Context-sensitive rewriting is a simple restriction of rewriting which is formalized by imposing fixed restrictions on replacements. Such a restriction is given on a purely syntactic basis: it is (explicitly or automatically) specified on the arguments of symbols of the signature and inductively extended to arbitrary positions of terms built from those symbols. Termination is not only preserved but usually improved and several methods have been developed to formally prove it. In this paper, we investigate the definition, properties, and use of context-sensitive rewriting strategies, i.e., particular, fixed sequences of context-sensitive rewriting steps. We study how to define them in order to obtain efficient computations and to ensure that context-sensitive computations terminate whenever possible. We give conditions enabling the use of these strategies for root-normalization, normalization, and infinitary normalization. We show that this theory is suitable for formalizing ...

Context-sensitive rewriting is a computational restriction of term rewriting used to model non-strict (lazy) evaluation in functional programming. The goal of this paper is the study and development of techniques to analyze the termination behavior of context-sensitive rewrite systems. For that purpose, several methods have been proposed in the literature which transform context-sensitive rewrite systems into ordinary rewrite systems such that termination of the transformed ordinary system implies termination of the original context-sensitive system. In this way, the huge variety of existing techniques for termination analysis of ordinary rewriting can be used for context-sensitive rewriting, too. We analyze the existing transformation techniques for proving termination of context-sensitive rewriting and we suggest two new transformations. Our first method is simple, sound, and more powerful than the previously proposed transformations. However, it is not complete, i.e., there are terminating context-sensitive rewrite systems that are transformed into non-terminating term rewrite systems. The second method that we present in this paper is both sound and complete. All these observations also hold for rewriting modulo associativity and commutativity.

Context-sensitive rewriting (CSR) is a simple restriction of rewriting which can be used e.g. for modelling non-eager evaluation in programming languages. Many times termination is a crucial property for program verification. Hence, developing tools for automatically proving termination of CSR is necessary. All known methods for...

Declarative languages such as OBJ*, CafeOBJ, and Maude use syntactic annotations to introduce replacement restrictions aimed at improving termination or efficiency of computations. Unfortunately, there is a lack of formal techniques for proving such benefits. We show that context-sensitive rewriting and on-demand rewriting provide a suitable framework to address this problem. We provide methods to analyze termination of on-demand rewriting and apply them to analyze termination of OBJ*, CafeOBJ, and Maude programs. Keywords Program analysis, replacement restrictions, term rewriting, termination 1.

We present two new transformation techniques for proving termination of context-sensitive rewriting. Our first method is simple, sound, and more powerful than previously suggested transformations. However, it is not complete, i.e., there are terminating context-sensitive rewrite systems that are transformed into non-terminating term rewrite systems. The second method that we present in this paper is both sound and complete. This latter result can be interpreted as stating that from a termination perspective there is no reason to study context-sensitive rewriting.

Abstract. Termination is one of the most interesting problems when dealing with context-sensitive rewrite systems. Although there is a good number of techniques for proving termination of context-sensitive rewriting (CSR), the dependency pair approach, one of the most powerful techniques for proving termination of rewriting, has not been investigated in connection with proofs of termination of CSR. In this paper, we show how to use dependency pairs in proofs of termination of CSR. The implementation and practical use of the developed techniques yield a novel and powerful framework which improves the current state-of-the-art of methods for proving termination of CSR.

We show how to generate well-founded and stable term orderings based on polynomial interpretations over the real numbers. Monotonicity (another usual requirement in termination proofs) can, then, be gradually introduced in the interpretations to deal with dierent applications.

We investigate termination of rewriting computations guided by strategy annotations. We show that proofs of termination can be obtained by proving (innermost) termination of context-sensitive rewriting (CSR). Hence, we investigate how to prove innermost termination of CSR using existing methods for proving termination of CSR.

Our main goal is automating termination proofs for programs in rewriting-based languages with features such as: (i) expressive type structures, (ii) conditional rules, (iii) matching modulo axioms, and (iv) contextsensitive rewriting. Specifically, we present a new operational termination method for membership equational programs with features (i)-(iv) that can be applied to programs in membership equational logic (MEL). The method first transforms a MEL program into a simpler, yet semantically equivalent, conditional order-sorted (OS) program. Subsequent trasformations make the OS-program unconditonal, and, finally, unsorted. In particular, we extend and generalize to this richer setting an order-sorted termination technique for unconditional OS programs proposed by Ölveczky and Lysne. An important advantage of our method is that it minimizes the use of conditional rules and produces simpler transformed programs whose termination is often easier to prove automatically.

Context-sensitive rewriting (CSR) is a restriction of rewriting which forbids reductions on selected arguments of functions. Proving termination of CSR is an interesting problem with several applications in the fields of term rewriting and programming languages. Several methods have been developed for proving termination of CSR. The new version of MU-TERM which we present here implements all currently known techniques. Furthermore, we show how to combine them to furnish MU-TERM with an expert which is able to automatically perform the termination proofs. Finally, we provide a first experimental evaluation of the tool.