Linear Head Reduction Research Articles

Krivine Machine and Taylor Expansion in a Non-uniform Setting

The Krivine machine is an abstract machine implementing the linear head reduction of lambda-calculus. Ehrhard and Regnier gave a resource sensitive version returning the annotated form of a lambda-term accounting for the resources used by the linear head reduction. These annotations take the form of terms in the resource lambda-calculus. We generalize this resource-driven Krivine machine to the case of the algebraic lambda-calculus. The latter is an extension of the pure lambda-calculus allowing for the linear combination of lambda-terms with coefficients taken from a semiring. Our machine associates a lambda-term M and a resource annotation t with a scalar k in the semiring describing some quantitative properties of the linear head reduction of M. In the particular case of non-negative real numbers and of algebraic terms M representing probability distributions, the coefficient k gives the probability that the linear head reduction actually uses exactly the resources annotated by t. In the general case, we prove that the coefficient k can be recovered from the coefficient of t in the Taylor expansion of M and from the normal form of t.

Linear β-reduction

Linear head reduction is a key tool for the analysis of reduction machines for lambda-calculus and for game semantics. Its definition requires a notion of redex at a distance named primary redex in the literature. Nevertheless, a clear and complete syntactic analysis of this rule is missing. We present here a general notion of beta-reduction at a distance and of linear reduction (i.e., not restricted to the head variable), and we analyse their relations and properties. This analysis rests on a variant of the so-called sigma-equivalence that is more suitable for the analysis of reduction machines, since the position along the spine of primary redexes is not permuted. We finally show that, in the simply typed case, the proof of strong normalisation of linear reduction can be obtained by a trivial tuning of Gandy's proof for strong normalisation of beta-reduction.

Bounding linear head reduction and visible interaction through skeletons

In this paper, we study the complexity of execution in higher-order programming languages. Our study has two facets: on the one hand we give an upper bound to the length of interactions between bounded P-visible strategies in Hyland-Ong game semantics. This result covers models of programming languages with access to computational effects like non-determinism, state or control operators, but its semantic formulation causes a loose connection to syntax. On the other hand we give a syntactic counterpart of our semantic study: a non-elementary upper bound to the length of the linear head reduction sequence (a low-level notion of reduction, close to the actual implementation of the reduction of higher-order programs by abstract machines) of simply-typed lambda-terms. In both cases our upper bounds are proved optimal by giving matching lower bounds. These two results, although different in scope, are proved using the same method: we introduce a simple reduction on finite trees of natural numbers, hereby called interaction skeletons. We study this reduction and give upper bounds to its complexity. We then apply this study by giving two simulation results: a semantic one measuring progress in game-theoretic interaction via interaction skeletons, and a syntactic one establishing a correspondence between linear head reduction of terms satisfying a locality condition called local scope and the reduction of interaction skeletons. This result is then generalized to arbitrary terms by a local scopization transformation.

Distilling abstract machines

It is well-known that many environment-based abstract machines can be seen as strategies in lambda calculi with explicit substitutions (ES). Recently, graphical syntaxes and linear logic led to the linear substitution calculus (LSC), a new approach to ES that is halfway between small-step calculi and traditional calculi with ES. This paper studies the relationship between the LSC and environment-based abstract machines. While traditional calculi with ES simulate abstract machines, the LSC rather distills them: some transitions are simulated while others vanish, as they map to a notion of structural congruence. The distillation process unveils that abstract machines in fact implement weak linear head reduction, a notion of evaluation having a central role in the theory of linear logic. We show that such a pattern applies uniformly in call-by-name, call-by-value, and call-by-need, catching many machines in the literature. We start by distilling the KAM, the CEK, and a sketch of the ZINC, and then provide simplified versions of the SECD, the lazy KAM, and Sestoft's machine. Along the way we also introduce some new machines with global environments. Moreover, we show that distillation preserves the time complexity of the executions, i.e. the LSC is a complexity-preserving abstraction of abstract machines.

