TitleOn some variants of second-order unification
Publication TypeBook
Year of Publication2005
AuthorsVillaret M
Series TitleMonografies del IA
PublisherIIIA - CSIC

In this thesis we present several results about Second-Order Unification. It is well known that the Second-Order Unification Problem is in general undecid- able; the frontier between its decidable and undecidable subclasses is thin and it still has not been completely defined. Our purpose is to shed some light on the Second-Order Unification problem and study some of its variants. We have mainly focused our attention on Context Unification and Linear Second-Order Unification. Roughly speaking, these problems are variants of Second-Order Unification where second-order variables are required to be linear. Context Uni- fication was defined more than ten years ago and its decidability has been an open question since then. Here we make relevant contributions to the study of this question. The first result that we present is a simplification on these problems thanks to “curryfication” (Levy and Villaret, 2002). We show that the Context Unification problem can be NP-reduced to the Context Unification problem where, apart from variables, just a single binary function symbol, and first-order constants, are used. We also show that a similar result also holds for Second-Order Unification. The main result of this thesis is the definition of a non-trivial sufficient and necessary condition on the unifiers, for the decidability of Context Unification. The condition is called rank-bound conjecture (Levy and Villaret, 2001) in or- der to enforce our belief about its truthness. It lies on a non-trivial measure of terms, the rank, and claims that, whenever an instance of the Context Uni- fication problem is satisfiable, there exists a unifier with a rank not exceeding a certain bound depending on the size of the problem. Under the assumption of this conjecture, we give a reduction of the satisfiability problem for Context Unification to the (decidable) satisfiability problem of Word Unification with regular constraints. Finally, in the same spirit of the extension of Word Unification with regular constraints, we also study the natural extension of Context Unification by means of tree-regular constraints on variable instantiations. We contribute with two more results: • firstly, we establish a relationship between Linear Second-Order Unification and Context Unification (Levy and Villaret, 2000). Mainly, we reduce Linear Second-Order Unification to Context Unification with tree-regular constraints, these constraints are used to avoid the capture of variables in v Page 10 this process. • Then, we also establish a relationship between Context Unification and the Constraint Language for Lambda Structures (Niehren and Villaret, 2002, 2003). This last formalism is broadly used in the treatment of ambiguous sentences of natural language, and there is currently an effort to quantify its power, and define its computational nature. Relating this constraints language with the unification framework can help us to apply the theoretic results from one side to the other. We also give a brief description of unification and introduce its main distinct kinds and variants. Although our thesis is not directly oriented to practical issues, we also illustrate what has been, and what is the main role of unifica- tion in computational logics and applications, mainly focusing on Higher-Order Unification.