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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.
A natural axiomatization of Church’s thesis
, 2007
"... The Abstract State Machine Thesis asserts that every classical algorithm is behaviorally equivalent to an abstract state machine. This thesis has been shown to follow from three natural postulates about algorithmic computation. Here, we prove that augmenting those postulates with an additional requ ..."
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The Abstract State Machine Thesis asserts that every classical algorithm is behaviorally equivalent to an abstract state machine. This thesis has been shown to follow from three natural postulates about algorithmic computation. Here, we prove that augmenting those postulates with an additional requirement regarding basic operations implies Church’s Thesis, namely, 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). In particular, this gives a natural axiomatization of Church’s Thesis, as Gödel and others suggested may be possible.
Proving Church’s Thesis (Abstract)
"... The talk reflects recent joint work with Nachum Dershowitz [4]. In 1936, Church suggested that the recursive functions, which had been defined by Gödel earlier that decade, adequately capture the intuitive notion of a computable (“effectively calculable”) numerical function1 [2]. Independently Turin ..."
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The talk reflects recent joint work with Nachum Dershowitz [4]. In 1936, Church suggested that the recursive functions, which had been defined by Gödel earlier that decade, adequately capture the intuitive notion of a computable (“effectively calculable”) numerical function1 [2]. Independently Turing argued that, for stringstostrings functions, the same goal is achieved by his machines [11]. The modern form of Church’s thesis is due to Church’s student Kleene. It asserts that every computable numerical partial function is partial recursive. (Originally Church spoke of total functions.) Kleene thought that the thesis as unprovable: “Since our original notion of effective calculability... is a somewhat vague intuitive one, the thesis cannot be proved ” [7]. But he presented evidence in favor of the thesis. By far the strongest argument was Turing’s analysis [11] of “the sorts of operations which a human computer could perform, working according to preassigned instructions ” [7]. The argument convinced Gödel who thought the idea “that this really is the correct
Effectiveness ∗
, 2011
"... We describe axiomatizations of several aspects of effectiveness: effectiveness of transitions; effectiveness relative to oracles; and absolute effectiveness, as posited by the ChurchTuring Thesis. Efficiency is doing things right; effectiveness is doing the right things. —Peter F. Drucker ..."
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We describe axiomatizations of several aspects of effectiveness: effectiveness of transitions; effectiveness relative to oracles; and absolute effectiveness, as posited by the ChurchTuring Thesis. Efficiency is doing things right; effectiveness is doing the right things. —Peter F. Drucker
Seamless Composition and Integration: A Perspective on Formal Methods Research †
, 2012
"... Have formal methods in computer science come of age? While the contributions to this volume of Mathematical Structures in Computer Science attest to their importance in the design and analysis of particular software systems, their relevance to the field as a whole is far wider. In recent years, form ..."
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Have formal methods in computer science come of age? While the contributions to this volume of Mathematical Structures in Computer Science attest to their importance in the design and analysis of particular software systems, their relevance to the field as a whole is far wider. In recent years, formal methods have become more accessible and easier to use, more directly related to practical problems, and more adaptable to imperfect and/or approximate specifications in reallife applications. They are now a central component of computerscience education and research. But perhaps ‘coming of age ’ is the wrong qualification for the success of a scientific endeavor. There will always be advances in mathematical logic – a.k.a. ‘formal methods’ among computer scientists – leading to advances in reliable, safe and secure computing. Others have already noted “the unusual effectiveness of [mathematical] logic ” in computer science (Halpern et al., 2001; Davis, 1988), while some others have gone as far as tying the future of mathematical logic (or the “best of it ” according to some) to its ability to contribute to advances in computer science (Buss et al., 2001). There are many research directions that will promote the impact of formal methods
Ordinary Membrane Machines versus Other Mathematical Models of Systems Realizing Massively Parallel Computations
"... Summary. A comparison of ordinary membrane machines, understood as certain recursive families of deterministic P systems, with some other mathematical models of systems realizing massively parallel computations is discussed. These mathematical models are those which respect recursiveness of computat ..."
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Summary. A comparison of ordinary membrane machines, understood as certain recursive families of deterministic P systems, with some other mathematical models of systems realizing massively parallel computations is discussed. These mathematical models are those which respect recursiveness of computational tasks of systems, i.e., the functions to be computed are recursive functions and the decision problems correspond to recursive sets. The comparison together with open problems is summarized in the enclosed tables, where open problems are indicated by question mark “?”. 1