Maria C. F. Ferreira

Total termination of term rewriting

We investigate proving termination of term rewriting systems by a compositional interpretation of terms in a total well-founded order. This kind of termination is calledtotal termination. Equivalently total termination can be characterized by the existence of an order on ground terms which is total, wellfounded and closed under contexts. For finite signatures, total termination implies simple termination. The converse does not hold. However, total termination generalizes most of the usual techniques for proving termination (including the well-known recursive path order). It turns out that for this kind of termination the only interesting orders below ɛ0 are built from the natural numbers by lexicographic product and the multiset construction. By examples we show that both constructions are essential. Most of the techniques used are based on ordinal arithmetic.

Dummy Elimination: Making Termination Easier

We investigate a technique whose goal is to simplify the task of proving termination of term rewriting systems. The technique consists of a transformation which eliminates function symbols considered “useless” and simplifies the rewrite rules. We show that the transformation is sound, i. e., termination of the original system can be inferred from termination of the transformed one. For proving this result we use a new notion of lifting of orders that is a generalization of the multiset construction.

Context-Sensitive AC-Rewriting

Context-sensitive rewriting was introduced in and consists of syntactical restrictions imposed on a Term Rewriting System indicating how reductions can be performed. So context-sensitive rewriting is a restriction of the usual rewrite relation which reduces the reduction space and allows for a finer control of the reductions of a term. In this paper we extend the concept of context-sensitive rewriting to the framework rewriting modulo an associative-commutative theory in two ways: by restricting reductions and restricting AC-steps, and we then study this new relation with respect to the property of termination.

Well-foundedness of Term Orderings

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 Kruskals theorem. Even more, our approach covers non-simplification orders like spo and gpo which can not be dealt with by Kruskals theorem.

Dummy Elimination in Equational Rewriting

In [5] we introduced the concept of dummy elimination in term rewriting: a transformation on terms which eliminates function symbols simplifying the rewrite rules and making, in general, the task of proving termination easier. Here we consider the more general setting of rewriting modulo an equational theory; we show that, in contrast with most techniques developed for proving termination of rewrite systems, dummy elimination remains valid in the presence of equational theories. Furthermore using the same proof technique, the soundness of a family of transformations (containing dummy elimination) can be shown. This work was motivated by an application in the area of Process Algebra.

Syntactical Analysis of Total Termination

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 Kruskals 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 usual techniques. One of these characterizations can be considered as a sound and complete description of totality of orderings on terms.

Total Termination of Term Rewriting

We investigate proving termination of term rewriting systems by interpretation of terms in a compositional way in a total wellfounded order. This kind of termination is called total termination. On one hand it is more restrictive than simple termination, on the other it generalizes most of the usual techniques for proving termination. For total termination it turns out that below Iµ0 the only orders of interest are built from the natural numbers by lexicographic product and the multiset construction. By examples we show that both constructions are essential. For a wide class of term rewriting systems we prove that total termination is a modular property. Most of our techniques are based on ordinal arithmetic.

λ-calculi with explicit substitutions and composition which preserve β-strong normalization

We study preservation of β-strong normalization by λ d and λ dn , two confluent λ-calculi with explicit substitutions defined in [10]; the particularity of these calculi is that both have a composition operator for substitutions. We develop an abstract simulation technique allowing to reduce preservation of β-strong normalization of one calculus to that of another one, and apply said technique to reduce preservation of β-strong normalization of λ d and λ dn to that of λ f , another calculus having no composition operator. Then, preservation of β-strong normalization of λ f is shown using the same technique as in [2]. As a consequence, λ d and λ dn become the first λ-calculi with explicit substitutions having composition and preserving β- strong normalization. We also apply our technique to reduce preservation of β-strong normalization of the calculus λ v in [14] to that of λ f .

λCalculi with Explicit Substitutions Preserving Strong Normalization

Abstract This paper studies preservation of β-strong normalization by two different confluent λ-calculi with explicit substitutions defined in [96]; the particularity of these calculi, called λd and λdn respectively, is that both have a (weak) composition operator for substitutions. We apply an abstract simulation technique to reduce preservation of β-strong normalization of λd and λdn to that of another calculus, called λf having no composition operator. Then, preservation of β-strong normalization of λf is shown using the same technique as in [2]. As a consequence, λd and λdn become the first λ-calculi with explicit substitutions having (weak) composition and preserving β-strong normalization. As an aside, we also show how to apply our technique to reduce preservation of β-strong normalization of the calculus λv in [20] to that of λf.