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

The dependency pair framework: Combining techniques for automated termination proofs

Abstract. The dependency pair approach is one of the most powerful techniques for automated termination proofs of term rewrite systems. Up to now, it was regarded as one of several possible methods to prove termination. In this paper, we show that dependency pairs can instead be used as a general concept to integrate arbitrary techniques for termination analysis. In this way, the benefits of different techniques can be combined and their modularity and power are increased significantly. We refer to this new concept as the “dependency pair framework ” to distinguish it from the old “dependency pair approach”. Moreover, this framework facilitates the development of new methods for termination analysis. To demonstrate this, we present several new techniques within the dependency pair framework which simplify termination problems considerably. We implemented the dependency pair framework in our termination prover AProVE and evaluated it on large collections of examples.

Innermost Termination of Context-Sensitive Rewriting

Innermost context-sensitive rewriting (CSR) has been proved useful for modeling the computational behavior of programs of algebraic languages like Maude, OBJ, etc, which incorporate an innermost strategy which is used to break down the nondeterminism which is inherent to reduction relations. Furthermore, innermost termination of rewriting is often easier to prove than termination. Thus, under appropriate conditions, a useful strategy for proving termination of rewriting is trying to prove termination of innermost rewriting. This phenomenon has also been investigated for context-sensitive rewriting. Up to now, only few transformation-based methods have been proposed and used to (specifically) prove termination of innermost CSR. Powerful and e cient techniques for proving (innermost) termination of (unrestricted) rewriting like the dependency pair framework have not been considered yet. In this work, we investigate the adaptation of the dependency pair framework to innermost CSR. We provide a suitable notion of innermost context-sensitive dependency pair and show how to extend and adapt the main notions which conform the framework (chain, termination problem, processor, etc.). Thanks to the innermost context-sensitive dependency pairs, we can now use powerful techniques for proving termination of innermost CSR. This is made clear by means of some benchmarks showing that our techniques dramatically improve over previously existing transformational techniques, thus establishing the new state-of-the-art in the area. We have implemented them as part of the termination tool MU-TERM.Alarcón Jiménez, B.; Lucas, S. (2011). Innermost Termination of Context-Sensitive Rewriting. http://hdl.handle.net/10251/1079

Modular termination proofs for rewriting using dependency pairs

AbstractRecently, Arts and Giesl developed the dependency pair approach which allows automated termination and innermost termination proofs for many term rewriting systems (TRSs) for which such proofs were not possible before. The motivation for this approach was that virtually all previous techniques for automated termination proofs of TRSs were based on simplification orderings. In practice, however, many rewrite systems are not simply terminating, i.e. their termination cannot be verified by any simplification ordering. In this paper we introduce a refinement of the dependency pair framework which further extends the class of TRSs for which termination or innermost termination can be shown automatically. By means of this refinement, one can now prove termination in a modular way. Thus, this refinement is inevitable in order to verify the termination of large rewrite systems occurring in practice. To be more precise, one may use several different orderings in one termination proof. Subsequently, we present several new modularity results based on dependency pairs. First, we show that the well-known modularity of simple termination for disjoint unions can be extended to DP quasi-simple termination, i.e. to the class of rewrite systems where termination can be shown automatically by the dependency pair technique in combination with quasi-simplification orderings. Under certain additional conditions, this new result also holds for constructor-sharing and composable systems. Second, the above-mentioned refinement of the dependency pair method yields new modularity criteria for innermost termination which extend previous results in this area considerably. In particular, existing results for modularity of innermost termination can easily be shown to be direct consequences of our new criteria

Context-Sensitive Dependency Pairs Framework

We show how to develop a dependency pair framework for proving termination of context-sensitive rewriting.Gutiérrez Gil, R. (2008). Context-Sensitive Dependency Pairs Framework. http://hdl.handle.net/10251/13625Archivo delegad

