272 research outputs found
The First-Order Theory of Ground Tree Rewrite Graphs
We prove that the complexity of the uniform first-order theory of ground tree
rewrite graphs is in ATIME(2^{2^{poly(n)}},O(n)). Providing a matching lower
bound, we show that there is some fixed ground tree rewrite graph whose
first-order theory is hard for ATIME(2^{2^{poly(n)}},poly(n)) with respect to
logspace reductions. Finally, we prove that there exists a fixed ground tree
rewrite graph together with a single unary predicate in form of a regular tree
language such that the resulting structure has a non-elementary first-order
theory.Comment: accepted for Logical Methods in Computer Scienc
Uniqueness of Normal Forms is Decidable for Shallow Term Rewrite Systems
Uniqueness of normal forms (UN=) is an important property of term rewrite systems. UN= is decidable for ground (i.e., variable-free) systems and undecidable in general. Recently it was shown to be decidable for linear, shallow systems. We generalize this previous result and show that this property is decidable for shallow rewrite systems, in contrast to confluence, reachability and other properties, which are all undecidable for flat systems. Our result is also optimal in some sense, since we prove that the UN= property is undecidable for two superclasses of flat systems: left-flat, left-linear systems in which right-hand sides are of depth at most two and right-flat, right-linear systems in which left-hand sides are of depth at most two
Some results on confluence: decision and what to do without
International audienceWe recall first some decidability results on the confluence of TRS, and related properties about unicity of normal forms. In particular we put it in perspective old proofs of undecidability of confluence for the class of flat systems with more recent results, in order to discuss the importance of linearity wrt these decision problems. Second, we describe a case study on musical rhythm notation involving modeling rewrite systems which are not confluent. In this case, instead of applying rewrite rules directly, we enumerate the equivalence class of a given term using automata-based representations and dynamic programming
Term rewriting systems from Church-Rosser to Knuth-Bendix and beyond
Term rewriting systems are important for computability theory of abstract data types, for automatic theorem proving, and for the foundations of functional programming. In this short survey we present, starting from first principles, several of the basic notions and facts in the area of term rewriting. Our treatment, which often will be informal, covers abstract rewriting, Combinatory Logic, orthogonal systems, strategies, critical pair completion, and some extended rewriting formats
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
Deciding Confluence and Normal Form Properties of Ground Term Rewrite Systems Efficiently
It is known that the first-order theory of rewriting is decidable for ground
term rewrite systems, but the general technique uses tree automata and often
takes exponential time. For many properties, including confluence (CR),
uniqueness of normal forms with respect to reductions (UNR) and with respect to
conversions (UNC), polynomial time decision procedures are known for ground
term rewrite systems. However, this is not the case for the normal form
property (NFP). In this work, we present a cubic time algorithm for NFP, an
almost cubic time algorithm for UNR, and an almost linear time algorithm for
UNC, improving previous bounds. We also present a cubic time algorithm for CR
The Confluence Problem for Flat TRSs
International audienceWe prove that the properties of reachability, joinability and confluence are undecidable for flat TRSs. Here, a TRS is flat if the heights of the left and right-hand sides of each rewrite rule are at most one
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