Abstract
In the last decades, the Satisfiability and Constraint Satisfaction Problem frameworks were extended to integrate aspects such as uncertainties, partial observabilities, or uncontrollabilities. The resulting formalisms, including Quantified Boolean Formulas (QBF), Quantified CSP (QCSP), Stochastic SAT (SSAT), or Stochastic CSP (SCSP), still rely on networks of local functions defining specific graphical models, but they involve queries defined by sequences of distinct elimination operators (∃ and ∀ for QBF and QCSP, max and + for SSAT and SCSP) preventing variables from being considered in an arbitrary order when the problem is solved (be it by tree search or by variable elimination).
In this paper, we show that it is possible to take advantage of the actual structure of such multi-operator queries to bring to light new ordering freedoms. This leads to an improved constrained induced-width and doing so to possible exponential gains in complexity. This analysis is performed in a generic semiring-based algebraic framework that makes it applicable to various formalisms. It is related with the quantifier tree approach recently proposed for QBF but it is much more general and gives theoretical bases to observed experimental gains.
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References
Mackworth, A.: Consistency in Networks of Relations. Artificial Intelligence 8, 99–118 (1977)
Bordeaux, L., Monfroy, E.: Beyond NP: Arc-consistency for Quantified Constraints. In: Van Hentenryck, P. (ed.) CP 2002. LNCS, vol. 2470. Springer, Heidelberg (2002)
Littman, M., Majercik, S., Pitassi, T.: Stochastic Boolean Satisfiability. Journal of Automated Reasoning 27, 251–296 (2001)
Walsh, T.: Stochastic Constraint Programming. In: Proc. of the 15th European Conference on Artificial Intelligence (ECAI 2002), Lyon, France (2002)
Benedetti, M.: Quantifier Trees for QBF. In: Bacchus, F., Walsh, T. (eds.) SAT 2005. LNCS, vol. 3569. Springer, Heidelberg (2005)
Kjaerulff, U.: Triangulation of Graphs - Algorithms Giving Small Total State Space. Technical Report R 90-09, Aalborg University, Denmark (1990)
Pearl, J.: Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, San Francisco (1988)
Giang, P., Shenoy, P.: A Qualitative Linear Utility Theory for Spohn’s Theory of Epistemic Beliefs. In: Proc. of the 16th International Conference on Uncertainty in Artificial Intelligence (UAI 2000), Stanford, California, USA, pp. 220–229 (2000)
Sabbadin, R.: A Possibilistic Model for Qualitative Sequential Decision Problems under Uncertainty in Partially Observable Environments. In: Proc. of the 15th International Conference on Uncertainty in Artificial Intelligence (UAI 1999), Stockholm, Sweden (1999)
Puterman, M.: Markov Decision Processes, Discrete Stochastic Dynamic Programming. John Wiley & Sons, Chichester (1994)
Bistarelli, S., Montanari, U., Rossi, F., Schiex, T., Verfaillie, G., Fargier, H.: Semiring-Based CSPs and Valued CSPs: Frameworks, Properties and Comparison. Constraints 4, 199–240 (1999)
Howard, R., Matheson, J.: Influence Diagrams. In: Readings on the Principles and Applications of Decision Analysis, Menlo Park, CA, USA, pp. 721–762 (1984)
Ndilikilikesha, P.: Potential Influence Diagrams. International Journal of Approximated Reasoning 10, 251–285 (1994)
Pralet, C., Verfaillie, G., Schiex, T.: From Influence Diagrams to Multioperator Cluster DAGs. In: Proc. of the 22nd International Conference on Uncertainty in Artificial Intelligence (UAI 2006), Cambridge, MA, USA (2006)
Dechter, R., Fattah, Y.E.: Topological Parameters for Time-Space Tradeoff. Artificial Intelligence 125, 93–118 (2001)
Jensen, F., Jensen, F., Dittmer, S.: From Influence Diagrams to Junction Trees. In: Proc. of the 10th International Conference on Uncertainty in Artificial Intelligence (UAI 1994), Seattle, WA, USA, pp. 367–373 (1994)
Park, J., Darwiche, A.: Complexity Results and Approximation Strategies for MAP Explanations. Journal of Artificial Intelligence Research 21, 101–133 (2004)
Marinescu, R., Dechter, R.: AND/OR Branch-and-Bound for Graphical Models. In: Proc. of the 19th International Joint Conference on Artificial Intelligence (IJCAI 2005), Edinburgh, Scotland (2005)
Jégou, P., Terrioux, C.: Hybrid Backtracking bounded by Tree-decomposition of Constraint Networks. Artificial Intelligence 146, 43–75 (2003)
Knuth, D., Moore, R.: An Analysis of Alpha-Beta Pruning. Artificial Intelligence 8, 293–326 (1975)
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Pralet, C., Schiex, T., Verfaillie, G. (2006). Decomposition of Multi-operator Queries on Semiring-Based Graphical Models. In: Benhamou, F. (eds) Principles and Practice of Constraint Programming - CP 2006. CP 2006. Lecture Notes in Computer Science, vol 4204. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11889205_32
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DOI: https://doi.org/10.1007/11889205_32
Publisher Name: Springer, Berlin, Heidelberg
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