Uniication in the Union of Disjoint Equational Theories: Combining Decision Procedures Uniication in the Union of Disjoint Equational Theories: Combining Decision Procedures

Most of the work on the combination of uniication algorithms for the union of disjoint equational theories has been restricted to algorithms which compute nite complete sets of uniiers. Thus the developed combination methods usually cannot be used to combine decision procedures, i.e., algorithms which just decide solvability of uniication problems without computing uniiers. In this paper we describe a combination algorithm for decision procedures which works for arbitrary equational theories, provided that solvability of so-called uniication problems with constant restrictions|a slight generalization of uniication problems with constants|is decidable for these theories. As a consequence of this new method, we can for example show that general A-uniiability, i.e., solvability of A-uniication problems with free function symbols, is decidable. Here A stands for the equational theory of one associative function symbol. Our method can also be used to combine algorithms which compute nite complete sets of uniiers. Manfred Schmidt-Schauu' combination result , the until now most general result in this direction, can be obtained as a consequence of this fact. We also get the new result that uniication in the union of disjoint equational theories is nitary, if general uniication| i.e., uniication of terms with additional free function symbols|is nitary in the single theories.