Term Rewriting Systems play an important role in various areas, such as abstract data type specifications, implementations of functional programming languages and automated deduction. In this chapter we introduce several of the basic comcepts and facts for TRS's. Specifically, we discuss Abstract Reduction 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 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...

Term rewriting systems are widely used throughout computer science as they provide an abstract model of computation while retaining a comparatively simple syntax and semantics. In order to reason within large term rewriting systems, structuring operations are used to build large term rewriting systems from smaller ones. Of particular interest is whether key properties are modular, thatis,ifthe components of a structured term rewriting system satisfy a property, then does the term rewriting system as a whole? A body of literature addresses this problem, but most of the results and proofs depend on strong syntactic conditions and do not easily generalize. Although many specific modularity results are known, a coherent framework which explains the underlying principles behind these results is lacking. This thesis posits that part of the problem is the usual, concrete and syntaxoriented semantics of term rewriting systems, and that a semantics is needed which on the one hand elides unnecessary syntactic details but on the other hand still possesses enough expressive power to model the key concepts arising from

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 result. Whenever the disjoint union R 1 \Phi R 2 of two (finite) 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 \Phi 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 sufficient syntactical criteria for modular termination of rewriting. Then we develop a specialized version of the `increasing 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