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.

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.

Abstract. Partial-inversion compilers generate programs which compute some unknown inputs of given programs from a given output and the rest of inputs whose values are already given. In this paper, we propose a partial-inversion compiler of constructor term rewriting systems. The compiler automatically generates a conditional term rewriting system, and then unravels it to an unconditional system. To improve the efficiency of inverse computation, we show that innermost strategy is usable to obtain all solutions if the generated system is right-linear. 1

The dependency pairs method is one of the most powerful technique for proving termination of rewriting and it is currently central in most automatic termination provers. Recently, it has been adapted to be used in proofs of termination of context-sensitive rewriting. The use of collapsing dependency pairs i.e., having a single variable in the right-hand side is a novel and essential feature to obtain a correct framework in this setting. Unfortunately, dependency pairs behave as a kind of glue in the context-sensitive dependency graph which makes the cycles bigger, thus making some proofs of termination harder. In this paper we show that this effect can be safely mitigated by removing some arcs from the graph, thus leading to faster and easier proofs. Narrowing dependency pairs is also introduced and used here to eventually simplify the treatment of the context-sensitive dependency graph. We show the practicality of the new techniques with some benchmarks.

Polynomial interpretations are a standard technique used in almost all tools for proving termination of term rewrite systems (TRSs) automatically. Traditionally, one applies interpretations with polynomials over the naturals. But recently, it was shown that interpretations with polynomials over the rationals can be significantly more powerful. However, searching for such interpretations is considerably more difficult than for natural polynomials. Moreover, while there exist highly efficient SAT-based techniques for finding natural polynomials, no such techniques had been developed for rational polynomials yet. In this paper, we tackle the two main problems when applying rational polynomial interpretations in practice: (1) We develop new criteria to decide when to use rational instead of natural polynomial interpretations. (2) Afterwards, we present SAT-based methods for finding rational polynomial interpretations and evaluate them empirically.

Despite the remarkable development of the theory of termination of rewriting, its application to high-level programming languages is far from being optimal. This is due to the need for features such as conditional equations and rules, types and subtypes, (possibly programmable) strategies for controlling the execution,

Abstract. Innermost context-sensitive rewriting has been proved useful for modeling computations of programs of algebraic languages like Maude, OBJ, etc. Furthermore, innermost termination of rewriting is often easier to prove than termination. Thus, under appropriate conditions, a useful strategy for proving termination of rewriting is trying to prove termination of innermost rewriting. This phenomenon has also been investigated for context-sensitive rewriting (CSR). Up to now, only few transformations have been proposed and used to prove termination of innermost CSR. In this paper, we investigate direct methods for proving termination of innermost CSR. We adapt the recently introduced context-sensitive dependency pairs approach to innermost CSR and show that they can be advantageously used for proving termination of innermost CSR. We have implemented them as part of the termination tool mu-term. 1

Abstract. Polynomial constraint-solving plays a prominent role in several areas of engineering and software verification. In particular, polynomial constraint solving has a long and successful history in the development of tools for proving termination of programs. Well-known and very efficient techniques, like SAT algorithms and tools, have been recently proposed and used for implementing polynomial constraint solving algorithms through appropriate encodings. However, powerful techniques like the ones provided by the SMT (SAT modulo theories) approach for linear arithmetic constraints (over the rationals) are underexplored to date. In this paper we show that the use of these techniques for developing polynomial constraint solvers outperforms the best existing solvers and provides a new and powerful approach for implementing better and more general solvers for termination provers.

Types (or sorts) are pervasive in computer science and in rewritingbased programming languages, which often support subtypes (subsorts) and subtype polymorphism. Programs in these languages can be modeled as order-sorted term rewriting systems (OS-TRSs). Often, termination of such programs heavily depends on sort information. But few techniques are currently available for proving termination of OS-TRSs; and they often fail for interesting OS-TRSs. In this paper we generalize the dependency pairs approach to prove termination of OS-TRSs. Preliminary experiments suggest that this technique can succeed where existing ones fail, yielding easier and simpler termination proofs.