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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.
GUALTIERO PICCININI COMPUTATIONALISM, THE CHURCH–TURING THESIS, AND THE CHURCH–TURING FALLACY
"... ABSTRACT. The Church–Turing Thesis (CTT) is often employed in arguments for computationalism. I scrutinize the most prominent of such arguments in light of recent work on CTT and argue that they are unsound. Although CTT does nothing to support computationalism, it is not irrelevant to it. By elimin ..."
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ABSTRACT. The Church–Turing Thesis (CTT) is often employed in arguments for computationalism. I scrutinize the most prominent of such arguments in light of recent work on CTT and argue that they are unsound. Although CTT does nothing to support computationalism, it is not irrelevant to it. By eliminating misunderstandings about the relationship between CTT and computationalism, we deepen our appreciation of computationalism as an empirical hypothesis. Computationalism, or the Computational Theory of Mind, is the view that mental capacities are explained by inner computations. In the case of human beings, computationalists typically assume that inner computations are realized by neural processes; I will borrow a term from current neuroscience and refer to them as neural computations. 1 Typically, computationalists also maintain that neural computations are Turingcomputable, that is, computable by Turing Machines (TMs). The Church–Turing thesis (CTT) says that a function is computable, in the intuitive sense, if and only if it is Turingcomputable (Church 1936; Turing 1936–7). CTT entails that TMs, and any formalism equivalent to TMs, capture the intuitive notion of computation. In other words, according to CTT, if a function is computable in the intuitive sense, then there is a TM that computes it (or equivalently, it is Turingcomputable). 2 This applies to neural computations as well. Suppose that, as computationalism maintains, neural activity is computation, and suppose that the functions computed by neural mechanisms are computable in the intuitive sense. Then, by CTT, for any function computed by a neural mechanism, there is a TM that computes the same function. This is a legitimate argument for a technical version of computationalism, according to which neural computations are Turingcomputable, from a generic one, according to which neural processes are computations in the intuitive sense, via CTT. But should we believe CTT? The initial proponents of CTT, and most of CTT’s supporters, appeal to a number of intuitive