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Logic and precognizable sets of integers
 Bull. Belg. Math. Soc
, 1994
"... We survey the properties of sets of integers recognizable by automata when they are written in pary expansions. We focus on Cobham’s theorem which characterizes the sets recognizable in different bases p and on its generalization to N m due to Semenov. We detail the remarkable proof recently given ..."
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Cited by 92 (4 self)
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We survey the properties of sets of integers recognizable by automata when they are written in pary expansions. We focus on Cobham’s theorem which characterizes the sets recognizable in different bases p and on its generalization to N m due to Semenov. We detail the remarkable proof recently given by Muchnik for the theorem of CobhamSemenov, the original proof being published in Russian. 1
On iterating linear transformations over recognizable sets of integers
 Theoretical Computer Science
"... It has been known for a long time that the sets of integer vectors that are recognizable by finitestate automata are those that can be defined in an extension of Presburger arithmetic. In this paper, we address the problem of deciding whether the closure of a linear transformation preserves the re ..."
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Cited by 20 (2 self)
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It has been known for a long time that the sets of integer vectors that are recognizable by finitestate automata are those that can be defined in an extension of Presburger arithmetic. In this paper, we address the problem of deciding whether the closure of a linear transformation preserves the recognizable nature of sets of integer vectors. We solve this problem by introducing an original extension of the concept of recognizability to sets of vectors with complex components. This generalization allows to obtain a simple necessary and sufficient condition over linear transformations, in terms of the eigenvalues of the transformation matrix. We then show that these eigenvalues do not need to be computed explicitly in order to evaluate the condition, and we give a full decision procedure based on simple integer arithmetic. The proof of this result is constructive, and can be turned into an algorithm for applying the closure of a linear transformation that satisfies the condition to a finitestate representation of a set. Finally, we show that the necessary and sufficient condition that we have obtained can straightforwardly be turned into a sufficient condition for linear transformations with linear guards. Key words: automata, iterations, Presburger arithmetic, recognizable sets of integers
CompileTime Cache Hint Generation for EPIC Architectures
 In 2nd Workshop on Explicitly Parallel Instruction Computing Architecture and Compilers (EPIC2
, 2002
"... One of the new possibilities oered by EPIC architectures is to allow the compiler to direct the cache placement and replacement policy through cache hints. In this work, a method to generate these cache hints at compile time is presented. The generation of appropriate cache hints is based on the loc ..."
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Cited by 6 (2 self)
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One of the new possibilities oered by EPIC architectures is to allow the compiler to direct the cache placement and replacement policy through cache hints. In this work, a method to generate these cache hints at compile time is presented. The generation of appropriate cache hints is based on the locality of the instructions they apply to, which is quanti ed by the reuse distance metric. Next to the static selection of the most appropriate cache hints, a dynamic selection of cache hints by predicates is proposed. The generation of static hints is based on a simple pro ling scheme, while the dynamic selection is based on an analytical model of the programs cache behavior.
Multidimensional extension of the MorseHedlund theorem
 To appear, Eur. J. of Comb
"... Abstract. A celebrated result of Morse and Hedlund, stated in 1938, asserts that a sequence x over a finite alphabet is ultimately periodic if and only if, for some n, the number of different factors of length n appearing in x is less than n+1. Attempts to extend this fundamental result, for example ..."
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Cited by 3 (0 self)
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Abstract. A celebrated result of Morse and Hedlund, stated in 1938, asserts that a sequence x over a finite alphabet is ultimately periodic if and only if, for some n, the number of different factors of length n appearing in x is less than n+1. Attempts to extend this fundamental result, for example, to higher dimensions, have been considered during the last fifteen years. Let d ≥ 2. A legitimate extension to a multidimensional setting of the notion of periodicity is to consider sets of Zd definable by a first order formula in the Presburger arithmetic 〈Z;<,+〉. With this latter notion and using a powerful criterion due to Muchnik, we exhibit a complete extension of the Morse–Hedlund theorem to an arbitrary dimension d and characterize sets of Zd definable in 〈Z;<,+〉 in terms of some functions counting recurrent blocks, that is, blocks occurring infinitely often. 1.
Building Satisfiability Procedures for Verification: The Case Study of Sorting Algorithms
 IN LOPSTR’03
, 2003
"... This paper describes the development of some decision procedures which are useful for the automatic verification of imperative algorithms manipulating arrays. Our approach—based on the superposition framework—consists of extending an available satisfiability for infinite arrays to procedures for f ..."
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This paper describes the development of some decision procedures which are useful for the automatic verification of imperative algorithms manipulating arrays. Our approach—based on the superposition framework—consists of extending an available satisfiability for infinite arrays to procedures for finite arrays, finite arrays with permutations, and their extensions with userdefined symbols. The Nelson and Oppen combination schema is used to incorporate a form of arithmetic reasoning and we propose an heuristic extension whereby symbols—defined by using both the theory of arrays and arithmetic—can be handled. The procedures so obtained are successfully put to work to automatically discharge the proof obligations arising in the correctness of the algorithms Find, Insertion sort, and Heap sort. While much existing research on decision procedures has been done in isolation or in the context of combination problems, the work described in this paper seems to be one of the few attempts to widen the scope of decision procedures aimed at building more flexible and extensible tools for verification.
Computing affine hulls over Q and Z from sets represented by Number Decision Diagrams
"... Abstract. Number Decision Diagrams (NDD) are finite automata representing sets of integer vectors and have recently been proposed as an efficient data structure for representing sets definable in Presburger arithmetic. In this context, some work has been done in order to generate formulas or sets of ..."
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Abstract. Number Decision Diagrams (NDD) are finite automata representing sets of integer vectors and have recently been proposed as an efficient data structure for representing sets definable in Presburger arithmetic. In this context, some work has been done in order to generate formulas or sets of generators from the NDDs. Taking another step in this direction, this paper present algorithms that takes as input an NDD and computes the affine hull over Q or over Z of the set represented by the NDD, i.e., the smallest set defined by a conjunction of equations or by a conjunction of equations and congruence relations that includes the set represented by the NDD. Our algorithms run in time O(Q  · Σ n r  · n) and O(Q  3 · Σ n r  · n 3) respectively, where n is the number of components of the vectors represented by the NDD, and Q  and Σ n r are the number of states and the alphabet of the NDD. On a prototype implementation, the computations of affine hulls of NDDs with more than 100000 states are done in seconds. 1
unknown title
"... Automatabased representations have recently been investigated as a tool for representing and manipulating sets of integer vectors. In this paper, we study some structural properties of automata accepting the encodings (most significant digit first) of the natural solutions of systems of linear Dio ..."
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Automatabased representations have recently been investigated as a tool for representing and manipulating sets of integer vectors. In this paper, we study some structural properties of automata accepting the encodings (most significant digit first) of the natural solutions of systems of linear Diophantine inequations, i.e., convex polyhedra in ¡£¢ Based on those structural properties, we develop an algorithm that takes as input an automaton and generates a quantifierfree formula that represents exactly the set of integer vectors accepted by the automaton. In addition, our algorithm generates the minimal Hilbert basis of the linear system. In experiments made with a prototype implementation, we have been able to synthesize in seconds formulas and Hilbert bases from automata with more than 10,000 states. 1.