Combinatory Reduction Systems Research Articles

Introducing ⦇ λ ⦈, a λ-calculus for effectful computation

We present ⦇λ⦈, a calculus with special constructions for dealing with effects and handlers. This is an extension of the simply-typed λ-calculus (STLC). We enrich STLC with a type for representing effectful computations alongside with operations to create and process values of this type. The calculus is motivated by natural language modelling, and especially semantic representation. Traditionally, the meaning of a sentence is calculated using λ-terms, but some semantic phenomena need more flexibility. In this article we introduce the calculus and show that the calculus respects the laws of algebraic structures and it enjoys strong normalisation. To do so, confluence is proven using the Combinatory Reduction Systems (CRSs) of Klop and termination using the Inductive Data Type Systems (IDTSs) of Blanqui.

Infinitary Combinatory Reduction Systems

We define infinitary Combinatory Reduction Systems (iCRSs), thus providing the first notion of infinitary higher-order rewriting. The systems defined are sufficiently general that ordinary infinitary term rewriting and infinitary λ -calculus are special cases. Furthermore, we generalise a number of known results from first-order infinitary rewriting and infinitary λ -calculus to iCRSs. In particular, for fully-extended, left-linear iCRSs we prove the well-known compression property, and for orthogonal iCRSs we prove that (1) if a set of redexes U has a complete development, then all complete developments of U end in the same term and that (2) any tiling diagram involving strongly convergent reductions S and T can be completed iff at least one of S / T and T / S is strongly convergent. We also prove an ancillary result of independent interest: a set of redexes in an orthogonal iCRS has a complete development iff the set has the so-called finite jumps property.

Infinitary Combinatory Reduction Systems: Confluence

We study confluence in the setting of higher-order infinitary rewriting, in particular for infinitary Combinatory Reduction Systems (iCRSs). We prove that fully-extended, orthogonal iCRSs are confluent modulo identification of hypercollapsing subterms. As a corollary, we obtain that fully-extended, orthogonal iCRSs have the normal form property and the unique normal form property (with respect to reduction). We also show that, unlike the case in first-order infinitary rewriting, almost non-collapsing iCRSs are not necessarily confluent.

Lambda calculus with patterns

In this paper we revisit the λ -calculus with patterns, originating from the practice of functional programming language design. We treat this feature in a framework ranging from pure λ -calculus to orthogonal combinatory reduction systems.

A Higher-Order Calculus for Graph Transformation

This paper presents a formalism for defining higher-order systems based on the notion of graph transformation (by rewriting or interaction). The syntax is inspired by the Combinatory Reduction Systems of Klop. The rewrite rules can be used to define first-order systems, such as graph or term-graph rewriting systems, Lafont's interaction nets, the interaction systems of Asperti and Laneve, the non-deterministic nets of Alexiev, or a process calculus. They can also be used to specify higher-order systems such as hierarchical graphs and proof nets of Linear Logic, or to specify the operational semantics of graph-based languages.

Expressing combinatory reduction systems derivations in the rewriting calculus

The last few years have seen the development of the rewriting calculus (also called rho-calculus or ?-calculus) that uniformly integrates first-order term rewriting and the ?-calculus. The combination 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 (CRS), or by adding to the ?-calculus algebraic features. The various higher-order rewriting systems and the rewriting calculus share similar concepts and have similar applications, and thus, it is important to compare these formalisms to better understand their respective strengths and differences. We show in this paper that we can express Combinatory Reduction Systems derivations in terms of rewriting calculus derivations. The approach we present is based on a translation of each possible CRS-reduction into a corresponding ?-reduction. Since for this purpose we need to make precise the matching used when evaluating CRS, the second contribution of the paper is to present an original matching algorithm for CRS terms that uses a simple term translation and the classical matching of lambda terms.

Translating Combinatory Reduction Systems into the Rewriting Calculus

The last few years have seen the development of the rewriting calculus (or rho-calculus, ρCal) that extends first order term rewriting and λ-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 λ-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.

Higher-Order Rewriting and Partial Evaluation

We demonstrate the usefulness of higher-order rewriting techniques for specializing programs, i.e., for partial evaluation. More precisely, we demonstrate how casting program specializers as combinatory reduction systems (CRSs) makes it possible to formalize the corresponding program transformations as meta-reductions, i.e., reductions in the internal "substitution calculus." For partial-evaluation problems, this means that instead of having to prove on a case-by-case basis that one's "two-level functions" operate properly, one can concisely formalize them as a combinatory reduction system and obtain as a corollary that static reduction does not go wrong and yields a well-formed residual program.<br />We have found that the CRS substitution calculus provides an adequate expressive power to formalize partial evaluation: it provides sufficient termination strength while avoiding the need for additional restrictions such as types that would complicate the description unnecessarily (for our purpose). We also review the benefits and penalties entailed by more expressive higher-order formalisms. In addition, partial evaluation provides a number of examples of higher-order rewriting where being higher order is a central (rather than an occasional or merely exotic) property. We illustrate this by demonstrating how standard but non-trivial partial-evaluation examples are<br />handled with higher-order rewriting.

