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

The First-Order Theory of One Step Rewriting in Linear Noetherian Systems is Undecidable

. We construct a finite linear finitely terminating rewrite rule system with undecidable theory of one step rewriting. 1 Preliminaries Given a functional signature \Sigma with constants and a finite rewrite rule system R, consider the model M = hT (\Sigma ); Ri, where T (\Sigma ) is the Herbrand universe over \Sigma and R = fhs; ti j s; t 2 T (\Sigma ) s !R tg ` T (\Sigma ) \Theta T (\Sigma ) is the one step rewrite relation on T (\Sigma ) generated by R. The presence of constants in \Sigma is necessary to guarantee that T (\Sigma ) is not empty. Let L be the first-order language without equality containing the only binary relation symbol R. The first-order theory of one step rewriting in R is the set of all sentences (closed formulas) of L true in M . Notice that the only non-logical symbol used in formulas of the theory is R, and the symbols of \Sigma are not allowed in formulas. The aim of this note is to give an example of a finite linear finitely terminating system with unde..

The first-order theory of one step rewriting in linear noetherian systems is undecidable

We construct a finite linear finitely terminating rewrite rule system with undecidable theory of one step rewriting