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. It states that a function on the natural numbers can be calculated by an effective method if and only if it is computable by a Turing machine. The thesis is named after American mathematician Alonzo Church and the British mathematician Alan Turing. Before the precise definition of computable function, mathematicians often used the informal term effectively calculable to describe functions that are computable by paper-and-pencil methods. In the 1930s, several independent attempts were made to formalize the notion of computability:

  • In 1933, Kurt Gödel, with Jacques Herbrand, formalized the definition of the class of general recursive functions: the smallest class of functions (with arbitrarily many arguments) that is closed under composition, recursion, and minimization, and includes zero, successor, and all projections.
  • 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. A function on the natural numbers is called λ-computable if the corresponding function on the Church numerals can be represented by a term of the λ-calculus.
  • Also in 1936, before learning of Church's work,[6] 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. Given a suitable encoding of the natural numbers as sequences of symbols, a function on the natural numbers is called Turing computable if some Turing machine computes the corresponding function on encoded natural numbers.

Church,[7] Kleene,[8] and Turing[9][11] proved that these three formally defined classes of computable functions coincide: a function is λ-computable if and only if it is Turing computable, and if and only if it is general recursive. This has led mathematicians and computer scientists to believe that the concept of computability is accurately characterized by these three equivalent processes. Other formal attempts to characterize computability have subsequently strengthened this belief (see below).

On the other hand, the Church–Turing thesis states that the above three formally defined classes of computable functions coincide with the informal notion of an effectively calculable function. Although the thesis has near-universal acceptance, it cannot be formally proven, as the concept of effective calculability is only informally defined.

Since its inception, variations on the original thesis have arisen, including statements about what can physically be realized by a computer in our universe (physical Church-Turing thesis) and what can be efficiently computed (Church–Turing thesis (complexity theory)). These variations are not due to Church or Turing, but arise from later work in complexity theory and digital physics. The thesis also has implications for the philosophy of mind (see below).

  1. ^ Soare, Robert I. (2009-09-01). "Turing oracle machines, online computing, and three displacements in computability theory". Annals of Pure and Applied Logic. Computation and Logic in the Real World: CiE 2007. 160 (3): 368–399. doi:10.1016/j.apal.2009.01.008. ISSN 0168-0072.
  2. ^ Conrad, Michael (May 1985). "On design principles for a molecular computer". Communications of the ACM. 28 (5): 464–480. doi:10.1145/3532.3533.
  3. ^ Correspondence between Max Newman and Church in Alonzo Church papers
  4. ^ Turing, Alan (2004). The essential Turing : seminal writings in computing, logic, philosophy, artificial intelligence, and artificial life, plus the secrets of Enigma (PDF). Oxford: Clarendon Press. p. 44. ISBN 9780198250791. Retrieved 2021-12-06.
  5. ^ Cite error: The named reference Soare was invoked but never defined (see the help page).
  6. ^ Church's paper was presented to the American Mathematical Society on 19 April 1935 and published on 15 April 1936. Turing, who had made substantial progress in writing up his own results, was disappointed to learn of Church's proof upon its publication.[3][4] Turing quickly completed his paper and rushed it to publication; it was received by the Proceedings of the London Mathematical Society on 28 May 1936, read on 12 November 1936, and published in series 2, volume 42 (1936–1937); it appeared in two sections: in Part 3 (pages 230–240), issued on 30 November 1936 and in Part 4 (pages 241–265), issued on 23 December 1936; Turing added corrections in volume 43 (1937), pp. 544–546.[5]: 45 
  7. ^ Church 1936a
  8. ^ Kleene 1936
  9. ^ Turing 1937a
  10. ^ Kleene 1936
  11. ^ Turing 1937b. Proof outline on p. 153: [10]
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