Context unification is in PSPACE

Contexts are terms with one `hole', i.e. a place in which we can substitute an argument. In context unification we are given an equation over terms with variables representing contexts and ask about the satisfiability of this equation. Context unification is a natural subvariant of second-order unification, which is undecidable, and a generalization of word equations, which are decidable, at the same time. It is the unique problem between those two whose decidability is uncertain (for already almost two decades). In this paper we show that the context unification is in PSPACE. The result holds under a (usual) assumption that the first-order signature is finite. This result is obtained by an extension of the recompression technique, recently developed by the author and used in particular to obtain a new PSPACE algorithm for satisfiability of word equations, to context unification. The recompression is based on performing simple compression rules (replacing pairs of neighbouring function symbols), which are (conceptually) applied on the solution of the context equation and modifying the equation in a way so that such compression steps can be in fact performed directly on the equation, without the knowledge of the actual solution.Comment: 27 pages, submitted, small notation changes and small improvements over the previous tex

Decidability of Bounded Higher-Order Unification

It is shown that unifiability of terms in the simply typed lambda calculus with beta and eta rules becomes decidable if there is a bound on the number of bound variables and lambdas in a unifier in eta-expanded beta-normal form

Decidability of bounded higher order unification

It is shown that unifiability of terms in the simply zyped lambda calculus with β and η rules becomes decidable if there is a bound on the number of bound variables and lambdas in an unifier in η-long β-normal form

Decidability of bounded higher order unification

It is shown that unifiability of terms in the simply zyped lambda calculus with β and η rules becomes decidable if there is a bound on the number of bound variables and lambdas in an unifier in η-long β-normal form