Beatriz Alarcón

Proving termination properties with MU-TERM

MU-TERM is a tool which can be used to verify a number of termination properties of (variants of) Term Rewriting Systems (TRSs): termination of rewriting, termination of innermost rewriting, termination of order-sorted rewriting, termination of context-sensitive rewriting, termination of innermost context-sensitive rewriting and termination of rewriting modulo specific axioms. Such termination properties are essential to prove termination of programs in sophisticated rewriting-based programming languages. Specific methods have been developed and implemented in mu-term in order to efficiently deal with most of them. In this paper, we report on these new features of the tool.

Proving Termination of Context-Sensitive Rewriting with MU-TERM

Context-sensitive rewriting (CSR) is a restriction of rewriting which forbids reductions on selected arguments of functions. Proving termination of CSR is an interesting problem with several applications in the fields of term rewriting and programming languages. Several methods have been developed for proving termination of CSR. The new version of MU-TERM which we present here implements all currently known techniques. Furthermore, we show how to combine them to furnish MU-TERM with an expert which is able to automatically perform the termination proofs. Finally, we provide a first experimental evaluation of the tool.

Improving the Context-sensitive Dependency Graph

The dependency pairs method is one of the most powerful technique for proving termination of rewriting and it is currently central in most automatic termination provers. Recently, it has been adapted to be used in proofs of termination of context-sensitive rewriting. The use of collapsing dependency pairs i.e., having a single variable in the right-hand side is a novel and essential feature to obtain a correct framework in this setting. Unfortunately, dependency pairs behave as a kind of glue in the context-sensitive dependency graph which makes the cycles bigger, thus making some proofs of termination harder. In this paper we show that this effect can be safely mitigated by removing some arcs from the graph, thus leading to faster and easier proofs. Narrowing dependency pairs is also introduced and used here to eventually simplify the treatment of the context-sensitive dependency graph. We show the practicality of the new techniques with some benchmarks.

Termination of Innermost Context-Sensitive Rewriting Using Dependency Pairs

Innermost context-sensitive rewriting has been proved useful for modeling computations of programs of algebraic languages like Maude, OBJ, etc. Furthermore, innermost termination of rewriting is often easier to prove than termination. Thus, under appropriate conditions, a useful strategy for proving termination of rewriting is trying to prove termination of innermost rewriting. This phenomenon has also been investigated for context-sensitive rewriting (CSR). Up to now, only few transformations have been proposed and used to prove termination of innermost CSR. In this paper, we investigate direct methods for proving termination of innermost CSR. We adapt the recently introduced context-sensitive dependency pairs approach to innermost CSRand show that they can be advantageously used for proving termination of innermost CSR. We have implemented them as part of the termination tool mu-term.

A dependency pair framework for A∨C-termination

The development of powerful techniques for proving termination of rewriting modulo a set of equations is essential when dealing with rewriting logic-based programming languages like CafeOBJ, Maude, OBJ, etc. One of the most important techniques for proving termination over a wide range of variants of rewriting (strategies) is the dependency pair approach. Several works have tried to adapt it to rewriting modulo associative and commutative (AC) equational theories, and even to more general theories. However, as we discuss in this paper, no appropriate notion of minimality (and minimal chain of dependency pairs) which is well-suited to develop a dependency pair framework has been proposed to date. In this paper we carefully analyze the structure of infinite rewrite sequences for rewrite theories whose equational part is a (free) combination of associative and commutative axioms which we call A∨C-rewrite theories. Our analysis leads to a more accurate and optimized notion of dependency pairs through the new notion of stably minimal term. Then, we have developed a suitable dependency pair framework for proving termination of A∨C-rewrite theories.

Using Context-Sensitive Rewriting for Proving Innermost Termination of Rewriting

Computational systems based on reducing expressions usually have a predefined reduction strategy to break down the nondeterminism which is inherent to reduction relations. The innermost strategy corresponds to call by value or eager computation, that is, the computational mechanism of several programming languages like Maude, OBJ, etc. where the arguments of a function call are always evaluated before calling the function. This strategy usually fails to terminate when nonterminating computations are possible in the programs and many eager programming languages also admit the explicit specification of a particular class of strategy annotations to (try to) avoid them. Context-Sensitive Rewriting provides an abstract model to describe and analyze the operational behavior of such programs. This paper aims at contributing to the development of appropriate techniques and tools for the verification of program termination in the aforementioned programming languages, so we focus on termination of innermost (context-sensitive) rewriting. We adapt the notion of usable argument introduced by Fernandez to prove innermost termination by proving termination of context-sensitive rewriting. Thanks to our recent developments for proving termination of (innermost) context-sensitive rewriting using dependency pairs, now we can also relax monotonicity requirements for proving innermost termination of (context-sensitive) rewriting. We have implemented these new improvements in the termination tool mu-term and evaluated the results with some benchmarks.