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Models of Computation  Exploring the Power of Computing
"... Theoretical computer science treats any computational subject for which a good model can be created. Research on formal models of computation was initiated in the 1930s and 1940s by Turing, Post, Kleene, Church, and others. In the 1950s and 1960s programming languages, language translators, and oper ..."
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Cited by 57 (7 self)
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Theoretical computer science treats any computational subject for which a good model can be created. Research on formal models of computation was initiated in the 1930s and 1940s by Turing, Post, Kleene, Church, and others. In the 1950s and 1960s programming languages, language translators, and operating systems were under development and therefore became both the subject and basis for a great deal of theoretical work. The power of computers of this period was limited by slow processors and small amounts of memory, and thus theories (models, algorithms, and analysis) were developed to explore the efficient use of computers as well as the inherent complexity of problems. The former subject is known today as algorithms and data structures, the latter computational complexity. The focus of theoretical computer scientists in the 1960s on languages is reflected in the first textbook on the subject, Formal Languages and Their Relation to Automata by John Hopcroft and Jeffrey Ullman. This influential book led to the creation of many languagecentered theoretical computer science courses; many introductory theory courses today continue to reflect the content of this book and the interests of theoreticians of the 1960s and early 1970s. Although
Computability and recursion
 BULL. SYMBOLIC LOGIC
, 1996
"... We consider the informal concept of “computability” or “effective calculability” and two of the formalisms commonly used to define it, “(Turing) computability” and “(general) recursiveness.” We consider their origin, exact technical definition, concepts, history, general English meanings, how they b ..."
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Cited by 32 (0 self)
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We consider the informal concept of “computability” or “effective calculability” and two of the formalisms commonly used to define it, “(Turing) computability” and “(general) recursiveness.” We consider their origin, exact technical definition, concepts, history, general English meanings, how they became fixed in their present roles, how they were first and are now used, their impact on nonspecialists, how their use will affect the future content of the subject of computability theory, and its connection to other related areas. After a careful historical and conceptual analysis of computability and recursion we make several recommendations in section §7 about preserving the intensional differences between the concepts of “computability” and “recursion.” Specifically we recommend that: the term “recursive ” should no longer carry the additional meaning of “computable” or “decidable;” functions defined using Turing machines, register machines, or their variants should be called “computable” rather than “recursive;” we should distinguish the intensional difference between Church’s Thesis and Turing’s Thesis, and use the latter particularly in dealing with mechanistic questions; the name of the subject should be “Computability Theory” or simply Computability rather than
On Folk Theorems
, 1980
"... this paper is to refine this definition somewhat, adapting it to the purposes of the research community in computer science. Accordingly, we shall attempt to provide a reasonable definition of or, rather, criteria for folk theorems, followed by a detailed example illustrating the ideas. The latter e ..."
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Cited by 29 (0 self)
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this paper is to refine this definition somewhat, adapting it to the purposes of the research community in computer science. Accordingly, we shall attempt to provide a reasonable definition of or, rather, criteria for folk theorems, followed by a detailed example illustrating the ideas. The latter endeavor might take one of two possible forms. We could take a piece of folklore and show that it is a theorem, or take a theorem and show that it is folklore. As an example of the first form we could have shown that the statement P NP, which is folklore, is also a theorem. However, since we have resolved to introduce no new technical material in this paper, and moreover, since researchers in our community seem to be less familiar with folklore than with theorems, Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission
Introduction to Quantum Algorithms
, 2001
"... Abstract. These notes discuss the quantum algorithms we know of that can solve problems significantly faster than the corresponding classical algorithms. ..."
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Cited by 20 (0 self)
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Abstract. These notes discuss the quantum algorithms we know of that can solve problems significantly faster than the corresponding classical algorithms.
The many forms of hypercomputation
 Applied Mathematics and Computation
, 2006
"... This paper surveys a wide range of proposed hypermachines, examining the resources that they require and the capabilities that they possess. ..."
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Cited by 16 (0 self)
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This paper surveys a wide range of proposed hypermachines, examining the resources that they require and the capabilities that they possess.
The tractable cognition thesis
 Cognitive Science: A Multidisciplinary Journal
, 2008
"... The recognition that human minds/brains are finite systems with limited resources for computation has led some researchers to advance the Tractable Cognition thesis: Human cognitive capacities are constrained by computational tractability. This thesis, if true, serves cognitive psychology by constra ..."
