Confluence of nearly orthogonal infinitary term rewriting systems

We give a relatively simple coinductive proof of confluence, modulo equivalence of root-active terms, of nearly orthogonal infinitary term rewriting systems. Nearly orthogonal systems allow certain root overlaps, but no non-root overlaps. Using a slightly more complicated method we also show confluence modulo equivalence of hypercollapsing terms. The condition we impose on root overlaps is similar to the condition used by Toyama in the context of finitary rewriting

A new coinductive confluence proof for infinitary lambda calculus

We present a new and formal coinductive proof of confluence and normalisation of B\"ohm reduction in infinitary lambda calculus. The proof is simpler than previous proofs of this result. The technique of the proof is new, i.e., it is not merely a coinductive reformulation of any earlier proofs. We formalised the proof in the Coq proof assistant.Comment: arXiv admin note: text overlap with arXiv:1501.0435

An operational interpretation of coinductive types

We introduce an operational rewriting-based semantics for strictly positive nested higher-order (co)inductive types. The semantics takes into account the "limits" of infinite reduction sequences. This may be seen as a refinement and generalization of the notion of productivity in term rewriting to a setting with higher-order functions and with data specified by nested higher-order inductive and coinductive definitions. Intuitively, we interpret lazy data structures in a higher-order functional language by potentially infinite terms corresponding to their complete unfoldings. We prove an approximation theorem which essentially states that if a term reduces to an arbitrarily large finite approximation of an infinite object in the interpretation of a coinductive type, then it infinitarily (i.e. in the "limit") reduces to an infinite object in the interpretation of this type. We introduce a sufficient syntactic correctness criterion, in the form of a type system, for finite terms decorated with type information. Using the approximation theorem, we show that each well-typed term has a well-defined interpretation in our semantics

Termination and Productivity

Klop, J.W. [Promotor]Vrijer, R.C. de [Copromotor

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

Mechanised Uniform Interpolation for Modal Logics K, GL, and iSL

The uniform interpolation property in a given logic can be understood as the definability of propositional quantifiers. We mechanise the computation of these quantifiers and prove correctness in the Coq proof assistant for three modal logics, namely: (1) the modal logic K, for which a pen-and-paper proof exists; (2) Gödel-Löb logic GL, for which our formalisation clarifies an important point in an existing, but incomplete, sequent-style proof; and (3) intuitionistic strong Löb logic iSL, for which this is the first proof-theoretic construction of uniform interpolants. Our work also yields verified programs that allow one to compute the propositional quantifiers on any formula in this logic

Mechanised Uniform Interpolation for Modal Logics K, GL, and iSL

The uniform interpolation property in a given logic can be understood as the definability of propositional quantifiers. We mechanise the computation of these quantifiers and prove correctness in the Coq proof assistant for three modal logics, namely: (1) the modal logic K, for which a pen-and-paper proof exists; (2) Gödel-Löb logic GL, for which our formalisation clarifies an important point in an existing, but incomplete, sequent-style proof; and (3) intuitionistic strong Löb logic iSL, for which this is the first proof-theoretic construction of uniform interpolants. Our work also yields verified programs that allow one to compute the propositional quantifiers on any formula in this logic