Towards a Framework for Proving Termination of Maude Programs

Maude es un lenguaje de programación declarativo basado en la lógica de reescritura que incorpora muchas características que lo hacen muy potente. Sin embargo, a la hora de probar ciertas propiedades computacionales esto conlleva dificultades. La tarea de probar la terminación de sistemas de reesctritura es de hecho bastante dura, pero aplicada a lenguajes de programación reales se concierte en más complicada debido a estas características inherentes. Esto provoca que métodos para probar la terminación de este tipo de programas requieran técnicas específicas y un análisis cuidadoso. Varios trabajos han intentado probar terminación de (un subconjunto de) programas Maude. Sin embargo, todos ellos siguen una aproximación transformacional, donde el programa original es trasformado hasta alcanzar un sistema de reescritura capaz de ser manejado con las técnicas y herramientas de terminación existentes. En la práctica, el hecho de transformar los sistemas originales suele complicar la demostración de la terminación ya que esto introduce nuevos símbolos y reglas en el sistema. En esta tesis, llevamos a cabo el problema de probar terminación de (un subconjunto de) programas Maude mediante métodos directos. Por un lado, nos centramos en la estrategia de Maude. Maude es un lenguaje impaciente donde los argumentos de una función son evaluados siempre antes de la aplicación de la función que los usa. Esta estrategia (conocida como llamada por valor) puede provocar la no terminación si los programas no están escritos cuidadosamente. Por esta razón, Maude (en concreto) incorpora mecanismos para controlar la ejecución de programas como las anotaciones sintácticas que están asociadas a los argumentos de los símbolos. En reescritura, esta estrategia sería conocida como reescritura sensible al contexto innermost (RSCI). Por otro lado, Maude también incorpora la posibilidad de declarar atributos.Alarcón Jiménez, B. (2011). Towards a Framework for Proving Termination of Maude Programs [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/11003Palanci

Proof Theory at Work: Complexity Analysis of Term Rewrite Systems

This thesis is concerned with investigations into the "complexity of term rewriting systems". Moreover the majority of the presented work deals with the "automation" of such a complexity analysis. The aim of this introduction is to present the main ideas in an easily accessible fashion to make the result presented accessible to the general public. Necessarily some technical points are stated in an over-simplified way.Comment: Cumulative Habilitation Thesis, submitted to the University of Innsbruc

Termination of Narrowing: Automated Proofs and Modularity Properties

En 1936 Alan Turing demostro que el halting problem, esto es, el problema de decidir si un programa termina o no, es un problema indecidible para la inmensa mayoria de los lenguajes de programacion. A pesar de ello, la terminacion es un problema tan relevante que en las ultimas decadas un gran numero de tecnicas han sido desarrolladas para demostrar la terminacion de forma automatica de la maxima cantidad posible de programas. Los sistemas de reescritura de terminos proporcionan un marco teorico abstracto perfecto para el estudio de la terminacion de programas. En este marco, la evaluaci on de un t ermino consiste en la aplicacion no determinista de un conjunto de reglas de reescritura. El estrechamiento (narrowing) de terminos es una generalizacion de la reescritura que proporciona un mecanismo de razonamiento automatico. Por ejemplo, dado un conjunto de reglas que denan la suma y la multiplicacion, la reescritura permite calcular expresiones aritmeticas, mientras que el estrechamiento permite resolver ecuaciones con variables. Esta tesis constituye el primer estudio en profundidad de las propiedades de terminacion del estrechamiento. Las contribuciones son las siguientes. En primer lugar, se identican clases de sistemas en las que el estrechamiento tiene un comportamiento bueno, en el sentido de que siempre termina. Muchos metodos de razonamiento automatico, como el analisis de la semantica de lenguajes de programaci on mediante operadores de punto jo, se benefician de esta caracterizacion. En segundo lugar, se introduce un metodo automatico, basado en el marco teorico de pares de dependencia, para demostrar la terminacion del estrechamiento en un sistema particular. Nuestro metodo es, por primera vez, aplicable a cualquier clase de sistemas. En tercer lugar, se propone un nuevo metodo para estudiar la terminacion del estrechamiento desde un termino particular, permitiendo el analisis de la terminacion de lenguajes de programacion. El nuevo metodo generaliza losIborra López, J. (2010). Termination of Narrowing: Automated Proofs and Modularity Properties [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/19251Palanci