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Cited by 15 (2 self)
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The recognition that human minds/brains are finite systems with limited resources for computation has led some researchers to advance the Tractable Cognition thesis: Human cognitive capacities are constrained by computational tractability. This thesis, if true, serves cognitive psychology by constraining the space of computationallevel theories of cognition. To utilize this constraint, a precise and workable definition of “computational tractability ” is needed. Following computer science tradition, many cognitive scientists and psychologists define computational tractability as polynomialtime computability, leading to the PCognition thesis. This article explains how and why the PCognition thesis may be overly restrictive, risking the exclusion of veridical computationallevel theories from scientific investigation. An argument is made to replace the PCognition thesis by the FPTCognition thesis as an alternative formalization of the Tractable Cognition thesis (here, FPT stands for fixedparameter tractable). Possible objections to the Tractable Cognition thesis, and its proposed formalization, are discussed, and existing misconceptions are clarified.
Physical Hypercomputation and the Church–Turing Thesis
, 2003
"... We describe a possible physical device that computes a function that cannot be computed by a Turing machine. The device is physical in the sense that it is compatible with General Relativity. We discuss some objections, focusing on those which deny that the device is either a computer or computes a ..."
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Cited by 14 (1 self)
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We describe a possible physical device that computes a function that cannot be computed by a Turing machine. The device is physical in the sense that it is compatible with General Relativity. We discuss some objections, focusing on those which deny that the device is either a computer or computes a function that is not Turing computable. Finally, we argue that the existence of the device does not refute the Church–Turing thesis, but nevertheless may be a counterexample to Gandy’s thesis.
Notions of computability at higher types I
 In Logic Colloquium 2000
, 2005
"... We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a ..."
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Cited by 12 (5 self)
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We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a first step in this programme, we give an extended survey of the di#erent strands of research on higher type computability to date, bringing together material from recursion theory, constructive logic and computer science. The paper thus serves as a reasonably complete overview of the literature on higher type computability. Two sequel papers will be devoted to developing a more systematic account of the material reviewed here.
Complexity and Real Computation: A Manifesto
 International Journal of Bifurcation and Chaos
, 1995
"... . Finding a natural meeting ground between the highly developed complexity theory of computer science with its historical roots in logic and the discrete mathematics of the integers and the traditional domain of real computation, the more eclectic less foundational field of numerical analysis ..."
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Cited by 11 (0 self)
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. Finding a natural meeting ground between the highly developed complexity theory of computer science with its historical roots in logic and the discrete mathematics of the integers and the traditional domain of real computation, the more eclectic less foundational field of numerical analysis with its rich history and longstanding traditions in the continuous mathematics of analysis presents a compelling challenge. Here we illustrate the issues and pose our perspective toward resolution. This article is essentially the introduction of a book with the same title (to be published by Springer) to appear shortly. Webster: A public declaration of intentions, motives, or views. k Partially supported by NSF grants. y International Computer Science Institute, 1947 Center St., Berkeley, CA 94704, U.S.A., lblum@icsi.berkeley.edu. Partially supported by the LettsVillard Chair at Mills College. z Universitat Pompeu Fabra, Balmes 132, Barcelona 08008, SPAIN, cucker@upf.es. P...
Symbol Processing Systems, Connectionist Networks, and Generalized Connectionist Networks
 IN S. GOONATILAKE AND S.KHEBBAL, EDITORS INTELLIGENT HYBRID SYSTEMS
, 1990
"... Many authors have suggested that SP (symbol processing) and CN (connectionist network) models offer radically, or even fundamentally, different paradigms for modeling intelligent behavior (see Schneider, 1987) and the design of intelligent systems. Others have argued that CN models have little to co ..."
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Cited by 9 (6 self)
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Many authors have suggested that SP (symbol processing) and CN (connectionist network) models offer radically, or even fundamentally, different paradigms for modeling intelligent behavior (see Schneider, 1987) and the design of intelligent systems. Others have argued that CN models have little to contribute to our efforts to understand intelligence (Fodor & Pylyshyn, 1988). A critical examination of the popular characterizations of SP and CN models suggests that neither of these extreme positions is justified. There are many advantages to be gained by a synthesis of the best of both SP and CN approaches in the design of intelligent systems. The Generalized connectionist networks (GCN) (alternately called generalized neuromorphic systems (GNS)) introduced in this paper provide a framework for such a synthesis.