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Approximate Quantified Constraint Solving by Cylindrical Box Decomposition

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Reliable Computing

Abstract

This paper applies interval methods to a classical problem in computer algebra. Let a quantified constraint be a first-order formula over the real numbers. As shown by A. Tarski in the 1930's, such constraints, when restricted to the predicate symbols <, = and function 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 quantified constraints without equalities, it can efficiently give approximate information on problems that are too hard for current exact methods.

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  1. Research Institute for Symbolic Computation, Johannes Kepler University, Linz, Austria

    Stefan Ratschan

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Ratschan, S. Approximate Quantified Constraint Solving by Cylindrical Box Decomposition. Reliable Computing 8, 21–42 (2002). https://doi.org/10.1023/A:1014785518570

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