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

Step-Indexed Normalization for a Language with General Recursion

The Trellys project has produced several designs for practical dependently typed languages. These languages are broken into two fragments-a_logical_fragment where every term normalizes and which is consistent when interpreted as a logic, and a_programmatic_fragment with general recursion and other convenient but unsound features. In this paper, we present a small example language in this style. Our design allows the programmer to explicitly mention and pass information between the two fragments. We show that this feature substantially complicates the metatheory and present a new technique, combining the traditional Girard-Tait method with step-indexed logical relations, which we use to show normalization for the logical fragment.Comment: In Proceedings MSFP 2012, arXiv:1202.240

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

Polynomial Interpretations over the Natural, Rational and Real Numbers Revisited

Polynomial interpretations are a useful technique for proving termination of term rewrite systems. They come in various flavors: polynomial interpretations with real, rational and integer coefficients. As to their relationship with respect to termination proving power, Lucas managed to prove in 2006 that there are rewrite systems that can be shown polynomially terminating by polynomial interpretations with real (algebraic) coefficients, but cannot be shown polynomially terminating using polynomials with rational coefficients only. He also proved the corresponding statement regarding the use of rational coefficients versus integer coefficients. In this article we extend these results, thereby giving the full picture of the relationship between the aforementioned variants of polynomial interpretations. In particular, we show that polynomial interpretations with real or rational coefficients do not subsume polynomial interpretations with integer coefficients. Our results hold also for incremental termination proofs with polynomial interpretations.Comment: 28 pages; special issue of RTA 201

Synthesis of sup-interpretations: a survey

In this paper, we survey the complexity of distinct methods that allow the programmer to synthesize a sup-interpretation, a function providing an upper- bound on the size of the output values computed by a program. It consists in a static space analysis tool without consideration of the time consumption. Although clearly related, sup-interpretation is independent from termination since it only provides an upper bound on the terminating computations. First, we study some undecidable properties of sup-interpretations from a theoretical point of view. Next, we fix term rewriting systems as our computational model and we show that a sup-interpretation can be obtained through the use of a well-known termination technique, the polynomial interpretations. The drawback is that such a method only applies to total functions (strongly normalizing programs). To overcome this problem we also study sup-interpretations through the notion of quasi-interpretation. Quasi-interpretations also suffer from a drawback that lies in the subterm property. This property drastically restricts the shape of the considered functions. Again we overcome this problem by introducing a new notion of interpretations mainly based on the dependency pairs method. We study the decidability and complexity of the sup-interpretation synthesis problem for all these three tools over sets of polynomials. Finally, we take benefit of some previous works on termination and runtime complexity to infer sup-interpretations.Comment: (2012

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

Termination Casts: A Flexible Approach to Termination with General Recursion

This paper proposes a type-and-effect system called Teqt, which distinguishes terminating terms and total functions from possibly diverging terms and partial functions, for a lambda calculus with general recursion and equality types. The central idea is to include a primitive type-form "Terminates t", expressing that term t is terminating; and then allow terms t to be coerced from possibly diverging to total, using a proof of Terminates t. We call such coercions termination casts, and show how to implement terminating recursion using them. For the meta-theory of the system, we describe a translation from Teqt to a logical theory of termination for general recursive, simply typed functions. Every typing judgment of Teqt is translated to a theorem expressing the appropriate termination property of the computational part of the Teqt term.Comment: In Proceedings PAR 2010, arXiv:1012.455

Type-Based Termination, Inflationary Fixed-Points, and Mixed Inductive-Coinductive Types

Type systems certify program properties in a compositional way. From a bigger program one can abstract out a part and certify the properties of the resulting abstract program by just using the type of the part that was abstracted away. Termination and productivity are non-trivial yet desired program properties, and several type systems have been put forward that guarantee termination, compositionally. These type systems are intimately connected to the definition of least and greatest fixed-points by ordinal iteration. While most type systems use conventional iteration, we consider inflationary iteration in this article. We demonstrate how this leads to a more principled type system, with recursion based on well-founded induction. The type system has a prototypical implementation, MiniAgda, and we show in particular how it certifies productivity of corecursive and mixed recursive-corecursive functions.Comment: In Proceedings FICS 2012, arXiv:1202.317