A natural axiomatization of computability and proof of church's thesis

Nachum Dershowitz, Yuri Gurevich

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish
Pages (from-to)299-350
Number of pages52
JournalBulletin of Symbolic Logic
Volume14
Issue number3
DOIs
StatePublished - Sep 2008

Keywords

  • Abstract state machines
  • Algorithms
  • Church's thesis
  • Computable functions
  • Effective computation
  • Encodings
  • Recursiveness
  • Turing's thesis

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