## A NATURAL AXIOMATIZATION OF COMPUTABILITY AND PROOF OF CHURCH’S THESIS

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@MISC{Dershowitz_anatural,

author = {Nachum Dershowitz and Yuri Gurevich},

title = {A NATURAL AXIOMATIZATION OF COMPUTABILITY AND PROOF OF CHURCH’S THESIS},

year = {}

}

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### Abstract

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 Turingcomputable 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.

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Citation Context ...e will make use of the following variant of Corollary 4.6, which uses this slightly more general notion of recursive function and which allows for arbitrarily many recursive oracles: 30 See [3] (also =-=[86]-=-) for the fundamental weakness of a two-counter machine, as compared to a machine with three counters or more.NATURAL AXIOMATIZATION OF COMPUTABILITY 329 Corollary 4.10. Every partial function comput... |

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Citation Context ... of algorithm, such as interactive and distributed computations. To capture such non-sequential processes and non-classical algorithms, additional postulates are required. For these developments, see =-=[9, 10, 11, 12, 13, 15, 16, 37]-=-. We also do not address the question of the computational capabilities of the human mind, what Shagrir [87, p. 223] refers to as “The Human version of the Church–Turing Thesis” (more generally called... |

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4 |
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Citation Context ... be taken into account is finite. The reasons for 18 The idea of providing the basic operations of recursively-defined functions over arbitrary domains by means of a logical structure also appears in =-=[62]-=-.NATURAL AXIOMATIZATION OF COMPUTABILITY 319 this are of the same character as those which restrict the number of symbols. This finiteness requirement is expressed in more general terms by Kolmogorov... |

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4 |
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3 |
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Citation Context ...r conviction of the adequacy of these concepts for expressing the popular meaning of ‘effective calculability’.” 11 Levin was Kolmogorov’s student.306 NACHUM DERSHOWITZ AND YURI GUREVICH (along with =-=[88, 90, 60, 94, 6]-=-) argued in favor of the possibility of axiomatizing effectivity. Kreisel described the discovery of “evident axioms about constructive functions” as “one of the really important open problems” [58] a... |

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