A new kind of transformation of term rewriting systems (TRS) is proposed, depending on a choice for a model for the TRS. The labelled TRS is obtained from the original one by labelling operation symbols, possibly creating extra copies of some rules. This construction has the remarkable property that the labelled TRS is terminating if and only if the original TRS is terminating. Although the labelled version has more operation symbols and may have more rules (sometimes infinitely many), termination is often easier to prove for the labelled TRS than for the original one. This provides a new technique for proving termination, making classical techniques like path orders and polynomial interpretations applicable even for non-simplifying TRS's. The requirement of having a model can slightly be weakened, yielding a remarkably simple termination proof of the system SUBST of [11] describing explicit substitution in λ-calculus.

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 R1 \Phi R2 of two (finitely branching) terminating term rewriting systems R1 , R2 is non-terminating, then one of the systems, say R1 , enjoys an interesting (undecidable) property, namely it is not termination preserving under non-deterministic collapses, i.e. R1 \Phi fG(x; y) ! x; G(x; y) ! yg is non-terminating, and the other system R2 is collapsing, i.e. contains a rule with a variable right hand side. This result generalizes known sufficient 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 pote...

this paper we prove several new modularity results for unconditional and conditional term rewriting systems. Most of the known modularity results for the former systems hold for disjoint or constructor-sharing combinations. Here we focus on a more general kind of combination: so-called composable systems. As far as conditional term rewriting systems are concerned, all known modularity result but one apply only to disjoint systems. Here we investigate conditional systems which may share constructors. Furthermore, we refute a conjecture of Middeldorp (1990, 1993). 1. Introduction

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. 1. Introduction

It is well-known that termination is not a modular property of term rewriting systems, i.e., it is not preserved under disjoint union. The objective of this paper is to provide a "uniform framework" for sufficient conditions which ensure the modularity of termination. We will prove the following result. Whenever the disjoint union of two terminating term rewriting systems is non-terminating, then one of the systems is not C E -terminating (i.e., it looses its termination property when extended with the rules Cons(x; y) ! x and Cons(x; y) ! y) and the other is collapsing. This result has already been achieved by Gramlich [7] for finitely branching term rewriting systems. A more sophisticated approach is necessary, however, to prove it in full generality. Most of the known sufficient criteria for the preservation of termination [24, 15, 13, 7] follow as corollaries from our result, and new criteria are derived. This paper particularly settles the open question whether simple termination ...

We investigate restricted termination and confluence properties of term rewriting systems, in particular weak termination and innermost termination, and their interrelation. New criteria are provided which are sufficient for the equivalence of innermost / weak termination and uniform termination of term rewriting systems. These criteria provide interesting possibilities to infer completeness, i.e. termination plus confluence, from restricted termination and confluence properties. Using these basic results we are also able to prove some new results about modular termination of rewriting. In particular, we show that termination is modular for some classes of innermost terminating and locally confluent term rewriting systems, namely for non-overlapping and even for overlay systems. As an easy consequence this latter result also entails a simplified proof of the fact that completeness is a decomposable property of so-called constructor systems. Furthermore we show how to obtain similar re...

This paper is concerned with the impact of stepwise development methodologies on prototyping.

Conditional rewriting is universally recognized as being much more complicated than unconditional rewriting. In this paper we study how much of conditional rewriting can be automatically inferred from the simpler theory of unconditional rewriting. We introduce a new tool, called unraveling, to automatically translate a conditional term rewriting system (CTRS) into a term rewriting system (TRS). An unraveling enables to infer properties of a CTRS by studying the corresponding ultra-property on the corresponding TRS. We show how to rediscover properties like decreasingness, and to give easy proofs of some existing results on CTRSs. Moreover, we show how unravelings provide a valuable tool to study modularity of CTRSs, automatically giving a multitude of new results.

ion variables which are variables coming from an abstraction, either during preprocessing or during the algorithm itself. 3. Introduced variables which are variables introduced by the unification algorithms for each theory. We make the very natural assumption that the unification algorithm for each theory may recognize initial, abstraction and introduced variables and never assigns an introduced variable to a non-introduced one or an abstraction variable to an initial one. With this assumption, our combination algorithm will always make an introduced variable appear in at most one \Gamma i . We may thus also suppose that the domain of each solution does not contain an introduced variable. This does not compromise the soundness of our algorithm. The combination algorithm is described by the two rules given in figure 2. In the rule UnifSolve i , ae SF is obtained by abstracting aliens in the range of ae by fresh variables. ae F i is the substitution such that xae = xae SF ae F i for al...

We present a new approach for proving termination of rewrite systems by innermost termination. From the resulting abstract criterion we derive concrete conditions, based on critical peak properties, under which innermost termination implies termination (and confluence). Finally, we show how to apply the main results for providing new sufficient conditions for the modularity of termination.