An Efficient Decision Procedure for Functional Decomposable Theories Based on Dual Constraints

Abstract

Over the last decade, first-order constraints have been efficiently used in the artificial intelligence world to model many kinds of complex problems such as: scheduling, resource allocation, computer graphics and bio-informatics. Recently, a new property called decomposability has been introduced and many first-order theories have been proved to be decomposable such as finite or infinite trees, rational and real numbers, linear dense order,...etc. A decision procedure in the form of five rewriting rules has also been developed. It decides if a first-order formula without free variables (proposition) is true or not in any decomposable theory. Unfortunately, this later needs to normalize the initial proposition before starting the solving process. This transformation generates many nested negations and quantifications which greatly slow down the performances of this decision procedure. We present in this paper an efficient decision procedure for functional decomposable theories, i.e. theories whose set of relation is reduced to { = , ≠ }. This new decision procedure does not need to normalize the formulas and transforms any first-order proposition with any logical symbols into a boolean combination of basic formulas which are either equivalent to true or to false. We show the efficiency of our algorithm (in time and space) and compare its performances with those of the classical decision procedure for decomposable theories. Our algorithm is able to solve first-order propositions involving many nested alternated quantifiers of the form $\exists\bar{x}\forall\bar{y}$ over different functional decomposable theories.