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The Generic Model of Computation
"... Over the past two decades, Yuri Gurevich and his colleagues have formulated axiomatic foundations for the notion of algorithm, be it classical, interactive, or parallel, and formalized them in the new generic framework of abstract state machines. This approach has recently been extended to suggest a ..."
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Over the past two decades, Yuri Gurevich and his colleagues have formulated axiomatic foundations for the notion of algorithm, be it classical, interactive, or parallel, and formalized them in the new generic framework of abstract state machines. This approach has recently been extended to suggest a formalization of the notion of effective computation over arbitrary countable domains. The central notions are summarized herein. 1
Abstract SuperTasks, Accelerating Turing Machines and Uncomputability
"... Accelerating Turing machines are abstract devices that have the same computational structure as Turing machines, but can perform supertasks. I argue that performing supertasks alone does not buy more computational power, and that accelerating Turing machines do not solve the halting problem. To sh ..."
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Accelerating Turing machines are abstract devices that have the same computational structure as Turing machines, but can perform supertasks. I argue that performing supertasks alone does not buy more computational power, and that accelerating Turing machines do not solve the halting problem. To show this, I analyze the reasoning that leads to Thomson's paradox, point out that the paradox rests on a conflation of different perspectives of accelerating processes, and conclude that the same conflation underlies the claim that accelerating Turing machines can solve the halting problem.
Comparative Analysis of Hypercomputational Systems Submitted in partial fulfilment
"... In the 1930s, Turing suggested his abstract model for a practical computer, hypothetically visualizing the digital programmable computer long before it was actually invented. His model formed the foundation for every computer made today. The past few years have seen a change in ideas where philosoph ..."
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In the 1930s, Turing suggested his abstract model for a practical computer, hypothetically visualizing the digital programmable computer long before it was actually invented. His model formed the foundation for every computer made today. The past few years have seen a change in ideas where philosophers and scientists are suggesting models of hypothetical computing devices which can outperform the Turing machine, performing some calculations the latter is unable to. The ChurchTuring Thesis, which the Turing machine model embodies, has raised discussions on whether it could be possible to solve undecidable problems which Turing’s model is unable to. Models which could solve these problems, have gone further to claim abilities relating to quantum computing, relativity theory, even the modeling of natural biological laws themselves. These so called ‘hypermachines ’ use hypercomputational abilities to make the impossible possible. Various models belonging to different disciplines of physics, mathematics and philosophy, have been suggested for these theories. My (primarily researchoriented) project is based on the study and review of these different hypercomputational models and attempts to compare the different models in terms of computational power. The project focuses on the ability to compare these models of different disciplines on similar grounds and
From logic to physics: How the meaning of computation changed over time.
"... The intuition guiding the de…nition of computation has shifted over time, a process that is re‡ected in the changing formulations of the ChurchTuring thesis. The theory of computation began with logic and gradually moved to the capacity of …nite automata. Consequently, modern computer models rely o ..."
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The intuition guiding the de…nition of computation has shifted over time, a process that is re‡ected in the changing formulations of the ChurchTuring thesis. The theory of computation began with logic and gradually moved to the capacity of …nite automata. Consequently, modern computer models rely on general physical principles, with quantum computers representing the extreme case. The paper discusses this development, and the challenges to the ChurchTuring thesis in its physical form, in particular, Kieu’s quantum computer and relativistic hypercomputation. Finally, the robustness of the boundary between polynomial and exponential time complexity is considered in connection with quantum computers and quantum information theory. Key words: ChurchTuring thesis, hypercomputation, quantum computers 1 The ChurchTuring thesis and the meaning of ‘computable function’ The common formulation of the ChurchTuring thesis runs as follows: Every computable function is computable by a Turing machine
CONTENTS
"... What is computation? Philosophical views of computation `Computation ' is a cluster concept and has been ..."
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What is computation? Philosophical views of computation `Computation ' is a cluster concept and has been
CHAPTER 3 COMPUTATIONALISM UNDER ATTACK
"... Since the early eighties, computationalism in the study of the mind has been “under attack ” i by several critics of the socalled “classic ” or “symbolic ” approaches in AI and cognitive science. Computationalism was generically identified with such approaches. For example, it was identified with b ..."
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Since the early eighties, computationalism in the study of the mind has been “under attack ” i by several critics of the socalled “classic ” or “symbolic ” approaches in AI and cognitive science. Computationalism was generically identified with such approaches. For example, it was identified with both Allen Newell and Herbert Simon’s Physical Symbol System Hypothesis and Jerry Fodor’s theory of Language of Thought, usually without taking into account the fact,that such approaches are very different as to their methods and aims. ii Zenon Pylyshyn, in his influential book Computation and Cognition, claimed that both Newell and Fodor deeply influenced his ideas on cognition as computation. iii This probably added to the confusion, as many people still consider Pylyshyn’s book as paradigmatic of the computational approach in the study of the mind. Since then, cognitive scientists, AI researchers and also philosophers of the mind have been asked to take sides on different “paradigms ” that have from time to time been proposed as opponents of (classic or symbolic) computationalism. Examples of such oppositions are: computationalism vs. connectionism,
Concepts and Axioms
, 1998
"... The paper discusses the transition from informal concepts to mathematically precise notions; examples are given, and in some detail the case of lawless sequences, a concept of intuitionistic mathematics, is discussed. A final section comments on philosophical discussions concerning intuitionistic lo ..."
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The paper discusses the transition from informal concepts to mathematically precise notions; examples are given, and in some detail the case of lawless sequences, a concept of intuitionistic mathematics, is discussed. A final section comments on philosophical discussions concerning intuitionistic logic in connection with a "theory of meaning". What I have to tell here is not a new story, and it does not contain any really new ideas. The main difference with my earlier discussions of the same topics ([TD88, chapter16],[Tro91]) is in the emphasis. This paper starts with some examples of the transition from informal concepts to mathematically precise notions, followed by a more detailed discussion of one of these examples, the intuitionistic notion of a choice sequence, arguing for the lasting interest of this notion for the philosophy of mathematics. In a final section, I describe my own position relative to some of the philosophical discussions concerning intuitionistic logic in the wr...
Cognitive Modeling and the Analysis of Computation Processes
"... this paper. Research for this paper was partly supported by MURST, grant no. 9911263337_004 ..."
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this paper. Research for this paper was partly supported by MURST, grant no. 9911263337_004