Completeness of Combinations of Constructor Systems

Abstract

A term rewriting system is called complete if it is both confluent and strongly normalising. Barendregt and Klop showed that the disjoint union of complete term rewriting systems does not need to be complete. In other words, completeness is not a modular property of term rewriting systems. Toyama, Klop and Barendregt showed that completeness is a modular property of left-linear term rewriting systems. In this paper we show that it is sufficient to impose the constructor discipline for obtaining the modularity of completeness. This result is a simple consequence of a quite powerful divide and conquer technique for establishing completeness of such constructor systems. Our approach is not limited to systems which are composed of disjoint parts. The importance of our method is that we may decompose a given constructor system into parts which possibly share function symbols and rewrite rules in order to infer completeness. We obtain a similar technique for semi-completeness, i.e. the combination of confluence and weak normalisation.

Modular Properties of Composable Term Rewriting Systems

“…Using the strong Theorem 5.11, Gramlich (1994b) proved that completeness is modular for constructor-sharing overlay systems. Corollary 5.12 shows that this is true even for composable TRSs, so the result of Middeldorp and Toyama (1993) is actually a special case thereof. Lately, showed that completeness is modular for the class of disjoint conditional overlay systems with joinable critical pairs.…”

“…An overlay system is complete if and only if it is locally con uent and innermost terminating. The main result of Middeldorp and Toyama (1993) stating that completeness is a modular property of composable constructor systems follows from Corollary 5.12 because a constructor system is an overlay system and the combined system of two composable constructor systems is again a constructor system. Also, modularity of termination (or equivalently completeness) for non-overlapping TRSs is a consequence of Corollary 5.12 because those systems are locally con uent overlay systems and the combined system of two non-overlapping composable systems is non-overlapping.…”

“…Several recent papers deal with the modularity of completeness for constructor systems and the more general class of overlay systems, respectively. The investigation of constructor systems started with the work of Middeldorp and Toyama (1993). Their main result has been extended to certain classes of hierarchical combinations by Krishna Rao (1993) and Dershowitz (1994).…”

“…Among other things, they showed that con uence is not modular for constructor-sharing systems. Middeldorp and Toyama (1993) introduced composable systems which have to contain all rewrite rules that dene a de ned symbol whenever that symbol is shared. The authors, however, restricted their investigations to constructor systems (where no proper subterm of a left-hand side of a rewrite rule is allowed to contain de ned symbols).…”