Termination of rewrite relations on λ\lambda-terms based on Girard's notion of reducibility

In this paper, we show how to extend the notion of reducibility introduced by Girard for proving the termination of $\beta$-reduction in the polymorphic $\lambda$-calculus, to prove the termination of various kinds of rewrite relations on $\lambda$-terms, including rewriting modulo some equational theory and rewriting with matching modulo $\beta$$\eta$, by using the notion of computability closure. This provides a powerful termination criterion for various higher-order rewriting frameworks, including Klop's Combinatory Reductions Systems with simple types and Nipkow's Higher-order Rewrite Systems

Challenging variants of the Collatz Conjecture

The Collatz Conjecture (also known as the 3N + 1 problem) is simple to explain, yet proving that all positive integers following the Collatz Mapping must converge to 1 has eluded mathematicians for over half a century. Aaronson and Heule are exploring solving the Collatz Conjecture using an approach involving string rewrite systems: Aaronson transformed the Conjecture into a string rewrite system and Heule has been applying parallel SAT solvers on instances of this system. Similar approaches have been applied successfully to other mathematical problems. We started looking into simpler variants of the conjecture. This thesis defines some of these variants and investigates easily provable as well as very hard variants. We study the hardness of unsolved variants by computing the number of rewrite steps needed up to 1 billion. Our hardness prediction method suggests that proving termination of the challenging variants should be considerably easier compared to solving the original conjecture.Computer Science

CoLoR: a Coq library on well-founded rewrite relations and its application to the automated verification of termination certificates

Termination is an important property of programs; notably required for programs formulated in proof assistants. It is a very active subject of research in the Turing-complete formalism of term rewriting systems, where many methods and tools have been developed over the years to address this problem. Ensuring reliability of those tools is therefore an important issue. In this paper we present a library formalizing important results of the theory of well-founded (rewrite) relations in the proof assistant Coq. We also present its application to the automated verification of termination certificates, as produced by termination tools

Extending Context-Sensitivity in Term Rewriting

We propose a generalized version of context-sensitivity in term rewriting based on the notion of "forbidden patterns". The basic idea is that a rewrite step should be forbidden if the redex to be contracted has a certain shape and appears in a certain context. This shape and context is expressed through forbidden patterns. In particular we analyze the relationships among this novel approach and the commonly used notion of context-sensitivity in term rewriting, as well as the feasibility of rewriting with forbidden patterns from a computational point of view. The latter feasibility is characterized by demanding that restricting a rewrite relation yields an improved termination behaviour while still being powerful enough to compute meaningful results. Sufficient criteria for both kinds of properties in certain classes of rewrite systems with forbidden patterns are presented

Automated verification of termination certificates

In order to increase user confidence, many automated theorem provers provide certificates that can be independently verified. In this paper, we report on our progress in developing a standalone tool for checking the correctness of certificates for the termination of term rewrite systems, and formally proving its correctness in the proof assistant Coq. To this end, we use the extraction mechanism of Coq and the library on rewriting theory and termination called CoLoR

12th International Workshop on Termination (WST 2012) : WST 2012, February 19–23, 2012, Obergurgl, Austria / ed. by Georg Moser

This volume contains the proceedings of the 12th International Workshop on Termination (WST 2012), to be held February 19–23, 2012 in Obergurgl, Austria. The goal of the Workshop on Termination is to be a venue for presentation and discussion of all topics in and around termination. In this way, the workshop tries to bridge the gaps between different communities interested and active in research in and around termination. The 12th International Workshop on Termination in Obergurgl continues the successful workshops held in St. Andrews (1993), La Bresse (1995), Ede (1997), Dagstuhl (1999), Utrecht (2001), Valencia (2003), Aachen (2004), Seattle (2006), Paris (2007), Leipzig (2009), and Edinburgh (2010). The 12th International Workshop on Termination did welcome contributions on all aspects of termination and complexity analysis. Contributions from the imperative, constraint, functional, and logic programming communities, and papers investigating applications of complexity or termination (for example in program transformation or theorem proving) were particularly welcome. We did receive 18 submissions which all were accepted. Each paper was assigned two reviewers. In addition to these 18 contributed talks, WST 2012, hosts three invited talks by Alexander Krauss, Martin Hofmann, and Fausto Spoto

Proving Looping and Non-Looping Non-Termination by Finite Automata

A new technique is presented to prove non-termination of term rewriting. The basic idea is to find a non-empty regular language of terms that is closed under rewriting and does not contain normal forms. It is automated by representing the language by a tree automaton with a fixed number of states, and expressing the mentioned requirements in a SAT formula. Satisfiability of this formula implies non-termination. Our approach succeeds for many examples where all earlier techniques fail, for instance for the S-rule from combinatory logic

Guided Unfoldings for Finding Loops in Standard Term Rewriting

In this paper, we reconsider the unfolding-based technique that we have introduced previously for detecting loops in standard term rewriting. We improve it by guiding the unfolding process, using distinguished positions in the rewrite rules. This results in a depth-first computation of the unfoldings, whereas the original technique was breadth-first. We have implemented this new approach in our tool NTI and compared it to the previous one on a bunch of rewrite systems. The results we get are promising (better times, more successful proofs).Comment: Pre-proceedings paper presented at the 28th International Symposium on Logic-Based Program Synthesis and Transformation (LOPSTR 2018), Frankfurt am Main, Germany, 4-6 September 2018 (arXiv:1808.03326

Refinement Types as Higher Order Dependency Pairs

Refinement types are a well-studied manner of performing in-depth analysis on functional programs. The dependency pair method is a very powerful method used to prove termination of rewrite systems; however its extension to higher order rewrite systems is still the object of active research. We observe that a variant of refinement types allow us to express a form of higher-order dependency pair criterion that only uses information at the type level, and we prove the correctness of this criterion