Stéphane Kaplan

Conditional rewrite rules

Abstract The notion of conditional term rewriting systems that allows to simulate theories specified by conditional equations is presented. The paper considers general conditional rules, as opposed to related works usually dealing with restricted cases. Means for the investigation of conditional rewriting (i.e., iterative constructs) are proposed. Evaluation by rewriting is shown to be correct with respect to the underlying algebraic theory. A general evaluation procedure is given. Intractability results about one-step rewriting, even for canonical systems, and computation of the normal form function are established.

Rewrite, rewrite, rewrite, rewrite, rewrite, …

Abstract We study properties of rewrite systems that are not necessarily terminating, but allow instead for transfinite derivations that have a limit. In particular, we give conditions for the existence of a limit and for its uniqueness and relate the operational and algebraic semantics of infinitary theories. We also consider sufficient completeness of hierarchical systems.

A compiler for conditional term rewriting systems

In this paper, we present a compiler for conditional term rewriting systems. With respect to traditional interpreters, the gain in execution time that we obtain is of several orders of magnitude. We discuss several optimizations, among which a method to share code in the premises of the conditional rules, well-adapted to algebraic specifications.

Complexity analysis of term-rewriting systems

Algebraic specifications are now widely used for data structuring and they turn out to be quite useful for various aspects of program development, such as prototyping, assisted program construction, proving properties, etc. [3, 12, 13, 1.5, 16, 17, 181. Some of these applications require adding a notion of computation to algebraic specifications, for instance by providing a (convergent) rewrite rule system that expresses the properties of the operators. In this context, it may be of prime interest to define a notion of algorithmic complexity for an algebraic specification, or, more precisely, a notion of complexity for each operator defined in the specification. Computing operator complexity within a given specification helps understanding how evaluation costs are distributed; it may single out “costly” operators, and motivate the search for an equivalent, but “cheaper”, specification. In [5], the cost of a term is defined as the number of rewriting steps for reducing it to its normal form, and the cost of an operator is defined as the genera1 cost of a term obtained by applying this operator to terms in normal form. In this paper, we further formalize this notion of operator complexity and investigate its computation through analysis methods developed for instance in [24,9]. We show how these methods apply to the computation of the enumerative series related to the terms of an algebraic specification. We define the notion of regular rewriting systems, and consider cost series associated with operators that are described by such systems. We show how these analysis methods apply to compute such costs and provide an asymptotic evaluation of the average cost of an operator. Our results allow costs to be computed without any explicit manipulation of series. We provide the user with ready-to-use formulae, where the different parameters only depend on the “geometry” of the system, e.g. the number of constructors in the left-hand side of rules, number of occurrences of a derived operator in the right-hand side, etc. Quantitative evaluation of rewriting systems had not yet been studied under such an approach (except in [5]), to our knowledge. From a different point of view, complexity of algebraic implementations has been studied in [2,8 etc.] w.r.t computability issues.

Completion Algorithms for Conditional Rewriting Systems

Publisher Summary This chapter discusses completion algorithms for conditional rewriting systems. It provides an overview on a completion algorithm for hierarchical conditional term rewriting systems (CTRS). The algorithm starts by trying to orient as many equations as it can, using information given by the user about the precedence on the operators. When it has too little information, it asks the user for more. Moreover, it checks the rules against hierarchy hypotheses. The preconditions have to be primitive and the left-hand sides have to contain at least one nonprimitive operator. Then, the algorithm computes critical pairs and tries to prove that they are convergent on ground terms. First order theorem proving techniques are considered as definitely necessary to deal with the preconditions when one restrict the class of models to Log-models. To apply the obtained results to proofs in parameterized a specification, which implies carefully studying the properties in this domain. The chapter presents an additional degree of complexity in regard to the nonparameterized case.

Infinite normal forms

We continue here a study of properties of rewrite systems that are not necessarily terminating, but allow for infinite derivations that have a limit. In particular, we give algebraic semantics for theories described by such systems, consider sufficient completeness of hierarchical systems, suggest practical conditions for the existence of a limit and for its uniqueness, and extend the ideas to conditional rewriting.

Rewrite, rewrite, rewrite, rewrite, rewrite...

The theory of term rewriting systems has important applications in abstract data type specifications and functional programming languages. We begin here a study of properties of systems that are not necessarily terminating, but allow for infinite derivations that have a limit. In particular, we give conditions for the existence of a limit and for its uniqueness.

Algebraic specification of concurrent systems

Abstract This article presents an extension of the formalism of algebraic specifications to the specification of concurrent systems. The key concept is that of process specifications , which have two hierarchical layers: processes and data. Processes apply to data via an application operator , eventually yielding a set of data as a result. Syntax and semantics of process specifications are presented, with emphasis on methodological issues (how to write hierarchically consistent and complete specifications). A suitable notion of observational congruence is introduced and characterized. A notion of implementation is defined, and a general method for proving correction is considered, based on the notion of serializability proof. A primitive for putting specifications together, in parallel, is analyzed. Finally, richer primitives for building basic specifications are discussed. Our proposal is illustrated via several examples including the one of the systematic, stepwise development of a complex specification.