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

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Recently well-quasi orders in general, and homeomorphic embedding in particular, have gained popularity to ensure the termination of program analysis, specialisation and transformation techniques. In this paper,

A notation for lambda terms is described that is useful in contexts where the intensions of these terms need to be manipulated. This notation uses the scheme of de Bruijn for eliminating variable names, thus obviating ff-conversion in comparing terms. This notation also provides for a class of terms that can encode other terms together with substitutions to be performed on them. The notion of an environment is used to realize this `delaying' of substitutions. The precise mechanism employed here is, however, more complex than the usual environment mechanism because it has to support the ability to examine subterms embedded under abstractions. The representation presented permits a fi-contraction to be realized via an atomic step that generates a substitution and associated steps that percolate this substitution over the structure of a term. The operations on terms that are described also include ones for combining substitutions so that they might be performed simultaneously. Our notatio...

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 ...

Well-foundedness is the essential property of orderings for proving termination. We introduce a simple criterion on term orderings such that any term ordering possessing the subterm property and satisfying this criterion is well-founded. The usual path orders fulfil this criterion, yielding a much simpler proof of well-foundedness than the classical proof depending on Kruskal's theorem. Even more, our approach covers non-simplification orders like spo and gpo which can not be dealt with by Kruskal's theorem. For finite alphabets we present completeness results, i. e., a term rewriting system terminates if and only if it is compatible with an order satisfying the criterion. For infinite alphabets the same completeness results hold for a slightly different criterion.

Termination is an important issue in the theory of term rewriting. In general termination is undecidable. There are nevertheless several methods successful in special cases. In [5] we introduced the notion of total termination: basically terms are interpreted compositionally in a total well-founded order, in such a way that rewriting chains map to descending chains. Total termination is thus a semantic notion. It turns out that most of the usual techniques for proving termination fall within the scope of total termination. This paper consists of two parts. In the first part we introduce a generalization of recursive path order presenting a new proof of its well-foundedness without using Kruskal's theorem. We also show that the notion of total termination covers this generalization. In the second part we present some syntactical characterizations of total termination that can be used to prove that many term rewriting systems are not totally terminating and hence outside the scope of the...

In this paper we investigate the concept of simple termination. A term rewriting system is called simply terminating if its termination can be proved by means of a simplification order. The basic ingredient of a simplification order is the subterm property, but in the literature two different definitions are given: one based on (strict) partial orders and another one based on preorders (or quasi-orders). We argue that there is no reason to choose the second one, while the first one has certain advantages. Simplification orders are known to be well-founded orders on terms over a finite signature. This important result no longer holds if we consider infinite signatures. Nevertheless, well-known simplification orders like the recursive path order are also well-founded on terms over infinite signatures, provided the underlying precedence is well-founded. We propose a new definition of simplification order, which coincides with the old one (based on partial orders) in case of finite signatu...

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Abstract. Proofs of termination are essential for establishing the correct behavior of computing systems. There are various ways of establishing termination, but the most general involves the use of ordinals. An example of a theorem proving system in which ordinals are used to prove termination is ACL2. In ACL2, every function defined must be shown to terminate using the ordinals up to ɛ0. We use a compact notation for the ordinals up to ɛ0 (exponentially more succinct than the one used by ACL2) and define efficient algorithms for ordinal addition, subtraction, multiplication, and exponentiation. In this paper we describe our notation and algorithms, prove their correctness, and analyze their complexity. 1