Abstract. Context unification (CU) is the open problem of solving context equations for trees. We distinguish a new decidable variant of CU– well-nested CU – and present a new unification algorithm that solves well-nested context equations in non-deterministic polynomial time. We show that minimal well-nested solutions of context equations can be composed from the material present in the equation (see Theorem 1). This property is wishful when modeling natural language ellipsis in CU. 1

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.

Abstract. Context unification was originally defined by H. Comon in ICALP’92, as the problem of finding a unifier for a set of equations containing first-order variables and context variables. These context variables have arguments, and can be instantiated by contexts. In other words, they are second-order variables that are restricted to be instantiated by linear terms (a linear term is a λ-expression λx1 ···λxn.t where every xi occurs exactly once in t). In this paper, we prove that, if the so called rank-bound conjecture is true, then the context unification problem is decidable. This is done reducing context unification to solvability of traversal equations (a kind of word unification modulo certain permutations) and then, reducing traversal equations to word equations with regular constraints. 1

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

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. Parallelism constraints are logical descriptions of trees. Parallelism constraints subsume dominance constraints and are equal in expressive power to context unification. Parallelism constraints belong to the constraint language for lambda structures (CLLS) which serves for modeling natural language semantics. In this paper, we investigate the extension of parallelism constraints by tree regular constraints. This canonical extension is subsumed by the monadic secondorder logic over parallelism constraints. We analyze the precise expressiveness of this extension on basis of a new relationship between tree automata and logic. Our result is relevant for classifying different extensions of parallelism constraints, as in CLLS. Finally, we prove that parallelism constraints and context unification remain equivalent when extended with tree regular constraints.

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.

Second-Order Unification is undecidable even for very specialized fragments. The signature plays a crucial role in these fragments. If all symbols are monadic, then the problem is NP-complete, whereas it is enough to have just one binary constant to lose decidability. In this work we reduce Second-Order Unification to Second-Order Unification with a signature that contains just one binary function symbol and constants. The reduction is based on partially currying the equations by using the binary function symbol for explicit application @. Our work simplifies the study of Second-Order Unification and some of its variants, like Context Unification and Bounded Second-Order Unification.