Efficient variable elimination for semi-structured simple temporal networks with continuous domains

Neil Yorke-Smith; Hung H. Bui; Neil Yorke-Smith; Hung H. Bui

doi:10.1017/S0269888910000184

Abstract: The Simple Temporal Network (STN) is a widely used framework for reasoning about quantitative temporal constraints over variables with continuous or discrete domains. The inference tasks of determining consistency and deriving the minimal network are traditionally achieved by graph algorithms (e.g. Floyd-Warshall, Johnson) or by iteration of narrowing operators (e.g. ΔSTP). None of these methods exploits effectively the tree-decomposition structure of the constraint graph of an STN. Methods based on variable elimination (e.g. adaptive consistency) can exploit this structure, but have not been applied to STNs as far as they could, in part because it is unclear how to efficiently pass the ‘messages’ over continuous domains. We show that for an STN, these messages can be represented compactly as sub-STNs. We then present an efficient message-passing scheme for computing the minimal constraints of an STN. Analysis of this algorithm, Prop-STP, brings formal explanation of the performance of the existing STN solvers ΔSTP and SR-PC. Empirical results validate the efficiency of Prop-STP, demonstrating performance comparable to ΔSTP, in cases where the constraint graph is known to have small tree width, such as those that arise during Hierarchical Task Network planning.

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Efficient variable elimination for semi-structured simple temporal networks with continuous domains

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Corresponding authors: Neil Yorke-Smith ; Hung H. Bui

Published online: 01 September 2010

Abstract: Abstract: The Simple Temporal Network (STN) is a widely used framework for reasoning about quantitative temporal constraints over variables with continuous or discrete domains. The inference tasks of determining consistency and deriving the minimal network are traditionally achieved by graph algorithms (e.g. Floyd-Warshall, Johnson) or by iteration of narrowing operators (e.g. ΔSTP). None of these methods exploits effectively the tree-decomposition structure of the constraint graph of an STN. Methods based on variable elimination (e.g. adaptive consistency) can exploit this structure, but have not been applied to STNs as far as they could, in part because it is unclear how to efficiently pass the ‘messages’ over continuous domains. We show that for an STN, these messages can be represented compactly as sub-STNs. We then present an efficient message-passing scheme for computing the minimal constraints of an STN. Analysis of this algorithm, Prop-STP, brings formal explanation of the performance of the existing STN solvers ΔSTP and SR-PC. Empirical results validate the efficiency of Prop-STP, demonstrating performance comparable to ΔSTP, in cases where the constraint graph is known to have small tree width, such as those that arise during Hierarchical Task Network planning.

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Acknowledgments

We gratefully thank Berthe Choueiry, Karen Myers, Bart Peintner, Léon Planken, and Mabry Tyson for discussions and insights, the reviewers for constructive comments, and the participants of the two workshops where preliminary reports of this work were presented.

While one can consider STNs with discrete domains, we focus on the more difficult case of continuous domains. The theory in this paper can be readily specialized to the discrete case, and the algorithm we evaluate operates effectively for either case.

In some representations, such as the one used by Xu and Choueiry (2003), unary domain constraints are modelled as binary relations to a distinguished temporal reference time point, denoted by TR, which marks the start of time; thus, without loss of generality, all constraints may be taken to have the binary form.

We implicitly assume that the minimalization operator takes precedence over restriction so that .

Sophisticated decomposition-based methods developed for the general CSP, such as BTD (Jégou & Terrioux, 2003), can be applied to the STN supposing discrete domains for the time point variables. However, by exploiting the highly structured nature of simple temporal constraints, decomposition-based methods dedicated to the STN have an inherent advantage.

The difficulty for ΔSTP is not the triangulation of G as much as storing the list of all triangles; in contrast, although we make Prop-STP naively work over the same triangulated graph , only itself is required, not the list of all triangles.

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Neil Yorke-Smith, Hung H. Bui. 2010. Efficient variable elimination for semi-structured simple temporal networks with continuous domains. The Knowledge Engineering Review. 25:184 doi: 10.1017/S0269888910000184

Neil Yorke-Smith, Hung H. Bui. 2010. Efficient variable elimination for semi-structured simple temporal networks with continuous domains. The Knowledge Engineering Review. 25:184 doi: 10.1017/S0269888910000184

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Efficient variable elimination for semi-structured simple temporal networks with continuous domains

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