The Existential Theories of Term Algebras with the Knuth-Bendix Orderings are Decidable

We consider term algebras with simpliication orderings (which are mono-tonic, and well-founded). Solving constraints (quantiier-free formulas) in term algebras with this kind of orderings has several important applications like pruning search space in automated deduction and proving termination and connuence of term rewriting systems. Two kinds of ordering are normally used in automated deduction: Knuth-Bendix ordering and various versions of recursive path orderings. There exists extensive literature on solving recursive path ordering constraints 1, 4], but no algorithms for solving Knuth-Bendix ordering constraints are known. We proved that the problem of solving Knuth-Bendix ordering constraints is decidable and N P { hard. Let us brieey describe the proof, for the full version we refer to 3]. We consider term algebras in a nite signature with at least one constant, denoted TA((). Let us now deene Knuth-Bendix orderings on TA(() 2]. The deenition of Knuth-Bendix ordering is parametrized by a weight function on , i.e., a function w : ! N, and a linear ordering on. We require from the weight function the following: if w(f) = 0 and f is unary, then f must be the greatest w.r.t. in , and weights of constants are positive. We deene the weight of a ground term as a sum of weights of func-tors occurring in the term. Given a weight function w and a linear ordering on , the Knuth-Bendix ordering on TA(() is the binary relation > KB deened as follows. For any ground terms g(t 1