Salvador Lucas

Context-sensitive rewriting strategies

Contex-sensitive rewriting is a simple rewriting restriction which is formalized by imposing fixed restrictions on replacements. Such a restriction is given on a purely syntactic basis: it is given on the arguments of symbols of the signature and inductively extended to arbitrary positions of terms built from those symbols. The termination behavior is not only preserved but usually improved and several methods have been developed to formally prove it. In this paper, we investigate the definition, properties, and use of context-sensitive rewriting strategies, i.e., particular, fixed sequences of context-sensitive rewriting steps. We study how to define them in order to obtain efficient computations and to ensure that context-sensitive computations terminate whenever possible. We give conditions enabling the use of these strategies for root-normalization, normalization, and infinitary normalization. We show that this theory is suitable for formalizing the definition and analysis of real computational strategies which are used in programming languages such as OBJ or ELAN.

Proving operational termination of membership equational programs

Reasoning about the termination of equational programs in sophisticated equational languages such as Elan, Maude, OBJ, CafeOBJ, Haskell, and so on, requires support for advanced features such as evaluation strategies, rewriting modulo, use of extra variables in conditions, partiality, and expressive type systems (possibly including polymorphism and higher-order). However, many of those features are, at best, only partially supported by current term rewriting termination tools (for instance mu-term, CiME, AProVE, TTT, Termptation, etc.) while they may be essential to ensure termination. We present a sequence of theory transformations that can be used to bridge the gap between expressive membership equational programs and such termination tools, and prove the correctness of such transformations. We also discuss a prototype tool performing the transformations on Maude equational programs and sending the resulting transformed theories to some of the aforementioned standard termination tools.

Operational termination of conditional term rewriting systems

We describe conditional rewriting by means of an inference system and capture termination as the absence of infinite inference: that is, all proof attempts must either successfully terminate, or they must fail in finite time. We call this notion operational termination. Our notion of operational termination is parametric on the inference system. We prove that operational termination of CTRSs is, in fact, equivalent to a very general notion proposed for 3-CTRSs, namely the notion of quasi-decreasingness, which is currently the most general one which is intended to be checked by comparing parts of the CTRS by means of term orderings. Therefore, existing methods for proving quasi-decreasingness of CTRSs immediately apply to prove operational termination of CTRSs.

Polynomials over the reals in proofs of termination : from theory to practice

This paper provides a framework to address termination problems in term rewriting by using orderings induced by algebras over the reals. The generation of such orderings is parameterized by concrete monotonicity requirements which are connected with different classes of termination problems: termination of rewriting, termination of rewriting by using dependency pairs, termination of innermost rewriting, top-termination of infinitary rewriting, termination of context-sensitive rewriting, etc. We show how to define term orderings based on algebraic interpretations over the real numbers which can be used for these purposes. Prom a practical point of view, we show how to automatically generate polynomial algebras over the reals by using constraint-solving systems to obtain the coefficients of a polynomial in the domain of the real or rational numbers. Moreover, as a consequence of our work, we argue that software systems which are able to generate constraints for obtaining polynomial interpretations over the naturals which prove termination of rewriting (e.g., AProVE, CiME, and TTT), are potentially able to obtain suitable interpretations over the reals by just solving the constraints in the domain of the real or rational numbers.

Termination of on-demand rewriting and termination of OBJ programs

Declarative languages such as OBJ, CafeOBJ, and Maude use syntactic annotations to introduce replacement restrictions aimed at improving termination or efficiency of computations. Unfortunately, there is a lack of formal techniques for proving such benefits. We show that context-sensitive rewriting and on-demand rewriting provide a suitable framework to address this problem. We provide methods to analyze termination of on-demand rewriting and apply them to analyze termination of OBJ, CafeOBJ, and Maude programs.

mu-term: A Tool for Proving Termination of Context-Sensitive Rewriting

Restrictions of rewriting can eventually achieve termination by pruning all infinite rewrite sequences issued from every term. Context-sensitive rewriting (CSR) is an example of such a restriction. In CSR, the replacements in some arguments of the function symbols are permanently forbidden. This paper describes mu-term, a tool which can be used to automatically prove termination of CSR. The tool implements the generation of the appropriate orderings for proving termination of CSR by means of polynomial interpretations over the rational numbers. In fact, mu-term is the first termination tool which generates term orderings based on such polynomial interpretations. These orderings can also be used, in a number of different ways, for proving termination of ordinary rewriting. Proofs of termination of CSR are also possible via existing transformations to TRSs (without any replacement restriction) which are also implemented in mu-term.

Termination of Rewriting With Strategy Annotations

We investigate termination of rewriting computations guided by strategy annotations. We show that proofs of termination can be obtained by proving(innermost) termination of context-sensitive rewriting( CSR). Hence, we investigate how to prove innermost termination of CSR usingexistingmetho ds for provingtermination of CSR.

Proving termination of membership equational programs

Advanced typing, matching, and evaluation strategy features, as well as very general conditional rules, are routinely used in equational programming languages such as, for example, <sc>ASF+SDF</sc>, <sc>OBJ</sc>, <sc>CafeOBJ</sc>, <sc>Maude</sc>, and equational subsets of <sc>ELAN</sc> and <sc>CASL</sc>. Proving termination of equational programs having such expressive features is important but nontrivial, because some of those features may not be supported by standard termination methods and tools, such as <sc>muterm</sc>, <sc>C<i>i</i>ME</sc>, <sc>AProVE</sc>, <sc>TTT</sc>, <sc>Termptation</sc>, etc. Yet, use of the features may be essential to ensure termination. We present a sequence of theory transformations that can be used to bridge the gap between expressive equational programs and termination tools, prove the correctness of such transformations, and discuss a prototype tool performing the transformations on <sc>Maude</sc> equational programs and sending the resulting transformed theories to some of the aforementioned tools.

Termination of Context-Sensitive Rewriting by Rewriting

Context-sensitive rewriting allows us to deal with certain context-replacing restrictions in performing reduction steps. This is useful in avoiding undesirable reductions. In this paper, we study termination of context-sensitive rewriting. We give some conditions under which it is possible to analyze the termination of the context-sensitive rewrite relation induced by a term rewriting system by means of any method to prove termination of rewriting. This allows us to apply all known (and forthcoming) results on termination to context-sensitive rewriting.

Abstract Diagnosis of Functional Programs

We present a generic scheme for the declarative debugging of functional programs modeled as term rewriting systems. We associate to our programs a semantics based on a (continuous) immediate consequence operator, \( T_\mathcal{R} \), which models the (values/normal forms) semantics of \( \mathcal{R} \) . Then, we develop an effective debugging methodology which is based on abstract interpretation: by approximating the intended specification of the semantics of \( \mathcal{R} \) we derive a finitely terminating bottom-up diagnosis method, which can be used statically. Our debugging framework does not require the user to either provide error symptoms in advance or answer questions concerning program correctness. We have made available a prototypical implementation in Haskell and have tested it on some non trivial examples.