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Approximate quantified constraint solving by cylindrical box decomposition
 RESEARCH INSTITUTE FOR SYMBOLIC COMPUTATION (RISC
, 2000
"... This paper applies interval methods to a classical problem in computer algebra. Let a quantified constraint be a firstorder formula over the real numbers. As shown by A. Tarski in the 1930's, such constraints, when restricted to the predicate symbols , are in general solvable. However, the problem ..."
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Cited by 14 (7 self)
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This paper applies interval methods to a classical problem in computer algebra. Let a quantified constraint be a firstorder formula over the real numbers. As shown by A. Tarski in the 1930's, such constraints, when restricted to the predicate symbols , are in general solvable. However, the problem becomes undecidable, when we add function symbols like sin. Furthermore, all exact algorithms known up to now are too slow for big examples, do not provide partial information before computing the total result, cannot satisfactorily deal with interval constants in the input, and often generate huge output. As a remedy we propose an approximation method based on interval arithmetic. It uses a generalization of the notion of cylindrical decomposition  as introduced by G. Collins. We describe an implementation of the method and demonstrate that, for quantied constraints without equalities, it can efficiently give approximate information on problems that are too hard for current exact methods.
RISCCLP(Tree(Delta))  A Constraint Logic Programming System with Parametric Domain
, 1995
"... Current implementations of constraint logic programming languages (like CLP(!), CHIP or RISCCLP(Real) support constraint solving over a certain fixed domain. In this paper a system is presented which gives the possibility to instantiate a constraint logic programming language with an arbitrary cons ..."
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Cited by 3 (1 self)
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Current implementations of constraint logic programming languages (like CLP(!), CHIP or RISCCLP(Real) support constraint solving over a certain fixed domain. In this paper a system is presented which gives the possibility to instantiate a constraint logic programming language with an arbitrary constraint domain. The interface between the system and such a constraint domain is given and the extensions to the Warren Abstract Machine (WAM) for this implementation are presented. 1 Introduction There already exist several implementations of constraint logic programming languages which are highly optimized for a certain constraint domain (for example PROLOG III [9], CHIP [1], CLP(!) [16], CLP(FD) [23], LIFE [4]), RISCCLP(Real) [12], RISCCLP(RTrees) [7] and RISCCLP(CF) [13]). An alternative approach would be a system which does not involve a fixed constraint domain but is parametric in that it can be instantiated with an arbitrary constraint domain. There already exist several systems ...
Structure and Efficient Computation of Multiplication Tables and Associated Quadratic Forms
, 1995
"... We characterize the multiplication table of an algebra with a multiplicative unity and derive properties of multiplication tables and associated quadratic forms, which allow efficient computation. We compare several ways of computing a multiplication table and associated quadratic forms. By index pe ..."
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Cited by 1 (1 self)
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We characterize the multiplication table of an algebra with a multiplicative unity and derive properties of multiplication tables and associated quadratic forms, which allow efficient computation. We compare several ways of computing a multiplication table and associated quadratic forms. By index permutation the complexity of the computation time of associated quadratic forms can be reduced. Exploiting the structure concerning equal entries in the multiplication table gives a further speedup. We apply our results to the case of the factor ring K[¯x]=I and give speedups in the appendix. 1 Domain and Motivation Let A be a finite dimensional vector space over a field K and let \Theta be a commutative and associative operation ("multiplication") such that (A; \Theta) becomes a commutative algebra (associativity is silently assumed here). Furthermore let \Theta have a unity ("1") throughout. Instead of a \Theta b, for a; b 2 A, we shall simply write ab. Given a basis V = fv 1 ; : : : v d g...