On one-pass term rewriting

Abstract

Reducing a term with a term rewriting system (TRS) is a highly nondeterministic process and usually no bound for the lengths of the possible reduction sequences can be given in advance. Here we consider two very restrictive strategies of term rewriting, one-pass root-started rewriting and one-pass leaf-started rewriting. If the former strategy is followed, rewriting starts at the root of the given term t and proceeds continuously towards the leaves without ever rewriting any part of the current term which has been produced in a previous rewrite step. When no more rewriting is possible, a one-pass root-started normal form of the term t has been reached. The leaf-started version is similar, but the rewriting is initiated at the leaves and proceeds towards the root. The requirement that rewriting should always concern positions immediately adjacent to parts of the term rewritten in previous steps distinguishes our rewriting strategies from the IO and OI rewriting schemes considered in [5] or [2]. It also implies that the top-down and bottom-up cases are different even for a linear TRS. Let ~ = (E, R) be a TRS over a ranked alphabet E. For any E-tree language T, we denote the sets of one-pass root-started sententiM forms, one-pass root-started normal forms, one-pass leaf-started sentential forms and one-pass leaf-started normal forms of trees in T by lrSn(T), lrNT~(T), I~Sn(T) and I~Nn(T), respectively. We show that the following inclusion problems, where T~ = (E, R) is a left-linear TRS and T1 and T2 are two regular E-tree languages, are decidable. The one-pass root-started sentential form inclusion problem: lrSn(T1) c T2? The one-pass root-started normal form inclusion problem: lrNn(T1) c T2? The one-pass leaf-started sentential form inclusion problem: 1~ Sn(T1) c_ T2? The one-pass leaf-started normal form inclusion problem: 1iNn(T1) c T2? In [9] the inclusion problem for ordinary sentential forms is called the secondorder reachability problem and the problem is shown to be decidable for a TRS T~ which preserves recognizability, i.e. if the set of sentential forms of the trees of