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

The Certification Problem Format

We provide an overview of CPF, the certification problem format, and explain some design decisions. Whereas CPF was originally invented to combine three different formats for termination proofs into a single one, in the meanwhile proofs for several other properties of term rewrite systems are also expressible: like confluence, complexity, and completion. As a consequence, the format is already supported by several tools and certifiers. Its acceptance is also demonstrated in international competitions: the certified tracks of both the termination and the confluence competition utilized CPF as exchange format between automated tools and trusted certifiers.Comment: In Proceedings UITP 2014, arXiv:1410.785

Ranking Functions for Size-Change Termination II

Size-Change Termination is an increasingly-popular technique for verifying program termination. These termination proofs are deduced from an abstract representation of the program in the form of "size-change graphs". We present algorithms that, for certain classes of size-change graphs, deduce a global ranking function: an expression that ranks program states, and decreases on every transition. A ranking function serves as a witness for a termination proof, and is therefore interesting for program certification. The particular form of the ranking expressions that represent SCT termination proofs sheds light on the scope of the proof method. The complexity of the expressions is also interesting, both practicaly and theoretically. While deducing ranking functions from size-change graphs has already been shown possible, the constructions in this paper are simpler and more transparent than previously known. They improve the upper bound on the size of the ranking expression from triply exponential down to singly exponential (for certain classes of instances). We claim that this result is, in some sense, optimal. To this end, we introduce a framework for lower bounds on the complexity of ranking expressions and prove exponential lower bounds.Comment: 29 pages

Size-Change Termination, Monotonicity Constraints and Ranking Functions

Size-Change Termination (SCT) is a method of proving program termination based on the impossibility of infinite descent. To this end we may use a program abstraction in which transitions are described by monotonicity constraints over (abstract) variables. When only constraints of the form x>y' and x>=y' are allowed, we have size-change graphs. Both theory and practice are now more evolved in this restricted framework then in the general framework of monotonicity constraints. This paper shows that it is possible to extend and adapt some theory from the domain of size-change graphs to the general case, thus complementing previous work on monotonicity constraints. In particular, we present precise decision procedures for termination; and we provide a procedure to construct explicit global ranking functions from monotonicity constraints in singly-exponential time, which is better than what has been published so far even for size-change graphs.Comment: revised version of September 2

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

The Size-Change Principle and Dependency Pairs for Termination of Term Rewriting

In [24], a new size-change principle was proposed to verify termination of functional programs automatically. We extend this principle in order to prove termination and innermost termination of arbitrary term rewrite systems (TRSs). Moreover, we compare this approach with existing techniques for termination analysis of TRSs (such as recursive path orders or dependency pairs). It turns out that the size-change principle on its own fails for many examples that can be handled by standard techniques for rewriting, but there are also TRSs where it succeeds whereas existing rewriting techniques fail. Moreover, we also compare the complexity of the respective methods. To this end, we develop the first complexity analysis for the dependency pair approach. While the size-change principle is PSPACE-complete, we prove that the dependency pair approach (in combination with classical path orders) is only NP-complete. To benefit from their respective advantages, we show how to combine the size-change principle with classical orders and with dependency pairs. In this way, we obtain a new approach for automated termination proofs of TRSs which is more powerful than previous approaches. We also show that the combination with dependency pairs does not increase the complexity of the size-change principle, i.e., the combined approach is still PSPACE-complete

Una aproximación offline a la evaluación parcial dirigida por narrowing

La evaluación parcial dirigida por narrowing (NPE: Narrowing-driven Partial Evaluation) es una técnica potente para la especialización de sistemas de reescritura, i.e., para el componente de primer orden de muchos lenguajes declarativos (lógico) funcionales como Haskell, Curry o Toy. Los evaluadores parciales se clasifican en dos grandes categorías: online y offline, de acuerdo al momento temporal en que se consideran los aspectos de terminación del proceso de especialización. Los evaluadores parciales online son usualmente más precisos ya que tienen más información disponible. Los evaluadores parciales offline proceden comúnmente en dos etapas; la primera etapa procesa un programa (e.g., para identificar aquellas llamadas a función que se pueden desplegar sin riesgo de no terminación) e incluye anotaciones para guiar las computaciones parciales; entonces, una segunda etapa, la de evaluación parcial propiamente dicha, sólo tiene que obedecer las anotaciones y por tanto el especializador es mucho más rápido que en la aproximación online. En esta tesis se presenta un nuevo esquema de evaluación parcial dirigido por narrowing, más eficiente y que asegura la terminación siguiendo el estilo offline. Para ello, identificamos una caracterización de programas cuasi-terminantes a los que llamamos "no crecientes". En tales programas, las computaciones por narrowing necesario presentan sólo un conjunto finito de términos diferentes (módulo renombramiento de variables). La propiedad de la cuasi-terminación es importante toda vez que su presencia es regularmente una condición suficiente para la terminación del proceso de especialización. Sin embargo, la clase de programas cuasi-terminantes es muy restrictiva, por lo que introducimos un algoritmo que acepta programas inductivamente secuenciales---una clase mucho más amplia sobre la que está definido el narrowing necesario---y anota aquellas partes que violan la caracterización de programas no crecientes. Para procesar de maneRamos Díaz, JG. (2007). Una aproximación offline a la evaluación parcial dirigida por narrowing [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/1888Palanci