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  The Church-Turing Thesis, Turing Machines

The Church-Turing thesis:

Turing machines:

Alan M. Turing: On computable numbers, with application to the Entscheidungs problem, Computability and λ-definability,

The other models of computation, like:

a) Effective calculability, λ-calculus:

Alonzo Church: The unsolvable trouble of elementary number theory, and a note on the Entscheidungs problem

b) Canonical systems:

Emil Post: The formal reductions of general combinatorial decision problem,

c) Recursive functions:

Stephen C. Kleene: Common recursive functions of the natural numbers and λ-definability and recursiveness.

d) Markov algorithms:

A.A. Markov: The theory of algorithms, Doklady Akademii Nauk SSSR and Trudy Math.

The above turn out to be mathematically equal: Any one of such models of computation can simulate the other. This equivalence strengthens the argument that all of them capture in a rigorous way the intuitive concept of ‘algorithm’.

The question of efficient computability was suddenly increased in 30s and investigated by many logicians by using various formalisms since of the crisis in the foundation of mathematics generated by Godel’s incompleteness theorem: Any system of logic powerful adequate to state elementary arithmetic comprises true statements which can’t be proven in that system.

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