Static termination analysis for Prolog using term rewriting and SAT solving

The most fundamental decision problem in computer science is the halting problem, i.e., given a description of a program and an input, decide whether the program terminates after finitely many steps or runs forever on that input. While Turing showed this problem to be undecidable in general, developing static analysis techniques that can automatically prove termination for many pairs of programs and inputs is of great practical interest. This is true in particular for logic programming, as the inherent lack of direction in the computation virtually guarantees that any non-trivial program terminates only for certain classes of inputs. Thus, termination of logic programs is widely studied and significant advances have been made during the last decades. Nowadays, there are fully-automated tools that try to prove termination of a given logic program w.r.t. a given class of inputs. Nevertheless, there still remain many logic programs that cannot be handled by any current termination technique for logic programs that is amenable to automation. Another area where termination has been studied even more intensively is term rewriting. This basic computation principle underlies the evaluation mechanism of many programming languages. Significant advances towards powerful automatable termination techniques during the last decade have yielded a plethora of powerful tools for proving termination automatically. In this thesis, we show that techniques developed for proving termination of term rewriting can successfully be adapted and applied to analyze logic programs. The new techniques developed significantly extend the applicability and the power of automated termination analysis for logic programs. The work presented here ranges from adapting techniques to work directly on logic programs to transformations from logic programs to a specialized version of term rewriting. On the logic programming side we also present a new pre-processing approach to handle logic programs with cuts. On the term rewriting side we show how to search for certain popular classes of well-founded orders on terms more efficiently by encoding the search into satisfiability problems of propositional logic. The contributions developed in this thesis are implemented in tools for automated termination analysis – mostly in our fully automated termination prover AProVE. The significance of our results is demonstrated by the fact that AProVE has reached the highest score both for term rewriting and logic programming at the annual international Termination Competitions in all years since 2004, where the leading automated tools try to analyze termination of programs from different areas of computer science

Static termination analysis for Prolog using term rewriting and SAT solving

The most fundamental decision problem in computer science is the halting problem, i.e., given a description of a program and an input, decide whether the program terminates after finitely many steps or runs forever on that input. While Turing showed this problem to be undecidable in general, developing static analysis techniques that can automatically prove termination for many pairs of programs and inputs is of great practical interest. This is true in particular for logic programming, as the inherent lack of direction in the computation virtually guarantees that any non-trivial program terminates only for certain classes of inputs. Thus, termination of logic programs is widely studied and significant advances have been made during the last decades. Nowadays, there are fully-automated tools that try to prove termination of a given logic program w.r.t. a given class of inputs. Nevertheless, there still remain many logic programs that cannot be handled by any current termination technique for logic programs that is amenable to automation. Another area where termination has been studied even more intensively is term rewriting. This basic computation principle underlies the evaluation mechanism of many programming languages. Significant advances towards powerful automatable termination techniques during the last decade have yielded a plethora of powerful tools for proving termination automatically. In this thesis, we show that techniques developed for proving termination of term rewriting can successfully be adapted and applied to analyze logic programs. The new techniques developed significantly extend the applicability and the power of automated termination analysis for logic programs. The work presented here ranges from adapting techniques to work directly on logic programs to transformations from logic programs to a specialized version of term rewriting. On the logic programming side we also present a new pre-processing approach to handle logic programs with cuts. On the term rewriting side we show how to search for certain popular classes of well-founded orders on terms more efficiently by encoding the search into satisfiability problems of propositional logic. The contributions developed in this thesis are implemented in tools for automated termination analysis – mostly in our fully automated termination prover AProVE. The significance of our results is demonstrated by the fact that AProVE has reached the highest score both for term rewriting and logic programming at the annual international Termination Competitions in all years since 2004, where the leading automated tools try to analyze termination of programs from different areas of computer science

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