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## Church’s Thesis for Turing Machine

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.

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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## What is The Church-Turing Thesis in TOC?

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

## The extended Church-Turing thesis for decision problems

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- 1. Church Turing Thesis Prepared by : Sharma Hemant [email protected]
- 2. Turing Machine Alan Turing has created Turing Machine Model. This model has computing of general purpose computer. The Turing Machine is a collection of following components: M = (Q, ∑, Г, δ, q0, Δ or B, F) 1. Q is a finite set of states. 2. Г is finite set of external symbols. 3. ∑ is a finite set of input symbols. 4. Δ or b or B Є Г is a blank symbol majorly used as end marker for input. 5. δ is a transition or a mapping function.
- 3. Turing Machine A Turing Machine (TM) is a theoretical symbol manipulating device. A TM can simulate any computer algorithm (this is a simple formation of what came to be known as the Church-Turing Thesis, a version of Church’s Thesis) The combination of the current symbol and the state determines what the device does next. TMs are useful for simulating and understanding how computer CPUs work.
- 4. Church Turing Thesis In 1936, Alonzo Church created a method for defining functions called the λ-calculus. Within λ-calculus, he defined an encoding of the natural numbers called the Church numerals. Also in 1936, before learning of Church's work, Alan Turing created a theoretical model for machines, now called Turing machines, that could carry out calculations from inputs by manipulating symbols on a tape.
- 5. Church Turing Thesis A Turing machine is an abstract representation of a computing device. It is more like a computer hardware than a computer software. LCMs [Logical Computing Machines: Turing’s expression for Turing machines] were first proposed by Alan Turing, in an attempt to give a mathematically precise definition of "algorithm" or "mechanical procedure".
- 6. Church Turing Thesis The Church-Turing thesis concerns an effective or mechanical method in logic and mathematics. A method, M, is called ‘effective’ or ‘mechanical’ just in case: 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, always produce the desired result in a finite number of steps;
- 7. Church Turing Thesis M can (in practice or in principle) be carried out by a human being unaided by any machinery except for paper and pencil; M demands no insight or ingenuity on the part of the human being carrying it out. They gave an hypothesis which means proposing certain facts. The Church’s hypothesis or Church’s turing thesis can be stated as:
- 8. Church Turing Thesis The assumption that the intuitive notion of computable functions can be identified with partial recursive functions. This statement was first formulated by Alonzo Church in the 1930s and is usually referred to as Church’s thesis, or the Church-Turing thesis. However, this hypothesis cannot be proved.
- 9. Church Turing Thesis The computability of recursive functions is based on following assumptions: 1. Each elementary function is computable. 2. Let f be the computable function & g be the another function which can be obtained by applying the elementary operation to f, then g becomes a computable function. 3. Any function becomes computable if it is obtained by rule 1 & 2.
- 10. Example Construct a TM for language consisting of strings having any number of 0’s and only even numbers of 1’s over the input set ∑ = {0,1} . The FSM can be draw as: 𝑞0 𝑞1 0 0 1 1
- 11. Example Now the same idea can be used to draw TM. 𝑞1 (0,0, R) (0,0, R) (1,1, 𝑅) (1,1, 𝑅) 𝑠𝑡𝑎𝑟𝑡 ℎ𝑎𝑙𝑡 (Δ, Δ, 𝐿)
- 12. Example Let us simulate the above TM for the input 110101 which has even number of 1’s. Thus this input is accepted by TM.

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## What is the Church-Turing thesis?

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We describe a Turing machine as an abstract representation of a computing device.

## Church-Turing thesis

Alan Turing gave the idea for logical computing machines that worked on Turing expressions.

## Requirements for the Church-Turing thesis

This defined method M M M should pass the following:

- It shouldn't require any complex processing requirements.
- We obtain output after performing a finite number of steps.
- We can implement the method M M M in the real world.
- The number of instructions for M M M should be finite.

The assertion that partial recursive functions can be used to identify the intuitive concept of computable functions.

Despite this hypothesis, we can't prove this claim.

We describe this thesis under two main categories:

- Church-Turing thesis for decidability problems: A decidability problem A problem that is either true or false. can be solved effectively if there exists a Turing machine that halts for all of its input strings and calculates the solution.
- Extended Church-Turing thesis for decidability problems: A decidability problem is partially solvable if there exists a Turing machine that accepts the elements of the problem whose answer is "yes."

