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

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We study the complexity and expressive power of conjunctive queries over unranked labeled trees, where the tree structures are represented using "axis relations" such as "child", "descendant", and "following" (we consider a superset of the XPath axes) as well as unary relations for node labels. (Cyclic) conjunctive queries over trees occur in a wide range of data management scenarios related to XML, the Web, and computational linguistics. We establish a framework for characterizing structures representing trees for which conjunctive queries can be evaluated efficiently. Then we completely chart the tractability frontier of the problem for our axis relations, i.e., we find all subsetmaximal sets of axes for which query evaluation is in polynomial time. All polynomial-time results are obtained immediately using the proof techniques from our framework. Finally, we study the expressiveness of conjunctive queries over trees and compare it to the expressive power of fragments of XPath. We show that for each conjunctive query, there is an equivalent acyclic positive query (i.e., a set of acyclic conjunctive queries), but that in general this query is not of polynomial size.

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Most of the work on the combination of uni cation algorithms for the union of disjoint equational theories has been restricted to algorithms which compute nite complete sets of uni ers. Thus the developed combination methods usually cannot be used to combine decision procedures, i.e., algorithms which just decide solvability of uni cation problems without computing uni ers. In this paper we describe a combination algorithm for decision procedures which works for arbitrary equational theories, provided that solvability of so-called uni cation problems with constant restrictions|a slight generalization of uni cation problems with constants|is decidable for these theories. As a consequence of this new method, we can for example show that general A-uni ability, i.e., solvability of A-uni cation 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 uni ers. Manfred Schmidt-Schau ' 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 uni cation in the union of disjoint equational theories is nitary, if general uni cation|i.e., uni cation of terms with additional free function symbols|is nitary in the single theories.