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76
Iterated function systems and permutation representations of the Cuntz algebra
, 1996
"... We study a class of representations of the Cuntz algebras ON, N = 2, 3,..., acting on L 2 (T) where T = R�2πZ. The representations arise in wavelet theory, but are of independent interest. We find and describe the decomposition into irreducibles, and show how the ONirreducibles decompose when rest ..."
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Cited by 81 (19 self)
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We study a class of representations of the Cuntz algebras ON, N = 2, 3,..., acting on L 2 (T) where T = R�2πZ. The representations arise in wavelet theory, but are of independent interest. We find and describe the decomposition into irreducibles, and show how the ONirreducibles decompose when restricted to the subalgebra UHFN ⊂ ON of gaugeinvariant elements; and we show that the whole structure is accounted for by arithmetic and combinatorial properties of the integers Z. We have general results on a class of representations of ON on Hilbert space H such that the generators Si as operators permute the elements in some orthonormal basis for H. We then use this to extend our results from L 2 (T) to L 2 ( T d) , d> 1; even to L 2 (T) where T is some fractal version of the torus which carries more of the algebraic
Closedform Analytic Maps in One and Two Dimensions Can Simulate Turing Machines
, 1996
"... We show closedform analytic functions consisting of a finite number of trigonometric terms can simulate Turing machines, with exponential slowdown in one dimension or in real time in two or more. 1 A part of this author's work was done when he was visiting DIMACS at Rutgers University. 1 Introduc ..."
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Cited by 29 (4 self)
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We show closedform analytic functions consisting of a finite number of trigonometric terms can simulate Turing machines, with exponential slowdown in one dimension or in real time in two or more. 1 A part of this author's work was done when he was visiting DIMACS at Rutgers University. 1 Introduction Various authors have independently shown [9, 12, 4, 14, 1] that finitedimensional piecewiselinear maps and flows can simulate Turing machines. The construction is simple: associate the digits of the x and y coordinates of a point with the left and right halves of a Turing machine's tape. Then we can shift the tape head by halving or doubling x and y, and write on the tape by adding constants to them. Thus two dimensions suffice for a map, or three for a continuoustime flow. These systems can be thought of as billiards or optical ray tracing in three dimensions, recurrent neural networks, or hybrid systems. However, piecewiselinear functions are not very realistic from a physical p...
A generic theorem prover of CSP refinement
 In TACAS 2005, LNCS 3440
, 2005
"... Abstract. We describe a new tool called CspProver which is an interactive theorem prover dedicated to refinement proofs within the process algebra Csp. It aims specifically at proofs for infinite state systems, which may also involve infinite nondeterminism. Semantically, CspProver supports both ..."
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Cited by 17 (11 self)
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Abstract. We describe a new tool called CspProver which is an interactive theorem prover dedicated to refinement proofs within the process algebra Csp. It aims specifically at proofs for infinite state systems, which may also involve infinite nondeterminism. Semantically, CspProver supports both the theory of complete metric spaces as well as the theory of complete partial orders. Both these theories are implemented for infinite product spaces. Technically, CspProver is based on the theorem prover Isabelle. It provides a deep encoding of Csp. The tool’s architecture follows a generic approach which makes it easy to adapt it for various Csp models besides those studied here: the stable failures model F and the traces model T. 1
A General Approach to Performance Analysis and Optimization of Asynchronous Circuits
, 1995
"... A systematic approach for evaluating and optimizing the performance of asynchronous VLSI circuits is presented. Indexpriority simulation is introduced to efficiently find minimal cycles in the state graph of a given circuit. These minimal cycles are used to determine the causality relationships bet ..."
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Cited by 14 (0 self)
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A systematic approach for evaluating and optimizing the performance of asynchronous VLSI circuits is presented. Indexpriority simulation is introduced to efficiently find minimal cycles in the state graph of a given circuit. These minimal cycles are used to determine the causality relationships between all signal transitions in the circuit. Once these relationships are known, the circuit is then modeled as an extended eventrule system, which can be used to describe many circuits, including ones that are inherently disjunctive. An accurate indication of the performance of the circuit is obtained by analytically computing the period of the corresponding extended eventrule system.