## Proof for recursive functions

We can compute recursive functions by considering the following assumptions:

- All functions must be computable.
- Assume a computable function f f f . After performing elementary operations, the function will transform into a new function g g g . g g g is automatically a computable function.

## Applications of the thesis

This thesis finds its applications in many calculable and computational fields. Some of these are:

- Lambda calculus
- Single and multiple tape Turing machines
- Counter machine model
- Register machine, a machine similar to the computer
- Markov algorithms
- Combinatory logic
- Pointer machines

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Regular expression, turing machine.

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## Artificial Intelligence Models

The basic idea, theory, meet practice.

There are different models of artificial intelligence.

- Artificial intelligence models are the tools and algorithms used to train computers to process and analyze data – just as humans do.
- Machine learning is a broad category that falls under the artificial intelligence model label, in which computers are taught to think by themselves and develop their own algorithms after processing vast amounts of data.
- Other artificial intelligence models need an algorithm to be programmed into the computer and will learn to adjust the algorithm based on experience.
- Lastly, there are also models that do not have the ability to learn on their own at all – they only function according to the preprogrammed algorithm and need human input. 1

## Consequences

## Controversies

There are quite a few ethical controversies when it comes to artificial intelligence models.

## Where are artificial intelligence models used?

## Related Content

Combining AI and Behavioral Science Responsibly

- What is an AI model? Here’s what you need to know . (2021, July 6). viso.ai. https://viso.ai/deep-learning/ml-ai-models/
- Reeves, S. (2020, August 10). 8 Helpful Everyday Examples of Artificial Intelligence . IoT For All. https://www.iotforall.com/8-helpful-everyday-examples-of-artificial-intelligence
- Marr, B. (2017, July 25). 28 Best Quotes About Artificial Intelligence . Forbes. https://www.forbes.com/sites/bernardmarr/2017/07/25/28-best-quotes-about-artificial-intelligence/?sh=115b1a8a4a6f
- Deep learning . (2021, October 1). The Decision Lab. https://thedecisionlab.com/reference-guide/computer-science/deep-learning/
- What is the AI effect, and is it set to happen again? (2020, December 3). ThinkAutomation. https://www.thinkautomation.com/bots-and-ai/what-is-the-ai-effect-and-is-it-set-to-happen-again/
- Machine learning . (2021, October 7). The Decision Lab. https://thedecisionlab.com/reference-guide/computer-science/machine-learning/
- Mullins, R. (2012). Raspberry Pi . Department of Computer Science and Technology. https://www.cl.cam.ac.uk/projects/raspberrypi/tutorials/turing-machine/one.html
- Soha, G. (May 24). What is an AI model? Reveal Brainspace. Retrieved November 1, 2021, from https://resource.revealdata.com/en/blog/what-is-an-ai-model
- Rustagi, D. (2020, May 20). Church’s Thesis for Turing Machine . GeeksforGeeks. https://www.geeksforgeeks.org/churchs-thesis-for-turing-machine/
- Piccinini, G. (2004). The first computational theory of mind and brain: A close look at Mcculloch and Pitts’s “Logical calculus of ideas immanent in nervous activity”. Synthese , 141 (2), 175-215. https://doi.org/10.1023/b:synt.0000043018.52445.3e
- Anyoha, R. (2017, August 28). The History of Artificial Intelligence. Science in the News . https://sitn.hms.harvard.edu/flash/2017/history-artificial-intelligence/
- McCrea, N. (2014, August 8). An Introduction to Machine Learning Theory and Its Applications: A Visual Tutorial with Examples . Toptal Engineering Blog. https://www.toptal.com/machine-learning/machine-learning-theory-an-introductory-primer
- Press, G. (2016, December 30). A Very Short History Of Artificial Intelligence (AI) . Forbes. https://www.forbes.com/sites/gilpress/2016/12/30/a-very-short-history-of-artificial-intelligence-ai/?sh=398b8256fba2
- McKinsey. (2018, April 25). The real-world potential and limitations of artificial intelligence. McKinsey Podcast [Audio podcast episode]. https://www.mckinsey.com/featured-insights/artificial-intelligence/the-real-world-potential-and-limitations-of-artificial-intelligence
- Artificial Intelligence: examples of ethical dilemmas . (2020, October 2). UNESCO. Retrieved November 1, 2021, from https://en.unesco.org/artificial-intelligence/ethics/cases
- Najibi, A. (2020, October 24). Racial Discrimination in Face Recognition Technology. Science in the News . https://sitn.hms.harvard.edu/flash/2020/racial-discrimination-in-face-recognition-technology/
- Cruz, J. A., & Wishart, D. S. (2006). Applications of machine learning in cancer prediction and prognosis. Cancer Informatics , 2 , 117693510600200. https://doi.org/10.1177/117693510600200030

