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A Natural Axiomatization of Computability and Proof of Church's Thesis

Published online by Cambridge University Press:  15 January 2014

Nachum Dershowitz  and
Yuri Gurevich
Nachum Dershowitz
Affiliation:
Microsoft Research, Redmond, WA 98052, USAand School of Computer Science, Tel Aviv University, Ramat Aviv 69978, Israel, E-mail: nachum.dershowitz@cs.tau.ac.ilURL: www.cs.tau.ac.il/~nachum
Yuri Gurevich
Affiliation:
Microsoft Research, Redmond, WA 98052, USAE-mail: gurevich@microsoft.comURL: research.microsoft.com/~gurevich
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Abstract

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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.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 14 , Issue 3 , September 2008 , pp. 299 - 350
Copyright
Copyright © Association for Symbolic Logic 2008

References

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