Decidability of Reachability for Right-shallow Context-sensitive Term Rewriting Systems

Abstract

The reachability problem for an initial term, a goal term, and a rewrite system is to decide whether the initial term is reachable to goal one by the rewrite system or not. The innermost reachability problem is to decide whether the initial term is reachable to goal one by innermost reductions of the rewrite system or not. A context-sensitive term rewriting system (CS-TRS) is a pair of a term rewriting system and a mapping that specifies arguments of function symbols and determines rewritable positions of terms. In this paper, we show that both reachability for right-linear right-shallow CS-TRSs and innermost reachability for shallow CS-TRSs are decidable. We prove these claims by presenting algorithms to construct a tree automaton accepting the set of terms reachable from a given term by (innermost) reductions of a given CS-TRS.