Abstract. The dependency pair technique [1, 11, 12] is a powerful method for automated termination and innermost termination proofs of term rewrite systems (TRSs). For any TRS, it generates inequality constraints that have to be satisfied by well-founded orders. We improve the dependency pair technique by considerably reducing the number of constraints produced for (innermost) termination proofs. Moreover, we extend transformation techniques to manipulate dependency pairs which simplify (innermost) termination proofs significantly. In order to fully mechanize the approach, we show how transformations and the search for suitable orders can be mechanized efficiently. We implemented our results in the automated termination prover AProVE and evaluated them on large collections of examples.

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.

The dependency pair approach is one of the most powerful techniques for automated (innermost) termination proofs of term rewrite systems (TRSs). For any TRS, it generates inequality constraints that have to be satisfied by well-founded orders. However, proving innermost termination is considerably easier than termination, since the constraints for innermost termination are a subset of those for termination.

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

Abstract — A property is persistent if for any many-sorted term rewriting system, has the property if and only if term rewriting system, which results from by omitting its sort information, has the property. Zantema showed that termination is persistent for term rewriting systems without collapsing or duplicating rules. In this paper, we show that the Zantema’s result can be extended to term rewriting systems on ordered sorts, i.e., termination is persistent for term rewriting systems on ordered sorts without collapsing, decreasing or duplicating rules. Furthermore we give the example as application of this result. Also we obtain that completeness is persistent for this class of term rewriting systems. Keywords: Theory of computing, Model-based reasoning, term rewriting system, termination

Abstract — A property is called persistent if for any manysorted term rewriting system, has the property if and only if term rewriting system, which results from by omitting its sort information, has the property. In this paper, we show that termination is persistent for non-overlapping term rewriting systems and we give the example as application of this result. Furthermore we obtain that completeness is persistent for non-overlapping term rewriting systems. Keywords: Theory of computing, Model-based reasoning, term rewriting system, termination

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.

Abstract — A property is persistent if for any many-sorted term rewriting system, has the property if and only if term rewriting system, which results from by omitting its sort information, has the property. In this paper, we show that weak normalization is persistent for term rewriting systems. Furthermore we obtain that semi-completeness is persistent for term rewriting systems and we give the example as application of this result. Keywords: Theory of computing, Model-based reasoning, Term rewriting system, Semi-completeness