Decomposition of Multi-operator Queries on Semiring-Based Graphical Models

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.