Bulletin of Symbolic Logic

A natural axiomatization of computability and proof of Church's Thesis

Nachum Dershowitz and Yuri Gurevich

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Church's Thesis asserts that the only numeric functions that can be calculated by effective means are the recursive ones, which are the same, extensionally, as the Turing-computable numeric functions. The Abstract State Machine Theorem states that every classical algorithm is behaviorally equivalent to an abstract state machine. This theorem presupposes three natural postulates about algorithmic computation. Here, we show that augmenting those postulates with an additional requirement regarding basic operations gives a natural axiomatization of computability and a proof of Church's Thesis, as Gödel and others suggested may be possible. In a similar way, but with a different set of basic operations, one can prove Turing's Thesis, characterizing the effective string functions, and—in particular—the effectively-computable functions on string representations of numbers.

Article information

Bull. Symbolic Logic Volume 14, Issue 3 (2008), 299-350.

First available in Project Euclid: 4 January 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03D10: Turing machines and related notions [See also 68Q05]

effective computation recursiveness computable functions Church's Thesis Turing's Thesis abstract state machines algorithms encodings


Dershowitz, Nachum; Gurevich, Yuri. A natural axiomatization of computability and proof of Church's Thesis. Bull. Symbolic Logic 14 (2008), no. 3, 299--350. doi:10.2178/bsl/1231081370. http://projecteuclid.org/euclid.bsl/1231081370.

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