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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.
Church Without Dogma: Axioms for computability
"... Abstract. Church’s and Turing’s theses assert dogmatically that an informal notion of effective calculability is adequately captured by a particular mathematical concept of computabilty. I present analyses of calculability that are embedded in a rich historical and philosophical context, lead to pre ..."
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Abstract. Church’s and Turing’s theses assert dogmatically that an informal notion of effective calculability is adequately captured by a particular mathematical concept of computabilty. I present analyses of calculability that are embedded in a rich historical and philosophical context, lead to precise concepts, and dispense with theses. To investigate effective calculability is to analyze processes that can in principle be carried out by calculators. This is a philosophical lesson we owe to Turing. Drawing on that lesson and recasting work of Gandy, I formulate boundedness and locality conditions for two types of calculators, namely, human computing agents and mechanical computing devices (or discrete machines). The distinctive feature of the latter is that they can carry out parallel computations. Representing human and machine computations by discrete dynamical systems, the boundedness and locality conditions can be captured through axioms for Turing computors and Gandy machines; models of
Services as a Paradigm of Computation
"... Abstract. The recent success of serviceoriented architectures gives rise to some fundamental questions: To what extent do services constitute a new paradigm of computation? What are the elementary ingredients of this paradigm? What are adequate notions of semantics, composition, equivalence? How ca ..."
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Abstract. The recent success of serviceoriented architectures gives rise to some fundamental questions: To what extent do services constitute a new paradigm of computation? What are the elementary ingredients of this paradigm? What are adequate notions of semantics, composition, equivalence? How can services be modeled and analyzed? This paper addresses and answers those questions, thus preparing the ground for forthcoming software design techniques. Key words:models of computation, services, SOA, open workflow nets
On Gurevich’s Theorem for Sequential ASM
"... AbstractState Machines have been introduced as “a computation model that is more powerful and more universal than standard computation models”, by Yuri Gurevich in 1985 ([Gur85]). ASM gained much attention as a specification method, in particular for the description of the semantics of programmin ..."
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AbstractState Machines have been introduced as “a computation model that is more powerful and more universal than standard computation models”, by Yuri Gurevich in 1985 ([Gur85]). ASM gained much attention as a specification method, in particular for the description of the semantics of programming languages, communication protocols, distributed algorithms, etc. Gurevich proved recently that a sequential algorithm must only meet a few, liberal requirements, to be representable as an ASM. We reformulate Gurevich’s requirements for sequential algorithms, as well as the semantics of ASMprograms and the proof of his main theorem. A couple of examples support and explain intuition and motivation of ASM.
Quantum Principles and Mathematical Computability
, 2008
"... Taking the view that computation is after all physical, we argue that physics, particularly quantum physics, could help extend the notion of computability. Here, we list the important and unique features of quantum mechanics and then outline a quantum mechanical “algorithm” for one of the insoluble ..."
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Taking the view that computation is after all physical, we argue that physics, particularly quantum physics, could help extend the notion of computability. Here, we list the important and unique features of quantum mechanics and then outline a quantum mechanical “algorithm” for one of the insoluble problems of mathematics, the Hilbert’s tenth and equivalently the Turing halting problem. The key element of this algorithm is the computability and measurability of both the values of physical observables and of the quantummechanical probability distributions for these values. The fact is that quantum computers can prove theorems by methods that neither a human brain nor any other Turingcomputational arbiter will ever be able to reproduce. What if a quantum algorithm delivered a theorem that it was infeasible to prove classically. No such algorithm is yet known, but nor is anything known to rule out such a possibility, and this raises a question of principle: should we still accept such a theorem as undoubtedly proved? We believe that the rational answer ot this question is yes, for our confidence in quantum proofs rests upon the same foundation as our confidence in classical proofs: our acceptance of the physical laws underlying the computing operations. D. Deustch, A. Ekert and R. Lupacchini [1] 1
Contents
, 2008
"... Different types of physical unknowables are discussed. Provable unknowables are derived from reduction to problems which are known to be recursively unsolvable. Recent series solutions to the nbody problem and related to it, chaotic systems, may have no computable radius of convergence. Quantum unk ..."
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Different types of physical unknowables are discussed. Provable unknowables are derived from reduction to problems which are known to be recursively unsolvable. Recent series solutions to the nbody problem and related to it, chaotic systems, may have no computable radius of convergence. Quantum unknowables include the random occurrence of single events, complementarity and value indefiniteness.
Real numbers, chaos, and the principle of a bounded density of information
"... Abstract. Two definitions of the notion of a chaotic transformation are compared: sensitivity to initial conditions and sensitivity to perturbations. Only the later is compatible with the idea that information has a finite density. 1 The notion of information in Physics Information is not a new noti ..."
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Abstract. Two definitions of the notion of a chaotic transformation are compared: sensitivity to initial conditions and sensitivity to perturbations. Only the later is compatible with the idea that information has a finite density. 1 The notion of information in Physics Information is not a new notion in Physics, as Ludwig Boltzmann already defined the entropy of a system as the logarithm of the number of microscopic states corresponding to its macroscopic state, that is, in modern terms, as the amount of information necessary to describe its microscopic state when its macroscopic state is known. This definition presupposes that the number of microscopic states corresponding to a macroscopic state is finite, an hypothesis that would only be clarified later by quantum theory, and still in a very partial way. After Bolzmann, this idea of a bound on the number of possible states of a given system, that is on the amount of information contained in such a system, or
4.3. iProver Modulo 4
"... 2.3. From proofchecking to Interoperability 2 2.4. Automated theorem proving 2 ..."
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2.3. From proofchecking to Interoperability 2 2.4. Automated theorem proving 2
Quantum Computation: A Computer Science Perspective
, 2005
"... The theory of quantum computation is presented in a self contained way from a computer science perspective. The basics of classical computation and quantum mechanics is reviewed. The circuit model of quantum computation is presented aspects of computation and the interplay between them. This report ..."
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The theory of quantum computation is presented in a self contained way from a computer science perspective. The basics of classical computation and quantum mechanics is reviewed. The circuit model of quantum computation is presented aspects of computation and the interplay between them. This report is presented as a Master’s thesis at the department of Computer Science and Engineering at Göteborg University, Göteborg, Sweden. The text is part of a larger work that is planned to include chapters on quantum algorithms, the quantum Turing machine model and abstract approaches to quantum computation.