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A NATURAL AXIOMATIZATION OF COMPUTABILITY AND PROOF OF CHURCH’S THESIS
"... 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 e ..."
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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 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 effectivelycomputable functions on string representations of numbers.
What Is an Algorithm?
, 2000
"... Machines and Recursive Definitions 2.1 Abstract Machines The bestknown model of mechanical computation is (still) the first, introduced by Turing [18], and after half a century of study, few doubt the truth of the fundamental ChurchTuring Thesis : A function f : N # N on the natural numbers (o ..."
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Cited by 30 (3 self)
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Machines and Recursive Definitions 2.1 Abstract Machines The bestknown model of mechanical computation is (still) the first, introduced by Turing [18], and after half a century of study, few doubt the truth of the fundamental ChurchTuring Thesis : A function f : N # N on the natural numbers (or, more generally, on strings from a finite alphabet) is computable in principle exactly when it can be computed by a Turing Machine. The ChurchTuring Thesis grounds proofs of undecidability and it is essential for the most important applications of logic. On the other hand, it cannot be argued seriously that Turing machines model faithfully all algorithms on the natural numbers. If, for example, we code the input n in binary (rather than unary) notation, then the time needed for the computation of f(n) can sometimes be considerably shortened; and if we let the machine use two tapes rather than one, then (in some cases) we may gain a quadratic speedup of the computation, see [8]. This mea...
The problem of learning the semantics of quanti ers
 Logic, Language, and Computation, 6th International Tbilisi Symposium on Logic, Language, and Computation, TbiLLC 2005, volume 4363 of Lecture Notes in Computer Science
, 2007
"... This paper is concerned with a possible mechanism for learning the meanings of quantiers in natural language. The meaning of a natural language construction is identied with a procedure for recognizing its extension. Therefore, acquisition of natural language quantiers is supposed to consist in col ..."
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This paper is concerned with a possible mechanism for learning the meanings of quantiers in natural language. The meaning of a natural language construction is identied with a procedure for recognizing its extension. Therefore, acquisition of natural language quantiers is supposed to consist in collecting procedures for computing their denotations. A method for encoding classes of nite models corresponding to given quanti ers is shown. The class of nite models is represented by appropriate languages. Some facts describing dependencies between classes of quanti ers and classes of devices are presented. In the second part of the paper examples of syntaxlearning models are shown. According to these models new results in quantier learning are presented. Finally, the question of the adequacy of syntaxlearning tools for describing the process of semantic learning is stated. 1