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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 Quantum LatticeGas Model for Computational Fluid Dynamics
, 1999
"... Quantumcomputing ideas are applied to the practical and ubiquitous problem of fluid dynamics simulation. Hence, this paper addresses two separate areas of physics: quantum mechanics and fluid dynamics (or specially, the computational simulation of fluid dynamics). The quantum algorithm is called a ..."
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Cited by 14 (4 self)
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Quantumcomputing ideas are applied to the practical and ubiquitous problem of fluid dynamics simulation. Hence, this paper addresses two separate areas of physics: quantum mechanics and fluid dynamics (or specially, the computational simulation of fluid dynamics). The quantum algorithm is called a quantum lattice gas. An analytical treatment of the microscopic quantum latticegas system is carried out to predict its behavior at the mesoscopic and macroscopic scales. At the mesoscopic scale, a lattice Boltzmann equation, with a nonlocal collision term that depends on the entire system wavefunction, governs the dynamical system. Numerical results obtained from an exact simulation of a onedimensional quantum latticemodel are included to illustrate the formalism. A symbolic mathematical method is used to implement the quantum mechanical model on a conventional workstation. The numerical simulation indicates that classical viscous damping is not present in the onedimensional quantum la...
A latticegas with longrange interactions coupled to a heat bath
 in [52
"... Introduced is a latticegas with longrange 2body interactions. An effective interparticle force is mediated by momentum exchanges. There exists the possibility of having both attractive and repulsive interactions using finite impact parameter collisions. There also exists an interesting possibili ..."
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Cited by 13 (6 self)
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Introduced is a latticegas with longrange 2body interactions. An effective interparticle force is mediated by momentum exchanges. There exists the possibility of having both attractive and repulsive interactions using finite impact parameter collisions. There also exists an interesting possibility of coupling these longrange interactions to a heat bath. A fixed temperature heat bath induces a permanent net attractive interparticle potential, but at the expense of reversibility. Thus the longrange dynamics is a kind of a Monte Carlo Kawasaki updating scheme. The model has a PρT equation of state. Presented are analytical and numerical results for a latticegas fluid governed by a nonideal equation of state. The model’s complexity is not much beyond that of the FHP latticegas. It is suitable for massively parallel processing and may be used to study critical phenomena in large systems. 1
Decidability and universality in symbolic dynamical systems
 Fund. Inform
"... Abstract. Many different definitions of computational universality for various types of dynamical systems have flourished since Turing’s work. We propose a general definition of universality that applies to arbitrary discrete time symbolic dynamical systems. Universality of a system is defined as un ..."
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Abstract. Many different definitions of computational universality for various types of dynamical systems have flourished since Turing’s work. We propose a general definition of universality that applies to arbitrary discrete time symbolic dynamical systems. Universality of a system is defined as undecidability of a modelchecking problem. For Turing machines, counter machines and tag systems, our definition coincides with the classical one. It yields, however, a new definition for cellular automata and subshifts. Our definition is robust with respect to initial condition, which is a desirable feature for physical realizability. We derive necessary conditions for undecidability and universality. For instance, a universal system must have a sensitive point and a proper subsystem. We conjecture that universal systems have infinite number of subsystems. We also discuss the thesis according to which computation should occur at the ‘edge of chaos ’ and we exhibit a universal chaotic system. 1.
Digital Analog Simulation Of Uniform Motion In Representations Of Physical NSpace By LatticeWork MIMD Computer Architectures
, 1991
"... This doctoral dissertation is part of an ongoing research project with John Case, Dayanand S. Rajan and myself. We are investigating the possibility of solving problems in scientific computing involving the motion of objects in a bounded region of physical nspace by (a) representing points in the r ..."
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Cited by 8 (7 self)
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This doctoral dissertation is part of an ongoing research project with John Case, Dayanand S. Rajan and myself. We are investigating the possibility of solving problems in scientific computing involving the motion of objects in a bounded region of physical nspace by (a) representing points in the region of space by processors in a latticework mesh of processors with local connections for interprocessor communication, and (b) literally, analogically simulating the motion of objects by representing the particles of these objects by algorithms which move themselves about in the latticework of processors, much as the motion in real space of the particles making up real objects, in effect, constitutes the motion of those objects. The main contributions of this dissertation are (i) two provably correct algorithms to generate virtually perfectly shaped spherical wavefronts emanating from a point source at virtually constant radial speed, (ii) a provably correct algorithm template for simu...
Optimally Representing Euclidean Space Discretely for Analogically Simulating Physical Phenomena
, 1990
"... this paper is `case@cis.udel.edu'. 1 Introduction ..."
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Cited by 7 (6 self)
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this paper is `case@cis.udel.edu'. 1 Introduction
Spherical Wave Front Generation in Lattice Computers
 SPECIAL ISSUE: PROCEEDINGS OF THE 6TH INTERNATIONAL CONFERENCE ON COMPUTING AND INFORMATION
, 1994
"... For parallel processing computers based on geometric lattices (including root lattices) algorithms are presented for the "physical" propagation of constant speed, nondissipating, euclideanspherical wave fronts in euclidean n space. The motivation includes obtaining as close to perfec ..."
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Cited by 6 (5 self)
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For parallel processing computers based on geometric lattices (including root lattices) algorithms are presented for the "physical" propagation of constant speed, nondissipating, euclideanspherical wave fronts in euclidean n space. The motivation includes obtaining as close to perfect as possible, realtime, analogical simulation of wave phenomena and also of the uniform expansion /contraction of geometric figures about a point. Also shown is just how close to perfect is possible.
Quantum Computation Using Geometric Algebra
, 2002
"... vi LIST OF TABLES .....................................................................................................................x LIST OF FIGURES ............................................................................................................... xiii CHAPTER 1 ..."
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Cited by 6 (3 self)
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vi LIST OF TABLES .....................................................................................................................x LIST OF FIGURES ............................................................................................................... xiii CHAPTER 1
Lattice Computers for Approximating Euclidean Space
 Journal of the ACM
, 2001
"... In the context of meshlike, parallel processing computers for (i) approximating continuous space and (ii) analog simulation of the motion of objects and waves in continuous space, the present paper is concerned with which meshlike interconnection of processors might be particularly suitable for th ..."
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Cited by 6 (2 self)
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In the context of meshlike, parallel processing computers for (i) approximating continuous space and (ii) analog simulation of the motion of objects and waves in continuous space, the present paper is concerned with which meshlike interconnection of processors might be particularly suitable for the task and why. Processor interconnection schemes based on nearest neighbor connections in geometric lattices are presented along with motivation. Then two major threads are explored regarding which lattices would be good: the regular lattices, for their symmetry and other properties in common with continuous space, and the wellknown root lattices, for being, in a sense, the lattices required for physically natural basic algorithms for motion. The main theorem of the present paper implies that the wellknown lattice An is the regular lattice having the maximum number of nearest neighbors among the ndimensional regular lattices. It is noted that the only ndimensional lattices that are bo...
Obfuscated Ciphertext Mixing
 In IACR Eprint archive number 394
, 2005
"... Mixnets are a type of anonymous channel composed of a handful of trustees that, each in turn, shu#e and rerandomize a batch ciphertexts. For applications that require verifiability, each trustee provides a proof of correct mixing. Though mixnets have recently been made quite e#cient, they still r ..."
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Cited by 5 (0 self)
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Mixnets are a type of anonymous channel composed of a handful of trustees that, each in turn, shu#e and rerandomize a batch ciphertexts. For applications that require verifiability, each trustee provides a proof of correct mixing. Though mixnets have recently been made quite e#cient, they still require secret computation and proof generation after the mixing process.