Abstract. AProVE 1.2 is one of the most powerful systems for automated termination proofs of term rewrite systems (TRSs). It is the first tool which automates the new dependency pair framework [8] and therefore permits a completely flexible combination of different termination proof techniques. Due to this framework, AProVE 1.2 is also the first termination prover which can be fully configured by the user. 1

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.

The dependency pair technique is a powerful modular method for automated termination proofs of term rewrite systems (TRSs). We present two important extensions of this technique: First, we show how to prove termination of higher-order functions using dependency pairs. To this end, the dependency pair technique is extended to handle (untyped) applicative TRSs. Second, we introduce a method to prove non-termination with dependency pairs, while up to now dependency pairs were only used to verify termination. Our results lead to a framework for combining termination and non-termination techniques for firstand higher-order functions in a very flexible way. We implemented and evaluated our results in the automated termination prover AProVE.

This paper introduces a propositional encoding for lexicographic path orders (LPOs) and the corresponding LPO termination property of term rewrite systems. Given this encoding, termination analysis can be performed using a state-of-the-art Boolean satisfiability solver. Experimental results are unequivocal, indicating orders of magnitude speedups in comparison with previous implementations for LPO termination. The results of this paper have already had a direct impact on the design of several major termination analyzers for term rewrite systems. The contribution builds on a symbol-based approach towards reasoning about partial orders. The symbols in an unspecified partial order are viewed as variables that take integer values and are interpreted as indices in the order. For a partial order statement on n symbols, each index is represented in ⌈log 2 n ⌉ propositional variables and partial order constraints between symbols are modeled on the bit representations. The proposed encoding is general and relevant to other applications which involve propositional reasoning about partial orders.

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. This paper introduces a propositional encoding for lexicographic path orders in connection with dependency pairs. This facilitates the application of SAT solvers for termination analysis of term rewrite systems based on the dependency pair method. We address two main inter-related issues and encode them as satisfiability problems of propositional formulas that can be efficiently handled by SAT solving: (1) the combined search for a lexicographic path order together with an argument filtering to orient a set of inequalities; and (2) how the choice of the argument filtering influences the set of inequalities that have to be oriented. We have implemented our contributions in the termination prover AProVE. Extensive experiments show that by our encoding and the application of SAT solvers one obtains speedups in orders of magnitude as well as increased termination proving power. 1

2 CÉDRIC – Conservatoire national des arts et métiers Abstract. Nowadays, formal methods rely on tools of different kinds: proof assistants with which the user interacts to discover a proof step by step; and fully automated tools which make use of (intricate) decision procedures. But while some proof assistants can check the soundness of a proof, they lack automation. Regarding automated tools, one still has to be satisfied with their answers Yes/No/Donotknow, the validity of which can be subject to question, in particular because of the increasing size and complexity of these tools. In the context of rewriting techniques, we aim at bridging the gap between proof assistants that yield formal guarantees of reliability and highly automated tools one has to trust. We present an approach making use of both shallow and deep embeddings. We illustrate this approach with a prototype based on the CiME rewriting toolbox, which can discover involved termination proofs that can be certified by the COQ proof assistant, using the COCCINELLE library for rewriting. 1

Abstract. Polynomial interpretations are a useful technique for proving termination of term rewrite systems. We show how polynomial interpretations with negative coefficients, like x − 1 for a unary function symbol or x − y for a binary function symbol, can be used to extend the class of rewrite systems that can be automatically proved terminating. 1

Abstract. This paper presents two new approaches to prove termination of rewrite systems with the Knuth-Bendix order efficiently. The constraints for the weight function and for the precedence are encoded in (pseudo-)propositional logic and the resulting formula is tested for satisfiability. Any satisfying assignment represents a weight function and a precedence such that the induced Knuth-Bendix order orients the rules of the encoded rewrite system from left to right. 1

Semantic labeling is a powerful transformation technique for proving termination of term rewrite systems. The semantic part is given by a model or a quasi-model of the rewrite rules. A variant of semantic labeling is predictive labeling where the quasi-model condition is only required for the usable rules. In this paper we investigate how semantic and predictive labeling can be used to prove innermost termination. Moreover, we show how to reduce the set of usable rules for predictive labeling even further, both in the termination and the innermost termination case.