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

Generic Encodings of Constructor Rewriting Systems

Rewriting is a formalism widely used in computer science and mathematical logic. The classical formalism has been extended, in the context of functional languages, with an order over the rules and, in the context of rewrite based languages, with the negation over patterns. We propose in this paper a concise and clear algorithm computing the difference over patterns which can be used to define generic encodings of constructor term rewriting systems with negation and order into classical term rewriting systems. As a direct consequence, established methods used for term rewriting systems can be applied to analyze properties of the extended systems. The approach can also be seen as a generic compiler which targets any language providing basic pattern matching primitives. The formalism provides also a new method for deciding if a set of patterns subsumes a given pattern and thus, for checking the presence of useless patterns or the completeness of a set of patterns.Comment: Added appendix with proofs and extended example

Polynomial Path Orders: A Maximal Model

This paper is concerned with the automated complexity analysis of term rewrite systems (TRSs for short) and the ramification of these in implicit computational complexity theory (ICC for short). We introduce a novel path order with multiset status, the polynomial path order POP*. Essentially relying on the principle of predicative recursion as proposed by Bellantoni and Cook, its distinct feature is the tight control of resources on compatible TRSs: The (innermost) runtime complexity of compatible TRSs is polynomially bounded. We have implemented the technique, as underpinned by our experimental evidence our approach to the automated runtime complexity analysis is not only feasible, but compared to existing methods incredibly fast. As an application in the context of ICC we provide an order-theoretic characterisation of the polytime computable functions. To be precise, the polytime computable functions are exactly the functions computable by an orthogonal constructor TRS compatible with POP*

Proving Termination using Recursive Path Orders and SAT solving

We introduce a propositional encoding of the recursive path order with status (RPO). RPO is a combination of a multiset path order and a lexicographic path order which considers permutations of the arguments in the lexicographic comparison. Our encoding allows us to apply SAT solvers in order to determine whether a given term rewrite system is RPO-terminating. Furthermore, to apply RPO within the dependency pair framework, we combined our novel encoding for RPO with an existing encoding for argument filters. We implemented our contributions in the termination prover AProVE. Our experiments show that due to our encoding, combining termination provers with SAT solvers improves the performance of RPO-implementations by orders of magnitude

Generic Encodings of Constructor Rewriting Systems

International audienceRewriting is a formalism widely used in computer science and mathematical logic. The classical formalism has been extended, in the context of functional languages, with an order over the rules and, in the context of rewrite based languages, with the negation over patterns. We propose in this paper a concise and clear algorithm computing the difference over patterns which can be used to define generic encodings of constructor term rewriting systems with negation and order into classical term rewriting systems. As a direct consequence, established methods used for term rewriting systems can be applied to analyze properties of the extended systems. The approach can also be seen as a generic compiler which targets any language providing basic pattern matching primitives. The formalism provides also a new method for deciding if a set of patterns subsumes a given pattern and thus, for checking the presence of useless patterns or the completeness of a set of patterns

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