Abstract
Various formalisms for representing and reasoning about temporal information with qualitative constraints have been studied in the past three decades. The most known are definitely the Point Algebra \((\mathsf {PA})\) and the Interval Algebra \(({\mathsf {IA}})\) proposed by Allen. In this paper, for both calculi, we study a problem that we call the minimal consistency problem \((\mathsf {MinCons})\). Given a temporal qualitative constraint network \((\mathsf {TQCN})\) and a positive integer \(k\), this problem consists in deciding whether or not this \(\mathsf {TQCN}\) admits a solution using at most \(k\) distinct points on the line.We show that \(\mathsf {MinCons}\) for \(\mathsf {PA}\) can be encoded into the finitary versions of Gödel logic. Furthermore, we prove that the \(\mathsf {MinCons}\) problem is \(\mathsf {NP}\)-complete for both \(\mathsf {PA}\) and \({\mathsf {IA}}\), in the general case. However, we show that for \(\mathsf {TQCN}\)s defined on the convex relations, \(\mathsf {MinCons}\) is polynomial. For such \(\mathsf {TQCN}\)s, we give a polynomial method that allows one to obtain compact scenarios.
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Notes
Given a lattice \((E,\le )\), an interval is either the empty set or a set \(\{e \in E: {\textit{min}} \le e \le {\textit{max}}\}\) for some \({\textit{min}}, {\textit{max}} \in E\) with \({\textit{min}}\le {\textit{max}}\).
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Condotta, JF., Kaci, S. & Salhi, Y. Optimization in temporal qualitative constraint networks. Acta Informatica 53, 149–170 (2016). https://doi.org/10.1007/s00236-015-0228-z
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DOI: https://doi.org/10.1007/s00236-015-0228-z