Translating Combinatory Reduction Systems into the Rewriting Calculus

Long version. Colloque avec actes et comité de lecture. internationale.International audienceThe last few years have seen the development of the rewriting calculus (or rho-calculus, RHO) that extends first order term rewriting and lambda-calculus. The integration of these two latter formalisms has been already handled either by enriching first-order rewriting with higher-order capabilities, like in the Combinatory Reduction Systems, or by adding to lambda-calculus algebraic features. The different higher-order rewriting systems and the rewriting calculus share similar concepts and have similar applications, and thus, it seems natural to compare these formalisms. We analyze in this paper the relationship between the Rewriting Calculus and the Combinatory Reduction Systems and we present a translation of CRS-terms and rewrite rules into rho-terms and we show that for any CRS-reduction we have a corresponding rho-reduction

From nominal to higher-order rewriting and back again

We present a translation function from nominal rewriting systems (NRSs) to combinatory reduction systems (CRSs), transforming closed nominal rules and ground nominal terms to CRSs rules and terms, respectively, while preserving the rewriting relation. We also provide a reduction-preserving translation in the other direction, from CRSs to NRSs, improving over a previously defined translation. These tools, together with existing translations between CRSs and other higher-order rewriting formalisms, open up the path for a transfer of results between higher-order and nominal rewriting. In particular, techniques and properties of the rewriting relation, such as termination, can be exported from one formalism to the other.Comment: 41 pages, journa

Weak orthogonality implies confluence : the higher-order case

In this paper we prove confluence for weakly orthogonal Higher-Order Rewriting Systems. This generalises all the known `confluence by orthogonality' results

Definitions by Rewriting in the Calculus of Constructions

The main novelty of this paper is to consider an extension of the Calculus of Constructions where predicates can be defined with a general form of rewrite rules. We prove the strong normalization of the reduction relation generated by the beta-rule and the user-defined rules under some general syntactic conditions including confluence. As examples, we show that two important systems satisfy these conditions: a sub-system of the Calculus of Inductive Constructions which is the basis of the proof assistant Coq, and the Natural Deduction Modulo a large class of equational theories.Comment: Best student paper (Kleene Award

CRSX - Combinatory Reduction Systems with Extensions

Combinatory Reduction Systems with Extensions (CRSX) is a system available from http://crsx.sourceforge.net and characterized by the following properties: - Higher-order rewriting engine based on pure Combinatory Reduction Systems with full strong reduction (but no specified reduction strategy). - Rule and term syntax based on lambda-calculus and term rewriting conventions including Unicode support. - Strict checking and declaration requirements to avoid idiosyncratic errors in rewrite rules. - Interpreter is implemented in Java 5 and usable stand-alone as well as from an Eclipse plugin (under development). - Includes a custom parser generator (front-end to JavaCC parser generator) designed to ease parsing directly into higher-order abstract syntax (as well as permitting the use of custom syntax in rules files). - Experimental (and evolving) sort system to help rule management. - Compiler from (well-sorted deterministic subset of) CRSX to stand-alone C code

Computation in director string calculus

In this thesis we introduce a modified version of Director String Calculus (MDSC) which preserves the applicative structure of the original lambda terms and captures the strong reduction as opposed to weak reduction of the original Director String Calculus (DSC). Furthermore, our reduction system provides an environment which supports the nonatomic nature of substitution operation and hence can lend itself to parallel and optimal reduction. We shall compare our reduction method with other reduction methods, and discuss some of the advantages and disadvantages of our method

On Constructor Rewrite Systems and the Lambda Calculus

We prove that orthogonal constructor term rewrite systems and lambda-calculus with weak (i.e., no reduction is allowed under the scope of a lambda-abstraction) call-by-value reduction can simulate each other with a linear overhead. In particular, weak call-by- value beta-reduction can be simulated by an orthogonal constructor term rewrite system in the same number of reduction steps. Conversely, each reduction in a term rewrite system can be simulated by a constant number of beta-reduction steps. This is relevant to implicit computational complexity, because the number of beta steps to normal form is polynomially related to the actual cost (that is, as performed on a Turing machine) of normalization, under weak call-by-value reduction. Orthogonal constructor term rewrite systems and lambda-calculus are thus both polynomially related to Turing machines, taking as notion of cost their natural parameters.Comment: 27 pages. arXiv admin note: substantial text overlap with arXiv:0904.412