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## The Church-Turing Thesis

## Note on terminology

- 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 except paper and pencil;
- M demands no insight, intuition, or ingenuity, on the part of the human being carrying out the method.

This is sufficiently well established that it is now agreed amongst logicians that “calculable by means of an L.C.M.” is the correct accurate rendering of such phrases. (Ibid.)

Although the subject of this paper is ostensibly the computable numbers, it is almost equally easy to define and investigate computable functions … I have chosen the computable numbers for explicit treatment as involving the least cumbrous technique. (1936: 58)

- any number whose decimal representation consists of a finite number of digits (e.g., 109, 1.142)
- all rational numbers, such as one-third, two-sevenths, etc.
- some irrational real numbers, such as π and e.

[T]he “computable numbers” include all numbers which would naturally be regarded as computable. (Turing 1936: 58) It is my contention that these operations [the operations of an L.C.M.] include all those which are used in the computation of a number. (Turing 1936: 60)

## Church’s contribution

Church stated the Entscheidungsproblem more generally:

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)

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 λ-definable function of positive integers). (Church 1936a: 356)

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

This, then, is the ‘working hypothesis’ that, in effect, Church proposed:

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)

Some prefer the name Turing-Church thesis .

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

According to a November 29, 1935, letter from Church to me, Gödel “regarded as thoroughly unsatisfactory” Church’s proposal to use λ-definability as a definition of effective calculability. … It seems that only after Turing’s formulation appeared did Gödel accept Church’s thesis. (Kleene 1981: 59, 61)

We had not perceived the sharp concept of mechanical procedures sharply before Turing, who brought us to the right perspective. (Quoted in Wang 1974: 85)

The resulting definition of the concept of mechanical by the sharp concept of “performable by a Turing machine” is both correct and unique. … Moreover it is absolutely impossible that anybody who understands the question and knows Turing’s definition should decide for a different concept. (Quoted in Wang 1996: 203)

- (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.

largely due to the fact that with this concept one has for the first time succeeded in giving an absolute definition of an interesting epistemological notion, i.e., one not depending on the formalism chosen. (Gödel 1946: 150)

- Instead of using two-dimensional sheets of paper, the computer can do his or her work on paper tape of the same kind that a Turing machine uses—a one-dimensional tape, divided into squares.
- The computer is not able to recognize more than a finite number of different types of symbol.
- The computer is not able to observe an unlimited number of tape-squares all at once—if he or she wishes to observe more squares than can be taken in at one time, then successive observations of the tape must be made.
- When the computer makes a successive observation in order to view more squares, none of the newly observed squares will be more than a certain fixed distance away from the nearest previously observed square. In other words, successive observations do not involve unbounded leaps along the tape.
- When the computer makes changes to the contents of the tape (e.g., by deleting the symbol written in a particular square and replacing it by a different symbol), no more than one square can be altered at once. If the computer wishes to alter, say, 100 squares then he or she performs 100 successive operations.
- The computer’s behavior at any moment is determined by the symbols that he or she is observing and his or her ‘state of mind’ at that moment; and the number of ‘states of mind’ that need to be taken into account when describing the computer’s behavior is finite. (Turing noted that reference to the computer’s states of mind can be avoided by talking instead about configurations of symbols, these being “a more definite and physical counterpart” of states of mind.)

Turing’s argument II hinges on a proposition we may call

Argument II is the subject of the next section.

Since our original notion of effective calculability of a function … is a somewhat vague intuitive one, the thesis cannot be proved. … While we cannot prove Church’s thesis, since its role is to delimit precisely an hitherto vaguely conceived totality, we require evidence …. (Kleene 1952: 318)

[A] computation is a special form of mathematical argument. 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 . (Kripke 2013: 80)

Now, applying Gödel’s completeness theorem to this yields in turn:

every (human) computation can be done by Turing machine.

