Gurevich: "... Turing's informal argument in favor of his thesis justifies a stronger thesis: every algorithm can be simulated by a Turing machine ... according to Savage, an algorithm is a computational process defined by a Turing machine ".

In computability theory, the Church – Turing thesis ( also known as the Turing-Church thesis, the Church – Turing conjecture, Church's thesis, Church's conjecture, and Turing's thesis ) is a combined hypothesis (" thesis ") about the nature of functions whose values are effectively calculable ; or, in more modern terms, functions whose values are algorithmically computable.

Turing's " definitions " given in a footnote in his 1939 Ph. D. thesis Systems of Logic Based on Ordinals, supervised by Church, are virtually the same:

The same thesis is implicit in Turing's description of computing machines (< sup > 23 </ sup >).

A more mathematically oriented definition with a similar " universal " nature was introduced by Alonzo Church, whose work on lambda calculus intertwined with Turing's in a formal theory of computation known as the Church – Turing thesis.

These results led Stephen Kleene ( 1952 ) to coin the two names " Church's thesis " ( Kleene 1952: 300 ) and " Turing's Thesis " ( p. 376 ).

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.

The concept of ASMs is due to Yuri Gurevich, who first proposed it in the mid-1980s as a way of improving on Turing's thesis that every algorithm is simulated by an appropriate Turing machine.

It was not a Turing complete computer, which distinguishes it from more general machines, like contemporary Konrad Zuse's Z3 ( 1941 ), or later machines like the 1946 ENIAC, 1949 EDVAC, the University of Manchester designs, or Alan Turing's post-War designs at NPL and elsewhere.

The proof of this fact relies on an algorithm which, given the first n digits of Ω, solves Turing's halting problem for programs of length up to n. Since the halting problem is undecidable, Ω can not be computed.

* Omega and why maths has no TOEs article based on one written by Gregory Chaitin which appeared in the August 2004 edition of Mathematics Today, on the occasion of the 50th anniversary of Alan Turing's death.

With knowledge of Alan Turing's theoretical ' universal computing machine ' John von Neumann defined an architecture which uses the same memory both to store programs and data: virtually all contemporary computers use this architecture ( or some variant ).

This result preceded Alan Turing's famous work on the halting problem, which also demonstrated the existence of a problem unsolvable by mechanical means.

The notoriety of Turing's proposed test stimulated great interest in Joseph Weizenbaum's program ELIZA, published in 1966, which seemed to be able to fool users into believing that they were conversing with a real human.

* Penrose uses Gödel's incompleteness theorem ( which states that there are mathematical truths which can never be proven in a sufficiently strong mathematical system ; any sufficiently strong system of axioms will also be incomplete ) and Turing's halting problem ( which states that there are some things which are inherently non-computable ) as evidence for his position.

Fascinated by Alan Turing's imitation game, and considering creating a system himself to pass it, Loebner realised that even if he were to succeed in developing a computer that could pass the Turing test, no avenue existed in which to prove it.

Three of its most famous publications, arguably, are Lewis Carroll's " What the Tortoise Said to Achilles " ( 1895 ), Bertrand Russell's " On Denoting " ( 1905 ), and Alan Turing's " Computing Machinery and Intelligence " ( 1950 ), in which he first proposed the Turing test.

Turing's method of accumulating a score of a number of decibans allows the calculation of which of these situations is most likely to represent messages in depth.

In 1951, Alan Turing designed a primitive chess playing program, which assigned values for material and mobility ; the program " played " chess based on Turing's manual calculations.

Another relevant addition would be the discussions concerning Interactive computation, especially those related to the meaning and use of Turing's model ( Church-Turing, TM, etc.

Hartree recommended the grant but Darwin opposed it on the grounds that Turing's ACE at NPL would be sufficient to serve the needs of the country.

The notion of the Kolmogorov complexity can be used to state and prove impossibility results akin to Gödel's incompleteness theorem and Turing's halting problem.

It also has been suggested that Alan Turing's recommendation of imitating not a human adult consciousness, but a human child consciousness, should be taken seriously.

Generally, there is also a possibility that no value may be computable or observable in a finite time ( see Turing's halting problem ).

: In late 1936 Alan Turing's paper ( also proving that the Entscheidungsproblem is unsolvable ) was deliverd orally, but had not yet appeared in print.

Turing's argument is as follows.

Turing's paper is # 3 in this volume.

Turing's paper is in this volume.

No oracle machine is capable of solving its own halting problem ( a variation of Turing's proof applies ).

Some have taken Turing's question to have been " Can a computer, communicating over a teleprinter, fool a person into believing it is human?

Turing's reply is now known as the " other minds reply ".

' This formulation is both better defined and more physical than Turing's own way of expressing it.

It is unknown whether Stibitz and / or McPhail had any influence on this work of Turing's ; McPhail's implication is that Turing's " about a possible war with Germany " ( p. 138 ) caused him to become interested in cryptanalysis, and this interest led to discussions with McPhail, and these discussions led to the relay-multiplier experiments ( the pertinent part of McPhail's letter to Hodges is quoted in Hodges p. 138 ).

One classic work in this area is Alan Turing's paper on morphogenesis entitled The Chemical Basis of Morphogenesis, published in 1952 in the Philosophical Transactions of the Royal Society.

However, unlike the lambda calculus, Turing's code does not serve well as a basis for higher-level languages — its principal use is in rigorous analyses of algorithmic complexity.

Turing is shown holding an apple — a symbol classically used to represent forbidden love, as well as being the fruit of the tree of knowledge, the object that inspired Isaac Newton's theory of gravitation, and the means of Turing's own death.

His ' Model III ' was under way in the New York building at the time of Alan Turing's stay there, but it had not drawn his attention.

Turing adds another definition, Rosser equates all three: Within just a short time, Turing's 1936 – 37 paper " On Computable Numbers, with an Application to the Entscheidungsproblem " appeared.

Subsequent formalizations were framed as attempts to define " effective calculability " or " effective method "; those formalizations included the Gödel – Herbrand – Kleene recursive functions of 1930, 1934 and 1935, Alonzo Church's lambda calculus of 1936, Emil Post's " Formulation 1 " of 1936, and Alan Turing's Turing machines of 1936 – 7 and 1939.

Turing's homosexuality resulted in a criminal prosecution in 1952, when homosexual acts were still illegal in the United Kingdom.

His father's civil service commission was still active, and during Turing's childhood years his parents travelled between Hastings in England and India, leaving their two sons to stay with a retired Army couple.

In the literature concerning artificial intelligence ( AI ), Searle's essay has been second only to Turing's in the volume of debate it has generated.

On the other hand, Emil Post's 1936 paper had appeared and was certified independent of Turing's work.

Alan M. Turing's biography.