An Algebra for Search Problems and Their Solutions

Abstract

We present an algebraic model for search problems and their solutions. The model includes in its formalism dynamic programming, branch and bound, and other implicit enumeration techniques. Search problems are defined over a finite set of conjuncts, a nonassociative analogue of strings. The generalization to conjuncts is important because it allows the model to handle naturally problems with objects (e.g., binary trees) that combine in a nonassociative manner. The solution of a search problem requires the identification of a minimal element of the set of feasible conjuncts. We model the solution process in terms of two types of operators. Each of these operators enumerates a set of conjuncts, and in so doing infers the indentity of the only conjunct in the set that can possibly be extended to a minimal feasible conjunct. The operators themselves are based on computationally feasible dominance relations that order certain pairs of conjuncts, and whose axioms allow us to infer the orderings of the feasible extensions of such conjuncts.