Jean H. Gallier

Linear-time algorithms for testing the satisfiability of propositional horn formulae

New algorithms for deciding whether a (propositional) Horn formula is satisfiable are presented. If the Horn formula A contains K distinct propositional letters and if it is assumed that they are exactly P1,…, PK, the two algorithms presented in this paper run in time O(N), where N is the total number of occurrences of literals in A. By representing a Horn proposition as a graph, the satisfiability problem can be formulated as a data flow problem, a certain type of pebbling. The difference between the two algorithms presented here is the strategy used for pebbling the graph. The first algorithm is based on the principle used for finding the set of nonterminals of a context-free grammar from which the empty string can be derived. The second algorithm is a graph traversal and uses a “call-by-need” strategy. This algorithm uses an attribute grammar to translate a propositional Horn formula to its corresponding graph in linear time. Our formulation of the satisfiability problem as a data flow problem appears to be new and suggests the possibility of improving efficiency using parallel processors.

Quadratic Optimization Problems

In this chapter, we consider two classes of quadratic optimization problems that appear frequently in engineering and in computer science (especially in computer vision): 1. Minimizing

Constructive logics: Part I: a tutorial on proof systems and typed l-calculi

f(x)=\frac{1}{2}x^\top Az+x^\top b

Polymorphic rewriting conserves algebraic strong normalization

over all \(x \varepsilon \mathbb{R}^n\), or subject to linear or affine constraints. 2. Minimizing

Reductions in tree replacement systems

f(x)=\frac{1}{2}x^\top Az+x^\top b

Extending SLD resolution to equational horn clauses using E -unification

over the unit sphere.