A nonstandard standardization theorem

Standardization is a fundamental notion for connecting programming languages and rewriting calculi. Since both programming languages and calculi rely on substitution for defining their dynamics, explicit substitutions (ES) help further close the gap between theory and practice. This paper focuses on standardization for the linear substitution calculus, a calculus with ES capable of mimicking reduction in lambda-calculus and linear logic proof-nets. For the latter, proof-nets can be formalized by means of a simple equational theory over the linear substitution calculus. Contrary to other extant calculi with ES, our system can be equipped with a residual theory in the sense of Lévy, which is used to prove a left-to-right standardization theorem for the calculus with ES but without the equational theory. Such a theorem, however, does not lift from the calculus with ES to proof-nets, because the notion of left-to-right derivation is not preserved by the equational theory. We then relax the notion of left-to-right standard derivation, based on a total order on redexes, to a more liberal notion of standard derivation based on partial orders. Our proofs rely on Gonthier, Lévy, and Melliès' axiomatic theory for standardization. However, we go beyond merely applying their framework, revisiting some of its key concepts: we obtain uniqueness (modulo) of standard derivations in an abstract way and we provide a coinductive characterization of their key abstract notion of external redex. This last point is then used to give a simple proof that linear head reduction --a nondeterministic strategy having a central role in the theory of linear logic-- is standard.

Non-idempotent intersection types and strong normalisation

We present a typing system with non-idempotent intersection types, typing a term syntax covering three different calculi: the pure {\lambda}-calculus, the calculus with explicit substitutions {\lambda}S, and the calculus with explicit substitutions, contractions and weakenings {\lambda}lxr. In each of the three calculi, a term is typable if and only if it is strongly normalising, as it is the case in (many) systems with idempotent intersections. Non-idempotency brings extra information into typing trees, such as simple bounds on the longest reduction sequence reducing a term to its normal form. Strong normalisation follows, without requiring reducibility techniques. Using this, we revisit models of the {\lambda}-calculus based on filters of intersection types, and extend them to {\lambda}S and {\lambda}lxr. Non-idempotency simplifies a methodology, based on such filter models, that produces modular proofs of strong normalisation for well-known typing systems (e.g. System F). We also present a filter model by means of orthogonality techniques, i.e. as an instance of an abstract notion of orthogonality model formalised in this paper and inspired by classical realisability. Compared to other instances based on terms (one of which rephrases a now standard proof of strong normalisation for the {\lambda}-calculus), the instance based on filters is shown to be better at proving strong normalisation results for {\lambda}S and {\lambda}lxr. Finally, the bounds on the longest reduction sequence, read off our typing trees, are refined into an exact measure, read off a specific typing tree (called principal); in each of the three calculi, a specific reduction sequence of such length is identified. In the case of the {\lambda}-calculus, this complexity result is, for longest reduction sequences, the counterpart of de Carvalho's result for linear head-reduction sequences.

Evaluating functions as processes

A famous result by Milner is that the lambda-calculus can be simulated inside the pi-calculus. This simulation, however, holds only modulo strong bisimilarity on processes, i.e. there is a slight mismatch between beta-reduction and how it is simulated in the pi-calculus. The idea is that evaluating a lambda-term in the pi-calculus is like running an environment-based abstract machine, rather than applying ordinary beta-reduction. In this paper we show that such an abstract-machine evaluation corresponds to linear weak head reduction, a strategy arising from the representation of lambda-terms as linear logic proof nets, and that the relation between the two is as tight as it can be. The study is also smoothly rephrased in the call-by-value case, introducing a call-by-value analogous of linear weak head reduction.

Totality in arena games

We tackle the problem of preservation of totality by composition in arena games. We first explain how this problem reduces to a finiteness theorem on what we call pointer structures, similar to the parity pointer functions of Harmer, Hyland & Melliès and the interaction sequences of Coquand. We discuss how this theorem relates to normalization of linear head reduction in simply-typed λ -calculus, leading us to a semantic realizability proof à la Kleene of our theorem. We then present another proof of a more combinatorial nature. Finally, we discuss the exact class of strategies to which our theorems apply.

The differential lambda-calculus

We present an extension of the lambda-calculus with differential constructions. We state and prove some basic results (confluence, strong normalization in the typed case), and also a theorem relating the usual Taylor series of analysis to the linear head reduction of lambda-calculus.

Reversible, Irreversible and Optimal λ-machines: Extended abstract

There are two quite different possibilities for implementing linear head reduction in λ-calculus. Two ways which we are going to explain briefly here in the introduction and in details in the body of the paper. The paper itself is concerned with showing an unexpectedly simple relation between these two ways, which we term reversible and irreversible, namely that the latter may be obtained as a natural optimization of the former.