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Optimal Length Resolution Refutations of Difference Constraint Systems

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Abstract

This paper is concerned with determining the optimal length resolution refutation (OLRR) of a system of difference constraints over an integral domain. The problem of finding short explanations for unsatisfiable difference constraint systems (DCS) finds applications in a number of design domains including program verification, proof theory, real-time scheduling and operations research. It is well-known that resolution refutation is a sound and complete procedure to establish the unsatisfiability of boolean formulas in clausal form. The literature has also established that a variant of the resolution procedure called Fourier-Motzkin elimination is a sound and complete procedure for establishing the unsatisfiability of linear constraint systems (LCS). Our work in this paper first establishes that every DCS has a short (polynomial in the size of the input) resolution refutation and then shows that there exists a polynomial time algorithm to compute the optimal size refutation. One of the consequences of our work is that the Minimum Unsatisfiable Subset (MUS) of a DCS can be computed in polynomial time; computing the MUS of an unsatisfiable constraint set is an extremely important aspect of certifying algorithms.

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  1. LDCSEE, West Virginia University, Morgantown, WV, USA

    K. Subramani

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This research was supported in part by a research grant from the Air-Force Office of Scientific Research under contract FA9550-06-1-0050 and in part by the National Science Foundation through Award CCF-0827397. A portion of this research was conducted at the Stanford Research Institute, where the author was a Visiting Fellow.

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Subramani, K. Optimal Length Resolution Refutations of Difference Constraint Systems. J Autom Reasoning 43, 121–137 (2009). https://doi.org/10.1007/s10817-009-9139-4

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