We provide new proofs of two important theorems for proving termination of term rewrite systems (TRSs), including a full formalization in Isabelle/HOL. We first consider Dershowitz' theorem that termination starting from arbitrary terms is equivalent to termination starting from terms in the right-forward closures of right-hand sides, provided that the TRS is right-linear or orthogonal. Our new proof deviates from the original one in that no reorderings of steps in infinite derivations are required, making it more precise in its argumentation. It also subsumes a later result that one can weaken orthogonality to locally confluent overlay TRSs. The second theorem is about matrix interpretations. These were introduced by Hofbauer and Waldmann for proving termination of string rewrite systems (SRSs), internally using the concept of a core. Subsequently, Endrullis, Waldmann and Zantema developed matrix interpretations for TRSs without using the idea of a core. Whereas matrix interpretations for TRSs have already been formalized several times, so far this was not the case for core SRS matrix interpretations. We not only provide such a formalization, but also extend core SRS matrix interpretations to TRSs. These new core matrix interpretations for TRSs generalize previous approaches.

The termination and complexity competition ( termCOMP ) focuses on automated termination and complexity analysis for various kinds of programming paradigms, including categories for term rewriting, integer transition systems, imperative programming, logic programming, and functional programming. In all categories, the competition also welcomes the participation of tools providing certifiable output. The goal of the competition is to demonstrate the power and advances of the state-of-the-art tools in each of these areas.

We refine matrix interpretations for proving termination and complexity bounds of term rewrite systems we restricting them to domains that satisfy a system of linear inequalities. Admissibility of such a restriction is shown by certificates whose validity can be expressed as a constraint program. This refinement is orthogonal to other features of matrix interpretations (complexity bounds, dependency pairs), but can be used to improve complexity bounds, and we discuss its relation with the usable rules criterion. We present an implementation and experiments.

We adapt the TreeRePair tree compression algorithm and use it as an intermediate step in proving termination of term rewriting systems. We introduce a cost function that approximates the size of constraint systems that specify compatibility of matrix interpretations. We show how to integrate the compression algorithm with the Dependency Pairs transformation. Experiments show that compression reduces running times of constraint solvers, and thus improves the power of automated termination provers.

Matrix interpretations can be used to bound the derivational complexity of rewrite systems. We present a criterion that completely characterizes matrix interpretations that are polynomially bounded. It includes the method of upper triangular interpretations as a special case, and we prove that the inclusion is strict. The criterion can be expressed as a finite domain constraint system. It translates to a Boolean constraint system with a size that is polynomial in the dimension of the interpretation. We report on performance of an implementation.

For a given (terminating) term rewriting system one can often estimate its \emph{derivational complexity} indirectly by looking at the proof method that established termination. In this spirit we investigate two instances of the interpretation method: \emph{matrix interpretations} and \emph{context dependent interpretations}. We introduce a subclass of matrix interpretations, denoted as \emph{triangular matrix interpretations}, which induce polynomial derivational complexity and establish tight correspondence results between a subclass of context dependent interpretations and restricted triangular matrix interpretations. The thus obtained new results are easy to implement and considerably extend the analytic power of existing results. We provide ample numerical data for assessing the viability of the method.