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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 example

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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Church Turing Thesis in Theory of Computation

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.

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Stanford Encyclopedia of Philosophy

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. (Ibid.)
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.)
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.)
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: .)
[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.
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.)
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.)
[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.)
[The] Church/ Turing thesis ... equates the mathematically precise notion of "solvable by a Turing machine" with the informal, intuitive notion of "solvable effectively", which alludes to all real computers and all programming languages, those that we know about at present as well as those that we do not. (Harel 1992: 233.)
The Church-Turing thesis makes a bold claim about the theoretical limits to computation. (Cleland 1993: 283.)
The first aspect that we examine of Church’s Thesis ... [w]e can formulate, more precisely: The behaviour of any discrete physical system evolving according to local mechanical laws is recursive. (Odifreddi 1989: 107.)
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.)
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.
All computable functions are computable by Turing machine.
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.)
Turing proposed that a certain class of abstract machines could perform any ‘mechanical’ computing procedure. (Mendelson 1964: 229.)
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.)
Thesis S : Any process that can be given a systematic mathematical description (or a ‘precise enough characterization as a set of steps’, or that is scientifically describable or scientifically explicable) can be simulated by a Turing machine.
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 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 1950: 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 1951: 1.)
[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.)
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),
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.)

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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

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The author was partially supported by the Austrian Science Fund FWF through project P 27527.

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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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Davis, M. 2006. Why there is no such discipline as hypercomputation. Applied Mathematics and Computation 178(1): 4–7.

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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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Mario Piazza

Department of Mathematics, Universidade Nova de Lisboa, Caparica, Portugal

Gabriele Pulcini

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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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  1. Church-Turing thesis

    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 thesis about the nature of computable functions.

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    Courses 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.

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    no Turing machine exists. •According to Church-Turing thesis, no other formalism is more powerful than Turing machines. -Now, prove one of the most philosophically important theorems of the theory of computation: There is a specific problem (halting problem) that is algorithmically unsolvable.

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    characterizations. There is, however, quite a body of evidence in the thesis' favor. Let us now survey some of it. The biggest piece of evidence is simply that every known enumeration procedure generates a E set and every known decision procedure determines a ) set. It's more than that.

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  13. PDF 0.1 Extended Church-Turing Thesis

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    key insights. ˽ The term "Church-Turing thesis" is used today for numerous theses that diverge significantly from the one Alonzo Church and Alan Turing conceived in 1936. ˽ The range of algorithmic processes studied in modern computer science far transcends the range of processes a "human computer" could possibly carry out.

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