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A NATURAL AXIOMATIZATION OF COMPUTABILITY AND PROOF OF CHURCH’S THESIS
, 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 Turingcomputable numeric functions. The Abstract State Machine Theorem states that every classical algorithm is behaviorally equivalent ..."
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Cited by 42 (17 self)
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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.
Quantum Information Theory and the Foundations of Quantum Mechanics
, 2004
"... This thesis is a contribution to the debate on the implications of quantum information theory for the foundational problems of quantum mechanics. In Part I an attempt is made to shed some light on the nature of information and quantum information theory. It is emphasized that the everyday notion of ..."
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Cited by 28 (7 self)
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This thesis is a contribution to the debate on the implications of quantum information theory for the foundational problems of quantum mechanics. In Part I an attempt is made to shed some light on the nature of information and quantum information theory. It is emphasized that the everyday notion of information is to be firmly distinguished from the technical notions arising in information theory; noun, hence does not refer to a particular or substance. The popular claim ‘Information is Physical ’ is assessed and it is argued that this proposition faces a destructive dilemma. Accordingly, the slogan may not be understood as an ontological claim, but at best, as a methodological one. A novel argument is provided against Dretske’s (1981) attempt to base a semantic notion of information on ideas from information theory. The function of various measures of information content for quantum systems is explored and the applicability of the Shannon information in the quantum context maintained against the challenge of Brukner and Zeilinger (2001). The phenomenon of quantum teleportation is then explored as a case study serving to emphasize the value of
Computational Modeling vs. Computational Explanation: Is Everything a Turing Machine, and Does It Matter to the Philosophy of Mind?” Australasian
 Philosophy of Science. Piccinini, G. (forthcoming b). “Computation without Representation,” Philosophical
, 2007
"... According to pancomputationalism, everything is a computing system. In this paper, I distinguish between different varieties of pancomputationalism. I find that although some varieties are more plausible than others, only the strongest variety is relevant to the philosophy of mind, but only the most ..."
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Cited by 14 (5 self)
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According to pancomputationalism, everything is a computing system. In this paper, I distinguish between different varieties of pancomputationalism. I find that although some varieties are more plausible than others, only the strongest variety is relevant to the philosophy of mind, but only the most trivial varieties are true. As a side effect of this exercise, I offer a clarified distinction between computational modelling and computational explanation. I. Pancomputationalism and the Computational Theory of Mind The main target of this paper is pancomputationalism, according to which everything is a computing system. I have encountered two peculiar responses to pancomputationalism: some philosophers find it obviously false, too silly to be worth refuting; others find it obviously true, too trivial to require a defence. Neither camp sees the need for this paper. But neither camp seems aware of the other camp. The existence of both camps, together with continuing appeals to pancomputationalism in the literature, compel me to analyse the matter more closely. In this paper, I distinguish between different varieties of pancomputationalism. I find that although some are more plausible than others, only the strongest variety is relevant to the philosophy of mind, but only the most trivial varieties are true. As a side effect of this exercise, I offer a clarified distinction between computational modelling and computational explanation. The canonical formulation of pancomputationalism is due to Hilary Putnam: ‘everything is a Probabilistic Automaton under some Description’ [Putnam 1999: 31; ‘probabilistic automaton ’ is Putnam’s term for
The Interactive Nature of Computing: Refuting the Strong ChurchTuring Thesis
, 2007
"... The classical view of computing positions computation as a closedbox transformation of inputs (rational numbers or finite strings) to outputs. According to the interactive view of computing, computation is an ongoing interactive process rather than a functionbased transformation of an input to a ..."
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Cited by 13 (0 self)
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The classical view of computing positions computation as a closedbox transformation of inputs (rational numbers or finite strings) to outputs. According to the interactive view of computing, computation is an ongoing interactive process rather than a functionbased transformation of an input to an output. Specifically, communication with the outside world happens during the computation, not before or after it. This approach radically changes our understanding of what is computation and how it is modeled. The acceptance of interaction as a new paradigm is hindered by the Strong ChurchTuring Thesis (SCT), the widespread belief that Turing Machines (TMs) capture all computation, so models of computation more expressive than TMs are impossible. In this paper, we show that SCT reinterprets the original ChurchTuring Thesis (CTT) in a way that Turing never intended; its commonly assumed equivalence to the original is a myth. We identify and analyze the historical reasons for the widespread belief in SCT. Only by accepting that it is false can we begin to adopt interaction as an alternative paradigm of computation. We present Persistent Turing Machines (PTMs), that extend TMs to capture sequential interaction. PTMs allow us to formulate the Sequential Interaction Thesis, going beyond the expressiveness of TMs and of the CTT. The paradigm shift to interaction provides an alternative understanding of the nature of computing that better reflects the services provided by today’s computing technology.
What is a Random Sequence
 The Mathematical Association of America, Monthly
, 2002
"... there laws of randomness? These old and deep philosophical questions still stir controversy today. Some scholars have suggested that our difficulty in dealing with notions of randomness could be gauged by the comparatively late development of probability theory, which had a ..."
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there laws of randomness? These old and deep philosophical questions still stir controversy today. Some scholars have suggested that our difficulty in dealing with notions of randomness could be gauged by the comparatively late development of probability theory, which had a
Zeno machines and hypercomputation
 Theoretical Computer Science
"... This paper reviews the ChurchTuring Thesis (or rather, theses) with reference to their origin and application and considers some models of “hypercomputation”, concentrating on perhaps the most straightforward option: Zeno machines (Turing machines with accelerating clock). The halting problem is br ..."
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Cited by 8 (0 self)
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This paper reviews the ChurchTuring Thesis (or rather, theses) with reference to their origin and application and considers some models of “hypercomputation”, concentrating on perhaps the most straightforward option: Zeno machines (Turing machines with accelerating clock). The halting problem is briefly discussed in a general context and the suggestion that it is an inevitable companion of any reasonable computational model is emphasised. It is suggested that claims to have “broken the Turing barrier ” could be toned down and that the important and wellfounded rôle of Turing computability in the mathematical sciences stands unchallenged.
COMPUTATIONALISM, THE CHURCH–TURING THESIS, AND THE CHURCH–TURING FALLACY
, 2007
"... 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 mis ..."
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Cited by 7 (4 self)
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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.
Church’s Thesis and the Conceptual Analysis of Computability
 Notre Dame Journal of Formal Logic
, 2007
"... ..."