A Structural Analysis of Modular Termination of Term Rewriting Systems

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 connuence 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 R 1 R 2 of two ((nite) terminating term rewriting systems R 1 , R 2 is non-terminating, then one of the systems, say R 1 , enjoys an interesting (undecidable) property, namely it is not termination preserving under non-deterministic collapses, i.e. R 1 fG(x; y) ! x; G(x; y) ! yg is non-terminating, and the other system R 2 is collapsing, i.e. contains a rule with a variable right hand side. This result generalizes known suucient syntactical criteria for modular termination of rewriting. Then we develop a specialized version of thèincreasing interpretation method' for proving termination of combinations of term rewriting systems. This method is applied to establish modularity of termination for certain classes of term rewriting systems. In particular, termination turns out to be modular for the class of systems, for which termination can be shown by simpliication orderings (this result has recently been obtained by Kurihara & Ohuchi by a similar, but less general proof technique). Moreover, we show that the weaker property of being non-self-embedding which also implies termination is not modular. We prove that the niteness restrictions in our main results concerning the term rewriting systems involved can be considerably weakened. Furthermore, we prove that the minimal rank of potential counterexamples in disjoint unions may be arbitrarily high. Hence, a general analysis of arbitrarily complicated`mixed' term seems to be necessary when modularity of termination is investigated. Finally, we show that generalizations of our main results are possible for the cases of conditional term rewriting systems as well as for some restricted form of non-disjoint combinations of term rewriting systems involving common constructors.