Analyzing Satisfiability and Refutability in Selected Constraint Systems

Abstract

This dissertation is concerned with the satisfiability and refutability problems for several constraint systems. We examine both boolean constraint systems, in which each variable is limited to the values true and false, and polyhedral constraint systems, in which each variable is limited to the set of real numbers R in the case of linear polyhedral systems or the set of integers Z in the case of integer polyhedral systems. An important aspect of our research is that we focus on providing certificates. That is, we provide satisfying assignments or easily checkable proofs of infeasibility depending on whether the instance is feasible or not. Providing easily checkable certificates has become a much sought after feature in algorithms, especially in light of spectacular failures in the implementations of some well-known algorithms. There exist a number of problems in the constraint-solving domain for which efficient algorithms have been proposed, but which lack a certifying counterpart. When examining boolean constraint systems, we specifically look at systems of 2-CNF clauses and systems of Horn clauses. When examining polyhedral constraint systems, we specifically look at systems of difference constraints, systems of UTVPI constraints, and systems of Horn constraints. For each examined system, we determine several properties of general refutations and determine the complexity of finding restricted refutations. These restricted forms of refutation include read-once refutations, in which each constraint can be used at most once; literal-once refutations, in which for each literal at most one constraint containing that literal can be used; and unit refutations, in which each step of the refutation must use a constraint containing exactly one literal. The advantage of read-once refutations is that they are guaranteed to be short. Thus, while not every constraint system has a read-once refutation, the small size of the refutation guarantees easy checkability. we have collaborated several times. He has contributed greatly to my understanding of horn constraints and this understanding has made many of the results in this dissertation possible. West Virginia University has provided me with the education and reseach opportunities that made this work possible. In particular the Lane Department of Computer Science and Electrical Engineering, along with its chair Dr. Brian Woerner, have given me the knowledge and funding to be able to perform the research which resulted in this dissertation.

The Horn Non-Clausal Class and its Polynomiality

“…A comprehensive account of a variety of theses Horn constraint systems and their corresponding complexity is given in [198]. Whether or not these classes can be extended to NC form is an open issue for us.…”