Towards a modelchecker for counter systems
 In ATVA 2006, 4 th International Symposium on Automated Technology for Verification and Analysis, Beijing, Rep. of China, volume 4218 of Lecture Notes in Computer Science
, 2006
"... Abstract. This paper deals with modelchecking of fragments and extensions of CTL * on infinitestate Presburger counter systems, where the states are vectors of integers and the transitions are determined by means of relations definable within Presburger arithmetic. We have identified a natural cla ..."
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Cited by 13 (9 self)
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Abstract. This paper deals with modelchecking of fragments and extensions of CTL * on infinitestate Presburger counter systems, where the states are vectors of integers and the transitions are determined by means of relations definable within Presburger arithmetic. We have identified a natural class of admissible counter systems (ACS) for which we show that the quantification over paths in CTL * can be simulated by quantification over tuples of natural numbers, eventually allowing translation of the whole PresburgerCTL * into Presburger arithmetic, thereby enabling effective model checking. We have provided evidence that our results are close to optimal with respect to the class of counter systems described above. Finally, we design a complete semialgorithm to verify firstorder LTL properties over traceflattable counter systems, extending the previous underlying FAST semialgorithm to verify reachability questions over flattable counter systems. 1
The 2adic valuation of a sequence arising from a rational integral
, 2007
"... Abstract. We analyze properties of the 2adic valuations of an integer sequence that originates from an explicit evaluation of a quartic integral. We also give a combinatorial interpretation of the valuations of this sequence. Connections with the orbits arising from the Collatz problem are discusse ..."
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Cited by 11 (9 self)
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Abstract. We analyze properties of the 2adic valuations of an integer sequence that originates from an explicit evaluation of a quartic integral. We also give a combinatorial interpretation of the valuations of this sequence. Connections with the orbits arising from the Collatz problem are discussed. The sequence (1.1) Al,m = l! m! 2 m−l 1.
Halting Problem of One Binary Horn Clause is Undecidable.
, 1993
"... . This paper proposes a codification of the halting problem of any Turing machine in the form of only one rightlinear binary Horn clause as follows : p(t) / p(tt) : where t (resp. tt) is any (resp. linear) term. Recursivity is wellknown to be a crucial and fundamental concept in programming th ..."
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Cited by 10 (0 self)
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. This paper proposes a codification of the halting problem of any Turing machine in the form of only one rightlinear binary Horn clause as follows : p(t) / p(tt) : where t (resp. tt) is any (resp. linear) term. Recursivity is wellknown to be a crucial and fundamental concept in programming theory. This result proves that in Horn clause languages there is no hope to control it without additional hypotheses even for the simplest recursive schemes. Some direct consequences are presented here. For instance, there exists an explicitly constructible rightlinear binary Horn clause for which no decision algorithm, given a goal, always decides in a finite number of steps whether or not the resolution using this clause is finite. The halting problem of derivations w.r.t. one binary Horn clause had been shown decidable if the goal is ground [SS88] or if the goal is linear [Dev88, Dev90, DLD90]. The undecidability in the nonlinear case is an unexpected extension. The proof of the main r...
The HOL Light manual (1.1)
, 2000
"... ion is in a precise sense a converse operation to application. Given 49 50 CHAPTER 5. PRIMITIVE BASIS OF HOL LIGHT a variable x and a term t, which may or may not contain x, one can construct the socalled lambdaabstraction x: t, which means `the function of x that yields t'. (In HOL's ASCII concr ..."
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Cited by 6 (0 self)
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ion is in a precise sense a converse operation to application. Given 49 50 CHAPTER 5. PRIMITIVE BASIS OF HOL LIGHT a variable x and a term t, which may or may not contain x, one can construct the socalled lambdaabstraction x: t, which means `the function of x that yields t'. (In HOL's ASCII concrete syntax the backslash is used, e.g. \x. t.) For example, x: x + 1 is the function that adds one to its argument. Abstractions are not often seen in informal mathematics, but they have at least two merits. First, they allow one to write anonymous functionvalued expressions without naming them (occasionally one sees x 7! t[x] used for this purpose), and since our logic is avowedly higher order, it's desirable to place functions on an equal footing with rstorder objects in this way. Secondly, they make variable dependencies and binding explicit; by contrast in informal mathematics one often writes f(x) in situations where one really means x: f(x). We should give some idea of how ordina...