Bulletin of Symbolic Logic

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

Nachum Dershowitz and Yuri Gurevich

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Abstract

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

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

Dates
First available: 4 January 2009

Permanent link to this document
http://projecteuclid.org/euclid.bsl/1231081370

Digital Object Identifier
doi:10.2178/bsl/1231081370

Mathematical Reviews number (MathSciNet)
MR2440596

Zentralblatt MATH identifier
05532698

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

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

Citation

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


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