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On a decision procedure for quantified linear programs

On a decision procedure for quantified linear programs Quantified linear programming is the problem of checking whether a polyhedron specified by a linear system of inequalities is non-empty, with respect to a specified quantifier string. Quantified linear programming subsumes traditional linear programming, since in traditional linear programming, all the program variables are existentially quantified (implicitly), whereas, in quantified linear programming, a program variable may be existentially quantified or universally quantified over a continuous range. In this paper, the term linear programming is used to describe the problem of checking whether a system of linear inequalities has a feasible solution. On account of the alternation of quantifiers in the specification of a quantified linear program (QLP), this problem is non-trivial. QLPs represent a class of declarative constraint logic programs (CLPs) that are extremely rich in their expressive power. The complexity of quantified linear programming for arbitrary constraint matrices is unknown. In this paper, we show that polynomial time decision procedures exist for the case in which the constraint matrix satisfies certain structural properties. We also provide a taxonomy of quantified linear programs, based on the structure of the quantifier string and discuss the computational complexities of the constituent classes. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annals of Mathematics and Artificial Intelligence Springer Journals

On a decision procedure for quantified linear programs

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References (52)

Publisher
Springer Journals
Copyright
Copyright © 2007 by Springer Science+Business Media B.V.
Subject
Computer Science; Complexity; Computer Science, general ; Mathematics, general; Artificial Intelligence (incl. Robotics)
ISSN
1012-2443
eISSN
1573-7470
DOI
10.1007/s10472-007-9085-y
Publisher site
See Article on Publisher Site

Abstract

Quantified linear programming is the problem of checking whether a polyhedron specified by a linear system of inequalities is non-empty, with respect to a specified quantifier string. Quantified linear programming subsumes traditional linear programming, since in traditional linear programming, all the program variables are existentially quantified (implicitly), whereas, in quantified linear programming, a program variable may be existentially quantified or universally quantified over a continuous range. In this paper, the term linear programming is used to describe the problem of checking whether a system of linear inequalities has a feasible solution. On account of the alternation of quantifiers in the specification of a quantified linear program (QLP), this problem is non-trivial. QLPs represent a class of declarative constraint logic programs (CLPs) that are extremely rich in their expressive power. The complexity of quantified linear programming for arbitrary constraint matrices is unknown. In this paper, we show that polynomial time decision procedures exist for the case in which the constraint matrix satisfies certain structural properties. We also provide a taxonomy of quantified linear programs, based on the structure of the quantifier string and discuss the computational complexities of the constituent classes.

Journal

Annals of Mathematics and Artificial Intelligence – Springer Journals

Published: Oct 11, 2007

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