David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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Bulletin of Symbolic Logic 14 (3):299-350 (2008)
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
|Keywords||effective computation recursiveness computable functions Church's Thesis Turing's Thesis abstract state machines algorithms encodings|
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Nir Fresco (2013). Information Processing as an Account of Concrete Digital Computation. Philosophy and Technology 26 (1):31-60.
Hector Zenil (2013). What Is Nature-Like Computation? A Behavioural Approach and a Notion of Programmability. Philosophy and Technology (3):1-23.
Jörgen Sjögren (2010). A Note on the Relation Between Formal and Informal Proof. Acta Analytica 25 (4):447-458.
Michael Rescorla (2012). Copeland and Proudfoot on Computability. Studies in History and Philosophy of Science Part A 43 (1):199-202.
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