Sufficient and necessary conditions for solution finding in valuation-based systems

Abstract

Valuation algebras abstract a large number of formalisms for automated reasoning and enable the definition of generic inference procedures. Many of these formalisms provide some notion of solution. Typical examples are satisfying assignments in constraint systems, models in logics or solutions to linear equation systems.Dynamic programming algorithms rely on low treewidth decompositions and can be understood as particular cases of a single algorithmic scheme for finding solutions in a valuation algebra. The most encompassing description of this algorithmic scheme to date has been proposed by Pouly and Kohlas together with sufficient conditions for its correctness. Unfortunately, the formalization relies on a theorem for which we provide a counterexample. In spite of that, the mainline of Pouly and Kohlas' theory is correct, although some of the necessary conditions have to be revised. In this paper we analyze the impact that the counterexample has on the theory, and rebuild the theory providing correct sufficient conditions for the algorithms. Furthermore, we also provide necessary conditions for the algorithms, allowing for a sharper characterization of when the algorithmic scheme can be applied.The contribution of the paper is not limited to correcting Pouly and Kohlas' theory. First, we generalize their results from discrete tuples to tuple systems. This allows us to cover some classical algorithms, such as sparse Cholesky factorization, as well as more recently proposed algorithms, such as Snowball (sparse all-pairs shortest path problems), as particular cases of the generic valuation-based algorithms presented here.For the particular case of valuation algebras induced by selective semirings, Pouly and Kohlas presented an algorithm for finding all solutions without requiring any additional condition for correctness. We characterize a necessary and a sufficient condition for it. Furthermore, we also introduce a new algorithm that requires no necessary condition.