. In functional language implementation, there is a folklore belief that there is a conflict between implementing call-by-need semantics and parallel evaluation. In this note we illustrate this by proving that reduction algorithms of a certain general and commonly used form which give call-byneed semantics offer very little parallelism. The analysis of lazy pattern-matching which leads to the above result also suggests an efficient sequential algorithm for the evaluation of a class functional programs satisfying certain constraints, an algorithm which respects the mathematical semantics of the program considered as a term rewrite system. 1 Introduction Huet and L'evy [Huet and L'evy, 1979, Huet and L'evy, 1991] have considered the problem of call by need computation of normal forms in orthogonal term rewrite systems. Call by need here means that no redex is ever reduced unless it must be reduced in order to compute the normal form. In general, such a redex cannot be effectiv...

We present a new criterion for confluence of (possibly) non-terminating leftlinear term rewriting systems. The criterion is based on certain strong joinability properties of parallel critical pairs . We show how this criterion relates to other well-known results, consider some special cases and discuss some possible extensions. 1 Introduction and Overview Computation formalisms which are based on rewriting systems heavily rely on the fundamental properties of termination and confluence. For terminating and confluent systems normal forms exist and are unique, irrespective of the computation (rewriting) strategy. For non-terminating but confluent systems, normal forms need not exist, however, if a normal form exists, it is still unique. More generally, any (possibly infinite) diverging computations can be joined again. In some cases, non-termination is inherently unavoidable, in other cases it may be very difficult to verify this property. Hence the problem of proving confluence (with o...

this article we solve this open problem,showing that UN

We study language-theoretical properties of the set of reducible ground terms and its complement- the set of ground normal forms induced by agiven rewriting system. As a tool for our analysis we introduce the property of nite irreducibility of a term with respect to a variable and prove it to be decidable. It turns out that this property generalizes numerous interesting properties of the language of ground normal forms. In particular, we show that testing regularity of this language can be reduced to verifying this property. In this way we prove the decidability of the regularity of the set of ground normal forms, the problem mentioned in the list of open problems in rewriting [Dershowitz et al., 1991]. Also, the decidability of the existence of an equivalent ground term rewriting system and some other results are proved. 2 1

this paper. References

Knuth-Bendix completion is a technique for equational automated theorem proving based on term rewriting. This classic procedure is parametrized by an equational theory and a (well-founded) reduction order used at runtime to ensure termination of intermediate rewriting systems. Any reduction order can be used in principle, but modern completion tools typically implement only a few classes of such orders (e.g., recursive path orders and polynomial orders). Consequently, the theories for which completion can possibly succeed are limited to those compatible with an instance of an implemented class of orders. Finding and specifying a compatible order, even among a small number of classes, is challenging in practice and crucial to the success of the method. In this thesis, a new variant on the Knuth-Bendix completion procedure is developed in which no order is provided by the user. Modern termination-checking methods are instead used to verify termination of rewriting systems. We prove the new method correct and also present an implementation called Slothrop which obtains solutions for theories that do not admit typical orders and that have not

In [KM91], Klop and Middeldorp conjectured that deciding strong sequentiality for orthogonal term rewriting systems is NP-Complete. This problem appeared as "Problem 8" in the list of open problems in rewriting published in [DJK91]. In this article we show that the problem is in co-NP. If, as we conjecture, the problem is also in NP , this reduces its chances of being NP-Complete.

We provide a global technique, called neatening, for the study of modularity of left-linear Term Rewriting Systems. Objects called bubbles are identi ed as the responsibles of most of the problems occurring in modularity, and the concept of well-behaved (from the modularity point of view) reduction, said neat reduction, is introduced. Neating consists of two steps: the rst is proving a property is modular when only neat reductions are considered � the second is to `neaten ' a generic reduction so to obtain a neat one, thus showing that restricting to neat reductions is not limitative. This general technique is used to provide a unique, uniform method able to prove all the existing results on the modularity ofevery basic property of left-linear Term Rewriting Systems, and also to provide new results on the modularity of termination. 1

. We introduce simultaneous critical pairs, which account for simultaneous overlapping of several rewrite rules. Based on this, we introduce a new CR-criterion widely applicable to arbitrary left-linear term rewriting systems. Our result extends the well-known criterion given by Huet (1980), Toyama (1988), and Oostrom (1997) and incomparable with other well-known criteria for left-linear systems. 1 Introduction Church-Rosser (CR for short) property is one of the most important properties of rewrite systems. For (nite) strongly normalising systems, this property is decidable by testing joinability of critical pairs whereas for arbitrary systems, undecidable even if they are weak CR. For this reason, many researchers have been interested in various (decidable) sucient conditions ensuring this property. We refer to these conditions as CR-criteria. To obtain CR-criteria, the critical-pair-based approach is still available as far as left-linear systems are concerned (Note that non-left-...

A property P of term rewriting system is persistent if for any many-sorted term rewriting system R, R has the property P iff its underlying term rewriting system 2(R), which results from R by omitting its sort information, has the property P. It is shown that termination is a persistent property of many-sorted term rewriting systems that contain only variables of the same sorts. This is the positive solution to a problem of Zantema, which has been appeared as Rewriting Open Problem 60 in literature. 1