Generalized sufficient conditions for modular termination of rewriting

Abstract

Modular properties of term rewriting systems, i.e. properties which are preserved under disjoint unions, have attracted an increasing attention within the last few years. Whereas confluence is modular this does not hold true in general for termination. By means of a careful analysis of potential counterexamples we prove the following abstract result. Whenever the disjoint union $\mathcal{R}_1 \oplus \mathcal{R}_2$ of two (finite) terminating term rewriting systems $\mathcal{R}_1 ,\mathcal{R}_2$ is non-terminating, then one of the systems, say $\mathcal{R}_1$, enjoys an interesting (undecidable) property, namely it is not termination preserving under non-deterministic collapses, i.e. $\mathcal{R}_1$⊕G({x,y}) → x,G({x, y}) → y is non-terminating, and the other system $\mathcal{R}_2$ is collapsing, i.e. contains a rule with a variable right hand side. This result generalizes known sufficient syntactical criteria for modular termination of rewriting and provides the basis for a couple of derived modularity results. Furthermore, we prove that the minimal rank of potential counterexamples in disjoint unions may be arbitrarily high which shows that interaction of systems in such disjoint unions may be very subtle. Finally, extensions and generalizations of our main results in various directions are discussed and sketched.

This research was supported by the ‘Deutsche Forschungsgemeinschaft, SFB 314 (D4-Projekt)’.