Finite Constraint Satisfaction Research Articles

Satisfiability Checking in Łukasiewicz Logic as Finite Constraint Satisfaction

Although it is well-known that every satisfiable formula in Łukasiewicz’ infinite-valued logic \(\mathcal{L}_{\infty}\) can be satisfied in some finite-valued logic, practical methods for finding an appropriate number of truth degrees do currently not exist. Extending upon earlier results by Aguzzoli et al., which only take the total number of variable occurrences into account, we present a detailed analysis of what type of formulas require a large number of truth degrees to be satisfied. In particular, we reveal important links between this number of truth degrees and the dimension, and structure, of the cycle space of an associated bipartite graph. We furthermore propose an efficient, polynomial-time algorithm for establishing a strong upper bound on the required number of truth degrees, allowing us to check the satisfiability of sets of formulas in \(\mathcal{L}_{\infty}\), and more generally, sets of fuzzy clauses over Łukasiewicz logic formulas, by solving a small number of constraint satisfaction problems. In an experimental evaluation, we demonstrate the practical usefulness of this approach, comparing it with a state-of-the-art technique based on mixed integer programming.

Using OBDDs to handle dynamic constraints

For many real-life problems naturally modeled as constraints systems, we have to manage dynamically systems of constraints. So, a model based on the formalism of finite Constraint Satisfaction Problems (CSPs) (Montanari, 1974) has been proposed with Dynamic CSPs (DCSPs) to handle this kind of problem (Dechter and Dechter, 1988; Janssen et al., 1990). Some classical techniques defined in the field of CSPs are usable in DCSPs, but the management of dynamicity induces new problems such as management of over-constrained systems and consistency maintenance. At present, Constraint Programming tools generally do not offer a framework for integrating dynamic constraints. The purpose of this paper is to introduce an efficient way to solve DCSPs based on a logical approach. We use and extend Ordered Binary Decision Diagrams (OBDDs) (Bryant, 1986) and propose a particular coding for dynamicity. We show that our approach allows to solve some major questions in the field of DCSPs.

Neural networks for finite constraint satisfaction

Constraint Satisfaction Problems (CSPs) are in general NP-hard, and a general deterministic polynomial time algorithm is not known. They play a central role in real-life problems. The satisfaction of a Conjunctive Normal Form (CNF-SAT)is the core of any CSP. We present a new modelisation technique for any CSP with finite variable domains, and, in particular, for solving CNF-SAT. The knowledge representation is based on two fundamental types of constraint: the choice constraint, and the exclusion constraint. These models are then implemented by means of several different neural networks, some based on backpropagation learning and others on different procedures. All these networks are trained through a supervised procedure, and learn to efficiently solve CNF-SAT. The results of significant tests are described: they show that some networks can effectively solve the proposed problems.

Neural Networks for Constraint Satisfaction

Abstract Constraint satisfaction problems (CSPs) play a central role in the real world and in computer science. CSPs are in general NP-hard and a general deterministic polynomial time algorithm is not known. CSPs with finite domains for the variables (finite constraint satisfaction problems) are considered. They (and all NP-complete problems) can be reduced in polynomial time to the satisfaction of a conjunctive normal form (CNF-SAT): we present here techniques for solving CNF-SAT by means of several different simulated neural networks. The results of significant tests are described and the reason for the success of some networks over others discussed.