Stream Productivity by Outermost Termination

Streams are infinite sequences over a given data type. A stream specification is a set of equations intended to define a stream. A core property is productivity: unfolding the equations produces the intended stream in the limit. In this paper we show that productivity is equivalent to termination with respect to the balanced outermost strategy of a TRS obtained by adding an additional rule. For specifications not involving branching symbols balancedness is obtained for free, by which tools for proving outermost termination can be used to prove productivity fully automatically

Extending Context-Sensitivity in Term Rewriting

We propose a generalized version of context-sensitivity in term rewriting based on the notion of "forbidden patterns". The basic idea is that a rewrite step should be forbidden if the redex to be contracted has a certain shape and appears in a certain context. This shape and context is expressed through forbidden patterns. In particular we analyze the relationships among this novel approach and the commonly used notion of context-sensitivity in term rewriting, as well as the feasibility of rewriting with forbidden patterns from a computational point of view. The latter feasibility is characterized by demanding that restricting a rewrite relation yields an improved termination behaviour while still being powerful enough to compute meaningful results. Sufficient criteria for both kinds of properties in certain classes of rewrite systems with forbidden patterns are presented

Termination of Rewriting with and Automated Synthesis of Forbidden Patterns

We introduce a modified version of the well-known dependency pair framework that is suitable for the termination analysis of rewriting under forbidden pattern restrictions. By attaching contexts to dependency pairs that represent the calling contexts of the corresponding recursive function calls, it is possible to incorporate the forbidden pattern restrictions in the (adapted) notion of dependency pair chains, thus yielding a sound and complete approach to termination analysis. Building upon this contextual dependency pair framework we introduce a dependency pair processor that simplifies problems by analyzing the contextual information of the dependency pairs. Moreover, we show how this processor can be used to synthesize forbidden patterns suitable for a given term rewriting system on-the-fly during the termination analysis.Comment: In Proceedings IWS 2010, arXiv:1012.533

Termination and Productivity

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

Well-definedness of Streams by Transformation and Termination

Streams are infinite sequences over a given data type. A stream specification is a set of equations intended to define a stream. We propose a transformation from such a stream specification to a term rewriting system (TRS) in such a way that termination of the resulting TRS implies that the stream specification is well-defined, that is, admits a unique solution. As a consequence, proving well-definedness of several interesting stream specifications can be done fully automatically using present powerful tools for proving TRS termination. In order to increase the power of this approach, we investigate transformations that preserve semantics and well-definedness. We give examples for which the above mentioned technique applies for the ransformed specification while it fails for the original one

Faithful (meta-)encodings of programmable strategies into term rewriting systems

Rewriting is a formalism widely used in computer science and mathematical logic. When using rewriting as a programming or modeling paradigm, the rewrite rules describe the transformations one wants to operate and rewriting strategies are used to con- trol their application. The operational semantics of these strategies are generally accepted and approaches for analyzing the termination of specific strategies have been studied. We propose in this paper a generic encoding of classic control and traversal strategies used in rewrite based languages such as Maude, Stratego and Tom into a plain term rewriting system. The encoding is proven sound and complete and, as a direct consequence, estab- lished termination methods used for term rewriting systems can be applied to analyze the termination of strategy controlled term rewriting systems. We show that the encoding of strategies into term rewriting systems can be easily adapted to handle many-sorted signa- tures and we use a meta-level representation of terms to reduce the size of the encodings. The corresponding implementation in Tom generates term rewriting systems compatible with the syntax of termination tools such as AProVE and TTT2, tools which turned out to be very effective in (dis)proving the termination of the generated term rewriting systems. The approach can also be seen as a generic strategy compiler which can be integrated into languages providing pattern matching primitives; experiments in Tom show that applying our encoding leads to performances comparable to the native Tom strategies

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