Axiomatizing permutation equivalence

We axiomatizepermutation equivalencein term rewriting systems and Klop’s orthogonal Combinatory Reduction Systems (Klop 1980). The axioms for the former are provided by the general approach proposed by Meseguer (Meseguer 1992). The latter need extra axioms modelling the interplay between reductions and the operation of substitution.As a consequence of this work, the definition of permutation equivalence is rid of residual calculi, which are heavy in general.

Interaction Systems I: The theory of optimal reductions

We introduce a new class of higher order rewriting systems, called Interaction Systems (IS's). IS's are derived from Lafont's (Intuitionistic) Interaction Nets (Lafont 1990) by dropping the linearity constraint. In particular, we borrow from Interaction Nets the syntactical bipartitions of operators into constructors and destructors and the principle of binary interaction. As a consequence, IS's are a subclass of Klop's Combinatory Reduction Systems (Klop 1980), where the Curry-Howard analogy still ‘makes sense’. Destructors and constructors, respectively, correspond to left and right logical introduction rules: interaction is cut and reduction is cut-elimination.Interaction Systems have been primarily motivated by the necessity of extending the practice of optimal evaluators for λ-calculus (Lamping 1990; Gonthier et al. 1992a) to other computational constructs such as conditionals and recursion. In this paper we focus on the theoretical aspects of optimal reductions. In particular, we generalize the family relation in Lévy (1978; 1980), thus defining the amount of sharing an optimal evaluator is required to perform. We reinforce our notion of family by approaching it in two different ways (generalizing labelling and extraction in Levy (1980)) and proving their coincidence. The reader is referred to Asperti and Laneve (1993c) for the paradigmatic description of optimal evaluators of IS's.

Combinatory reduction systems: introduction and survey

Combinatory reduction systems, or CRSs for short, were designed to combine the usual first-order format of term rewriting with the presence of bound variables as in pure λ-calculus and various typed λ-calculi. Bound variables are also present in many other rewrite systems, such as systems with simplification rules for proof normalization. The original idea of CRSs is due to Aczel, who introduced a restricted class of CRSs and, under the assumption of orthogonality, proved confluence. Orthogonality means that the rules are nonambiguous (no overlap leading to a critical pair) and left-linear (no global comparison of terms necessary). We introduce the class of orthogonal CRSs, illustrated with many examples, discuss its expressive power and give an outline of a short proof of confluence. This proof is a direct generalization of Aczel's original proof, which is close to the well-known confluence proof for λ-calculus by Tait and Martin-Löf. There is a well-known connection between the parallel reduction featuring in the latter proof and the concept of “developments”, and a classical lemma in the theory of λ-calculus is that of “finite developments”, a strong normalization result. It turns out that the notion of “parallel reduction” used in Aczel's proof gives rise to a generalized form of developments which we call “superdevelopments” and on which we will briefly comment.

Sequential evaluation strategies for parallel-or and related reduction systems

It is known that for every recursive strongly sequential regular term rewrite system there is a recursive normalising one-step evaluation strategy (briefly, a good strategy), that is, a recursive function which chooses a redex in every term not in normal form, such that repeated reduction of chosen redexes will eventually arrive at the normal form of any term which has one. We extend this result to a large class of abstract reduction systems, whose possessing what we call the ‘recursively acyclic Church-Rosser’ property. We show that this class includes all weakly regular combinatory reduction systems. Examples of these are lambda calculus, all regular term rewrite systems, and all systems whose only ambiguities are of the sort exemplified by ‘parallel or’: porTx→T , porxT→T . This refutes the claim that is sometimes made, that parallel-or and similar operators require parallel evaluation.

Tree Rewriting Systems, Combinatory Weak Reduction Systems and Type Free Languages1

In this paper we consider combinators as tree transducers: this approach is based on the one-to-one correspondence between terms of Combinatory Logic and trees, and on the fact that combinators may be considered as transformers of terms. Since combinators are terms themselves, we will deal with trees as objects to be transformed and tree transformers as well. Methods for defining and studying tree rewriting systems inside Combinatory Weak Reduction Systems and Weak Combinatory Logic are also analyzed and particular attention is devoted to the problem of finiteness and infinity of the generated tree languages (here defined). This implies the study of the termination of the rewriting process (i.e. reduction) for combinators.