Modeling and Solving Constraint Satisfaction Problems through Petri Nets

Constraint satisfaction problems (CSP) represent one of the most studied areas in Artificial Intelligence and related disciplines. A lot of theoretical problems and applications, including computer vision, job-shop scheduling, planning, design and others, can be formulated as problems related to the satisfaction of constraints. Classical approaches to the solution of CSP are usually based on some form of backtracking search; such approaches suffer, in the general case, of some drawbacks essentially represented by the so-called thrashing behavior, a problem arising when the search algorithm repeatedly explores parts of the search space not leading to any solution. In this paper an alternative approach to backtracking search is proposed by modeling a system of constraints through an ordinary Petri net model. The Petri net model is able to capture every finite domain CSP, representing a wide and significant class of CSP used in applications. Algebraic analysis is then exploited for solving the CSP corresponding to the net model. In particular, even if a direct application of the fundamental equation of Petri nets would be theoretically possible, such a possibility could be unsatisfactory in practice. Because of that, a cyclic net model for CSP is introduced, such that the whole set of solutions can be characterized by means of a subset of the set of minimal support T-invariants of the net model. An algorithm able to directly compute only such a subset of invariants is then proposed as the core process for solving a CSP.