Lazy rewriting (LR) is intended to improve the termination behavior of TRSs. This is attempted by restricting reductions for selected arguments of functions. Similarly, context-sensitive rewriting (CSR) forbids any reduction on those arguments. We show that, under certain conditions, LR and CSR coincide. On the basis of this result, we also describe a transformation which permits proving termination of LR as termination of CSR for the transformed system. Since there is a number of different techniques for proving termination of CSR, this provides a (first) formal framework for proving termination of lazy rewriting.

On-demand rewriting provides an operational model for term rewriting controlled by annotations consisting of (sets of) positive and negative integers referring the arguments of function symbols. Contextsensitive rewriting and lazy rewriting provide (different) operational models for positive annotations. In this paper we prove that, under certain conditions, the three operational models coincide. In this case, it makes sense using context-sensitive rewriting as it is the simplest one. This fact also permits proving termination of lazy rewriting by proving termination of context-sensitive rewriting for a transformed rewrite system.

Lazy rewriting is a proper restriction of term rewriting that dynamically restricts the reduction of certain arguments of functions in order to obtain termination. In contrast to context-sensitive rewriting, reductions at such argument positions are not completely forbidden but delayed. Based on the observation that the only existing (non-trivial) approach to prove termination of such lazy rewrite systems is flawed, we develop a modified approach for transforming lazy rewrite systems into contextsensitive ones that is sound and complete with respect to termination. First experimental results with this transformation based technique are encouraging.

I'd like to conclude by emphasizing what a wonderful eld this is to work in. Logical reasoning plays such a fundamental role in the spectrum of intellectual activities that advances in automating logic will inevitably have a profound impact in many intellectual disciplines. Of course, these things take time. We tend to be impatient, but we need some historical perspective. The study of logic has a very long history, going back at least as far as Aristotle. During some of this time not very much progress was made. It's gratifying to realize how much has been accomplished in the less than fty years since serious e orts to mechanize logic began.