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 e#- ciently. 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.

Context and sequence variables allow matching to explore term-trees both in depth and in breadth. It makes context sequence matching a suitable computational mechanism for a rule-based language to query and transform XML, or to specify and verify web sites. Such a language would have advantages of both path-based and pattern-based languages. We develop a context sequence matching algorithm and its extension for regular expression matching, and prove their soundness, termination and completeness properties.

We describe a sound, terminating, and complete matching algorithm for terms built over flexible arity function symbols and context, function, sequence, and individual variables. Context and sequence variables allow matching to move in term trees to arbitrary depth and breadth, respectively. The values of variables can be constrained by regular expressions which are not necessarily linear. We describe heuristics for optimization, and discuss applications.

Abstract. Context Unification is the problem to decide for a given set of second-order equations E where all second-order variables are unary, whether there exists a unifier, such that for every second-order variable X, theabstractionλx.r instantiated for X has exactly one occurrence of the bound variable x in r. Stratified Context Unification is a specialization where the nesting of second-order variables in E is restricted. It is already known that Stratified Context Unification is decidable, NP-hard, and in PSPACE, whereas the decidability and the complexity of Context Unification is unknown. We prove that Stratified Context Unification is in NP by proving that a size-minimal solution can be represented in a singleton tree grammar of polynomial size, and then applying a generalization of Plandowski’s polynomial algorithm that compares compacted terms in polynomial time. This also demonstrates the high potential of singleton tree grammars for optimizing programs maintaining large terms. A corollary of our result is that solvability of rewrite constraints is NP-complete. 1

We study matching in flat theories both from theoretical and practical points of view. A flat theory is defined by the axiom f(x, f(y), z). = f(x, y, z) that indicates that nested occurrences of the function symbol f can be flattened out. From the theoretical side, we design a procedure to solve a system of flat matching equations and prove its soundness, completeness, and minimality. The minimal complete set of matchers for such a system can be infinite. The procedure enumerates this set and stops if it is finite. We identify a class of problems on which the procedure stops. From the practical point of view, we look into restrictions of the procedure that give an incomplete terminating algorithm. From this perspective, we give a set of rules that, in our opinion, describes the precise semantics for the flat matching algorithm implemented in the Mathematica system. 1.

Sequence variables play an interesting role in unification and matching when dealing with terms in an unranked signature. Sequence Unification generalizes Word Unification and seems to be appealing for information extraction in XML documents, program transformation, and rule-based programming. In this work we study a relation between Sequence Unification and another generalization of Word Unification: Context Unification. We introduce a variant of Context Unification, called Left-Hole Context Unification that

Abstract. We prove that linear second-order matching in the linear λ-calculus with linear occurrences of the unknowns is NP-complete. This result shows that context matching and second-order matching in the linear λ-calculus are, in fact, two different problems. 1

Both Sequence and Context Unification generalize the same problem: Word Unification. Besides that, Sequence Unification solves equations between unranked terms involving sequence variables, and seems to be appealing for information extraction in XML documents, program transformation, knowledge representation, and rule-based programming. It is decidable. Context Unification deals with the same problem for ranked terms involving context variables, and has applications in computational linguistics and program transformation. Its decidability is a long-standing open question. In this work we study a relation between these two problems. We introduce a variant (restriction) of Context Unification, called Left-Hole Context Unification (LHCU), to which Sequence Unification is P-reduced: We define a partial currying procedure to translate sequence unification problems into left-hole context unification problems, and prove soundness of the translation. Furthermore, a precise characterization of the shape of the unifiers allows us to easily reduce Left-Hole Context Unification to (the decidable problem of) Word Unification with Regular Constraints, obtaining then a new decidability proof for Sequence Unification. Finally, we define an extension of Sequence Unification (ESU) and, closing the circle, prove the inter P-reducibility of LHCU and ESU.

We describe a matching algorithm for terms built over flexible arity function symbols and context, function, sequence, and individual variables. The algorithm is called a context sequence matching algorithm. Context variables allow matching to descend in term-trees to arbitrary depth. Sequence variables allow matching to move in term-trees in arbitrary breadth. The ability to explore terms in two orthogonal directions in a uniform way may be useful for querying data available as a large term, like XML documents. We extend the algorithm to process regular constraints and discuss its possible application in XML querying.

Abstract. We prove that higher-order matching in the linear λ-calculus with pairing is decidable. We also establish its NP-completeness under the assumption that the right-hand side of the equation to be solved is given in normal form. 1