Raúl Gutiérrez

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.

Proving termination in the context-sensitive dependency pair framework

Termination of context-sensitive rewriting (CSR) is an interesting problem with several applications in the fields of term rewriting and in the analysis of programming languages like CafeOBJ, Maude, OBJ, etc. The dependency pair approach, one of the most powerful techniques for proving termination of rewriting, has been adapted to be used for proving termination of CSR. The corresponding notion of context-sensitive dependency pair (CSDP) is different from the standard one in that collapsing pairs (i.e., rules whose right-hand side is a variable) are considered. Although the implementation and practical use of CSDPs lead to a powerful framework for proving termination of CSR, handling collapsing pairs is not easy and often leads to impose heavy requirements over the base orderings which are used to achieve the proofs. A recent proposal removes collapsing pairs by transforming them into sets of new (standard) pairs. In this way, though, the role of collapsing pairs for modeling context-sensitive computations gets lost. This leads to a less intuitive and accurate description of the termination behavior of the system. In this paper, we show how to get the best of the two approaches, thus obtaining a powerful context-sensitive dependency pair framework which satisfies all practical and theoretical expectations.

Usable Rules for Context-Sensitive Rewrite Systems

Recently, the dependency pairs (DP) approach has been generalized to context-sensitive rewriting (CSR). Although the context-sensitive dependency pairs (CS-DP) approachprovides a very good basis for proving termination of CSR, the current developments basically correspond to a ten-years-old DP approach. Thus, the task of adapting all recently introduced dependency pairs techniques to get a more powerful approach becomes an important issue. In this direction, usable rulesare one of the most interesting and powerful notions. Actually usable rule have been investigated in connection with proofs of innermost terminationof CSR. However, the existing results apply to a quite restricted class of systems. In this paper, we introduce a notion of usable rules that can be used in proofs of termination of CSR with arbitrary systems. Our benchmarks show that the performance of the CS-DP approach is much better when such usable rules are considered in proofs of termination of CSR.

Order-sorted equality enrichments modulo axioms

We make the addition of equationally defined equality predicates effective and automatic by means of a transformation.The transformation is constructive and valid for a wide class of equational specifications.All the expected good properties of the input theory are preserved by the transformation.The transformation is implemented in Maude and integrated into Maude formal tools. Built-in equality and inequality predicates based on comparison of canonical forms in algebraic specifications are frequently used because they are handy and efficient. However, their use places algebraic specifications with initial algebra semantics beyond the pale of theorem proving tools based, for example, on explicit or inductionless induction techniques, and of other formal tools for checking key properties such as confluence, termination, and sufficient completeness. Such specifications would instead be amenable to formal analysis if an equationally-defined equality predicate enriching the algebraic data types were to be added to them. Furthermore, having an equationally-defined equality predicate is very useful in its own right, particularly in inductive theorem proving. Is it possible to effectively define a theory transformation E ? E ~ that extends an algebraic specification E to a specification E ~ having an equationally-defined equality predicate? This paper answers this question in the affirmative for a broad class of order-sorted conditional specifications E that are sort-decreasing, ground confluent, and operationally terminating modulo axioms B and have a subsignature of constructors. The axioms B can consist of associativity, or commutativity, or associativity-commutativity axioms, so that the constructors are free modulo B. We prove that the transformation E ? E ~ preserves all the just-mentioned properties of E . The transformation has been automated in Maude using reflection and is used as a component in many Maude formal tools.

Order-Sorted equality enrichments modulo axioms

Built-in equality and inequality predicates based on comparison of canonical forms in algebraic specifications are frequently used because they are handy and efficient. However, their use places algebraic specifications with initial algebra semantics beyond the pale of theorem proving tools based, for example, on explicit or inductionless induction techniques, and of other formal tools for checking key properties such as confluence, termination, and sufficient completeness. Such specifications would instead be amenable to formal analysis if an equationally-defined equality predicate enriching the algebraic data types were to be added to them. Furthermore, having an equationally-defined equality predicate is very useful in its own right, particularly in inductive theorem proving. Is it possible to effectively define a theory transformation

Automatic Synthesis of Logical Models for Order-Sorted First-Order Theories

{\cal E} \mapsto {\cal E}^{\mathsf{\:\simeq\:}}

Extending the 2D Dependency Pair Framework for Conditional Term Rewriting Systems

that extends an algebraic specification

Use of logical models for proving infeasibility in term rewriting

{\cal E}