In a similar way, but with a different set of basic operations, one can prove Turing’s Thesis, … . (Dershowitz and Gurevich 2008: 299)

fundamentally, appeals to intuition, and for this reason rather unsatisfactory mathematically. (Turing 1936: 74)

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. (Turing 1954: 588)

is a phrase which, like many others e.g., ‘vegetable’ one understands well enough in the ordinary way. But one can have difficulties when speaking to greengrocers or microbiologists or when playing ‘twenty questions’. 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? … The same sort of difficulty arises about question c) above [ Is there a systematic method by which I can answer such-and-such questions ?]. An ordinary sort of acquaintance with the meaning of the phrase ‘systematic method’ won’t do, because one has got to be able to say quite clearly about any kind of method that might be proposed whether it is allowable or not. (Turing in Copeland 2004b: 590)

## 2. Misunderstandings of the Thesis

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. (Richard Gregory writing in his 1987: 784)

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)

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)

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) 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) Church-Turing thesis: If there is a well defined procedure for manipulating symbols, then a Turing machine can be designed to do the procedure. (Henry 1993: 149) [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:

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)

[T]he Physical Church-Turing Thesis … is the conjecture that whatever physical computing device (in the broader sense) or physical thought-experiment will be designed by any future civilization, it will always be simulateable by a Turing machine. (Andréka, Németi, and Németi 2009: 500)

These writers go on to assert that what they call the ‘Physical Church-Turing Thesis’

was formulated and generally accepted in the 1930s. (Ibid.)

## 2.2 The maximality thesis

All computable functions are computable by Turing machine.

[T]he ‘computable numbers’ include all numbers which would naturally be regarded as computable. (Turing 1936: 58)

Entailments such as the following are sometimes offered in the literature:

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

No possible computing machine can generate a function that the universal Turing machine cannot.

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

## 1 st counterexample to the stronger form of the thesis: Extended Turing Machines

## 2 nd counterexample to the stronger form of the thesis: Accelerating Turing Machines

So, again, ATMs form counterexamples to the stronger form of the maximality thesis.

all attempts to … formulate … general notions of mechanism … lead to classes of machines that are equivalent in that they encompass in toto exactly the same set of input-output functions;

and, he says, the various equivalent analyses constitute a

large zoo of different formulations of maximal classes of machines. (Newell 1980: 150)

## 2.3 Some consequences of misunderstanding the Church-Turing thesis

There are certain behaviours that are ‘uncomputable’—behaviours for which no formal specification can be given for a machine that will exhibit that behaviour. The classic example of this sort of limitation is Turing’s famous Halting Problem : can we give a formal specification for a machine which, when provided with the description of any other machine together with its initial state, will … determine whether or not that machine will reach its halt state? Turing proved that no such machine can be specified. (Langton 1989: 12)

we can depend on there being a Turing machine that captures the functional relations of the brain,

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. (Sam Guttenplan writing in his 1994: 595)

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)

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) 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. (Johnson-Laird 1987: 252)

As previously mentioned, Churchland and Churchland say that Turing’s

results entail … that a standard digital computer, given only the right program, a large enough memory and sufficient time, can … display any systematic pattern of responses to the environment whatsoever. (1990: 26)

## 3. Some Key Remarks by Turing

Turing prefaced his first description of a Turing machine with the words:

We may compare a man in the process of computing a … number to a machine. (Turing 1936: 59)

Wittgenstein put this point in a striking way:

Turing’s ‘Machines’. These machines are humans who calculate. (Wittgenstein 1947 [1980]: 1096.)

It is a point that Turing was to emphasize, in various forms, again and again. For example:

A man provided with paper, pencil, and rubber, and subject to strict discipline, is in effect a universal machine. (Turing 1948: 416)

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 [the Automatic Computing Engine] … [T]he ACE will do the work of about 10,000 computers … Computers will still be employed on small calculations … (Turing 1947: 387, 391)

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: 444)

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 1945: 386)

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 c 1950: 1)

The expression ‘machine process’ of course means one which could be carried out by the type of machine I was considering [in “On Computable Numbers”]. (Turing 1947: 378–9)

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: 414)

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