" [the numbers whose decimal representations can be generated progressively by a Turing machine] include all numbers which would naturally be regarded as computable. (Turing 1936: 249.)

It is my contention that these operations [the primitive operations of a Turing machine] include all those which are used in the computation of a number. (Turing 1936: 232.)

To understand these assertions as Turing intended them it is essential to keep in mind that when he uses the words ‘computer’, ‘computable’ and ‘computation’ he employs them not in their modern sense as pertaining to machines but as pertaining to human calculators. Many passages make this obvious.

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 ... (Turing 1947: 116, 120.)

To define effectiveness as computability by an arbitrary machine, subject to restrictions of finiteness, would seem to be an adequate representation of the ordinary notion (Church 1937b: 43),

It is equally important to note also that when Turing uses the word ‘machine’ he often means not machine-in-general but, as we would now say, Turing machine. At one point he explicitly draws attention to this idiosyncratic usage:

The expression "machine process" of course means one which could be carried out by the type of machine I was considering [in Turing (1936)]. (Turing 1947: 107.)

The importance of the universal machine is clear. We do not need to have an infinity of different machines doing different jobs. A single one will suffice. The engineering problem of producing various machines for various jobs is replaced by the office work of "programming" the universal machine to do these jobs. (Turing 1948: 7.)

In context it is perfectly clear that these remarks concern machines equivalent to Turing machines (the passage is embedded in a discussion of LCMs).

Whether or not Turing would, if queried, have assented to thesis M is unknown. There is certainly no textual evidence in favour of the common belief that he did so assent.

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Truth, Existence and Explanation pp 225–248 Cite as

Church-Turing Thesis, in Practice

• Luca San Mauro 6  
• First Online: 25 October 2018

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Part of the Boston Studies in the Philosophy and History of Science book series (BSPS,volume 334)

We aim at providing a philosophical analysis of the notion of “proof by Church’s Thesis”, which is – in a nutshell – the conceptual device that permits to rely on informal methods when working in Computability Theory. This notion allows, in most cases, to not specify the background model of computation in which a given algorithm – or a construction – is framed. In pursuing such analysis, we carefully reconstruct the development of this notion (from Post to Rogers, to the present days), and we focus on some classical constructions of the field, such as the construction of a simple set. Then, we make use of this focus in order to support the following encompassing claim (which opposes to a somehow commonly received view): the informal side of Computability, consisting of the large class of methods typically employed in the proofs of the field, is not fully reducible to its formal counterpart.

• Church-Turing Thesis
• Informal Methods
• Informal Algorithm
• Acceptable Number

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A classical introduction to CTT can be found in Kleene ( 1952 ). See also Church ( 1936 ), Turing ( 1936 , 1948 ), Post ( 1936 ), and Gödel ( 1946 ). Soare ( 1987b ) contains, in its first part, an accurate reconstruction of the role of Turing’s work in the acceptance of the thesis. For recent philosophical work concerning CTT, the reader is referred, for instance, to Olszewski et al. ( 2006 ).

The standard interpretation is that CTT is indeed a thesis , or, in Post’s words, “a working hyphotesis” (Post 1936 ). That is to say, something that cannot be subject of a mathematical proof. Yet, it has been argued that CTT has not necessarily an hypothetical status, but rather that it can be susceptible of a rigorous mathematical proof, or even that such a proof is already contained in Turing ( 1936 ) (for this line of thought see, e.g., Mendelson ( 1990 ), Gandy ( 1988 ), Sieg ( 1994 ), and the discussion contained in Shapiro ( 2006 )). Responses to this latter position can be found in Folina ( 1998 ) and Black ( 2000 ).

See Welch ( 2007 ) for a rich survey on models of transfinite computation. On the other hand, Davis ( 2006 ) denies any theoretical significance to “hypercomputationalism” as such.

For instance, the following is the first proof-theoretic reference to CTT in Rogers ( 1967 ):

There are exactly ℵ 0 partial recursive functions, and there are exactly ℵ 0 recursive functions.

All constants functions are recursive, by Church’s Thesis. Hence there are at least ℵ 0 recursive functions. The Gödel numbering shows that there are at most ℵ 0 partial recursive functions.

To be fair, Post mainly speaks of computably enumerable sets, there introduced for the first time. But since, by definition, a set is computably enumerable if it is the range of a computable function, then one can trivially translate Post’s formulations in instances of our prototype.

It is worth noticing that the Leibnizian ideal is by no means archeological. Quite to the contrary. Hacking reports Voevodsky’s opinion that “in a few years, journal will accept only articles accompanied by their machine-verifiable equivalents”. More generally – and less radically – research on proof-assistants can be (partially) motivated as a way of improving automatic verification of proofs.

For an accurate reconstruction of Post’s thought see De Mol ( 2006 )

From now on, in describing the practical side of CTT, we will mostly refer to textbooks. This is a natural choice. Since, as already said, there are no philosophical studies concerning the practice of Computability, the most immediate source of observations regarding how such practice has to be intended comes from the kind of expository remarks that abound in books such as Rogers’.

The interested reader can consult Mancosu ( 2008 ) for an anthology of papers in Philosophy of Mathematical Practice.

\(\mathbb {I}\) does clearly correspond to a pre-theoretic object whose formalization would be far from trivial. For instance, there could be a worry concerning a sort of Berry-like paradox, inasmuch we admit a too relaxed notion on what counts as an informal description for an algorithm. Nonetheless, we can suppose to deal with sufficiently clear descriptions. This is because, although border-cases cannot arguably be expunged, we are more interested, as we will see, in a somewhat global tendency.

All of this is of course related to the philosophical problem of determining if one can possibly formulate a definition for algorithms that would be correct in the sense of Shore: “Find, and argue conclusively for, a formal definition of algorithm and the appropriate analog of the Church-Turing thesis. Here we want to capture the intuitive notion that, for example, two particular programs in perhaps different languages express the same algorithm, while other ones that compute the same function represent different algorithms for the function. Thus we want a definition that will up to some precise equivalence relation capture the notion that two algorithms are the same as opposed to just computing the same function”(Buss et al. 2001 ). See also Dean ( 2007 ) for a rich discussion on whether algorithms can be fairly regarded as abstract mathematical objects.

Indeed, Blass et al. ( 2009 ) argue that “one cannot give a precise equivalence relation capturing the intuitive notion of ‘the same algorithm.”’

The reader is referred to Rogers ( 1967 ) for the proof of this fact, and to Odifreddi ( 1989 ) for additional results concerning numberings.

Some equivalence between classical models of computation can be found in Odifreddi ( 1989 ).

For a classical defense of structuralism in philosophy of mathematics, the reader is referred to Resnik ( 1997 ).

Two noteworthy exception being Carter ( 2008 ) and McLarty ( 2008 ).

Awodey, S. 2014. Structuralism, invariance, and univalence. Philosophia Mathematica 22(1): 1–11.

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Acknowledgements

A preliminary version of this paper appeared as a chapter of my PhD thesis. I would like to thank my supervisors, Gabriele Lolli and Andrea Sorbi, for their guidance and support. I have presented this work at several conferences. In particular, I am grateful to the participants of APMP 2014, in Paris, and of FilMat 2016, in Chieti, for their comments. Finally, Richard Epstein’s remarks were fundamental in rethinking the organization of the present material.

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Mauro, L.S. (2018). Church-Turing Thesis, in Practice. In: Piazza, M., Pulcini, G. (eds) Truth, Existence and Explanation. Boston Studies in the Philosophy and History of Science, vol 334. Springer, Cham. https://doi.org/10.1007/978-3-319-93342-9_13

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Church-Turing Thesis

The Church-Turing thesis (formerly commonly known simply as Church's thesis) says that any real-world computation can be translated into an equivalent computation involving a Turing machine . In Church's original formulation (Church 1935, 1936), the thesis says that real-world calculation can be done using the lambda calculus , which is equivalent to using general recursive functions .

The Church-Turing thesis encompasses more kinds of computations than those originally envisioned, such as those involving cellular automata , combinators , register machines , and substitution systems . It also applies to other kinds of computations found in theoretical computer science such as quantum computing and probabilistic computing.

There are conflicting points of view about the Church-Turing thesis. One says that it can be proven, and the other says that it serves as a definition for computation. There has never been a proof, but the evidence for its validity comes from the fact that every realistic model of computation, yet discovered, has been shown to be equivalent. If there were a device which could answer questions beyond those that a Turing machine can answer, then it would be called an oracle .

Some computational models are more efficient, in terms of computation time and memory, for different tasks. For example, it is suspected that quantum computers can perform many common tasks with lower time complexity , compared to modern computers, in the sense that for large enough versions of these problems, a quantum computer would solve the problem faster than an ordinary computer. In contrast, there exist questions, such as the halting problem , which an ordinary computer cannot answer, and according to the Church-Turing thesis, no other computational device can answer such a question.

The Church-Turing thesis has been extended to a proposition about the processes in the natural world by Stephen Wolfram in his principle of computational equivalence (Wolfram 2002), which also claims that there are only a small number of intermediate levels of computing power before a system is universal and that most natural systems are universal.

This entry contributed by Todd Rowland

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What is The Church-Turing Thesis in TOC?

The Church-Turing thesis says that every solvable decision problem can be transformed into an equivalent Turing machine problem.

It can be explained in two ways, as given below −

The Church-Turing thesis for decision problems.

The extended Church-Turing thesis for decision problems.

Let us understand these two ways.

The Church-Turing thesis for decision problems

There is some effective procedure to solve any decision problem if and only if there is a Turing machine which halts for all input strings and solves the problem.

The extended Church-Turing thesis for decision problems

A decision problem Q is said to be partially solvable if and only if there is a Turing machine which accepts precisely the elements of Q whose answer is yes.

A proof by the Church-Turing thesis is a shortcut often taken in establishing the existence of a decision algorithm.

For any decision problem, rather than constructing a Turing machine solution, let us describe an effective procedure which solves the problem.

The Church-Turing thesis explains that a decision problem Q has a solution if and only if there is a Turing machine that determines the answer for every q ϵ Q. If no such Turing machine exists, the problem is said to be undecidable.

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The Church-Turing Thesis: Story and Recent Progress

The Church-Turing thesis is one of the foundations of computer science. The thesis heralded the dawn of the computer revolution by enabling the construct of the universal Turing machine which led the way, at least conceptually, to the von Neumann architecture and first electronic computers. One way to state the Church-Turing thesis is as follows:

A Turing Machine computes every numerical function that is computable by means of a purely mechanical procedure.

It is that remarkable and a priori implausible characterization that underlies the ubiquitous applicability of digital computers. But why do we believe the thesis? Careful analysis shows that the existing arguments are insufficient. Kurt Gödel surmised that it might be possible to state axioms which embody the generally accepted properties of computability, and to prove the thesis on that basis. That is exactly what we did in a recent paper with Nachum Dershowitz of Tel Aviv University.

Beyond our proof, the story of the Church-Turing thesis is fascinating and scattered in specialized and often obscure publications. I will try to do justice to that intellectual drama.

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Yuri Gurevich is Principal Researcher at Microsoft Research in Redmond, WA. He is also Prof. Emeritus at the University of Michigan, ACM Fellow, Guggenheim Fellow, a member of Academia Europaea, and Dr. Honoris Causa of a couple of universities.

Yuri Gurevich

Principal Researcher

Jeff Running

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