Other affiliations: Carnegie Mellon University, University of Bologna, École Normale Supérieure ...read more
Bio: Beniamino Accattoli is an academic researcher from École Polytechnique. The author has contributed to research in topics: Linear logic & Abstract machine. The author has an hindex of 19, co-authored 73 publications receiving 993 citations. Previous affiliations of Beniamino Accattoli include Carnegie Mellon University & University of Bologna.
19 Aug 2014
TL;DR: 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, and shows that the LSC is a complexity-preserving abstraction of abstract machines.
Abstract: 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.
08 Jan 2014
TL;DR: 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, and relies on Gonthier, Lévy, and Melliès' axiomatic theory for standardization.
Abstract: 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 Levy, 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, Levy, and Mellies' 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.
••28 May 2012
TL;DR: A simple form of standardization, here called factorization, for explicit substitutions calculi, i.e. lambda-calculi where beta-reduction is decomposed in various rules, is studied and an abstract theorem deducing factorization from some axioms on local diagrams is developed.
Abstract: We study a simple form of standardization, here called factorization, for explicit substitutions calculi, i.e. lambda-calculi where beta-reduction is decomposed in various rules. These calculi, despite being non-terminating and non-orthogonal, have a key feature: each rule terminates when considered separately. It is well-known that the study of rewriting properties simplifies in presence of termination (e.g. confluence reduces to local confluence). This remark is exploited to develop an abstract theorem deducing factorization from some axioms on local diagrams. The axioms are simple and easy to check, in particular they do not mention residuals. The abstract theorem is then applied to some explicit substitution calculi related to Proof-Nets. We show how to recover standardization by levels, we model both call-by-name and call-by-value calculi and we characterize linear head reduction via a factorization theorem for a linear calculus of substitutions.
••23 May 2012
TL;DR: The value-substitution lambda-calculus is introduced, a simple calculus borrowing ideas from Herbelin and Zimmerman's call-by-value λ CBV calculus and from Accattoli and Kesner's substitution calculus λ sub .
Abstract: In the call-by-value lambda-calculus solvable terms have been characterised by means of call-by-name reductions, which is disappointing and requires complex reasonings. We introduce the value-substitution lambda-calculus, a simple calculus borrowing ideas from Herbelin and Zimmerman's call-by-value λ CBV calculus and from Accattoli and Kesner's substitution calculus λ sub . In this new setting, we characterise solvable terms as those terms having normal form with respect to a suitable restriction of the rewriting relation.
••14 Jul 2014
TL;DR: The main technical contribution of the paper is indeed the definition of useful reductions and the thorough analysis of their properties, and the first complete positive answer to this long-standing problem of λ-calculus.
Abstract: Slot and van Emde Boas' weak invariance thesis states that reasonable machines can simulate each other within a polynomially overhead in time. Is λ-calculus a reasonable machine? Is there a way to measure the computational complexity of a λ-term? This paper presents the first complete positive answer to this long-standing problem. Moreover, our answer is completely machine-independent and based over a standard notion in the theory of λ-calculus: the length of a leftmost-outermost derivation to normal form is an invariant cost model. Such a theorem cannot be proved by directly relating λ-calculus with Turing machines or random access machines, because of the size explosion problem: there are terms that in a linear number of steps produce an exponentially long output. The first step towards the solution is to shift to a notion of evaluation for which the length and the size of the output are linearly related. This is done by adopting the linear substitution calculus (LSC), a calculus of explicit substitutions modelled after linear logic proof nets and admitting a decomposition of leftmost-outermost derivations with the desired property. Thus, the LSC is invariant with respect to, say, random access machines. The second step is to show that LSC is invariant with respect to the λ-calculus. The size explosion problem seems to imply that this is not possible: having the same notions of normal form, evaluation in the LSC is exponentially longer than in the λ-calculus. We solve such an impasse by introducing a new form of shared normal form and shared reduction, deemed useful. Useful evaluation avoids those steps that only unshare the output without contributing to β-redexes, i.e. the steps that cause the blow-up in size. The main technical contribution of the paper is indeed the definition of useful reductions and the thorough analysis of their properties.
01 Jan 2002
TL;DR: This chapter presents the basic concepts of term rewriting that are needed in this book and suggests several survey articles that can be consulted.
Abstract: In this chapter we will present the basic concepts of term rewriting that are needed in this book. More details on term rewriting, its applications, and related subjects can be found in the textbook of Baader and Nipkow [BN98]. Readers versed in German are also referred to the textbooks of Avenhaus [Ave95], Bundgen [Bun98], and Drosten [Dro89]. Moreover, there are several survey articles [HO80, DJ90, Klo92, Pla93] that can also be consulted.
01 Jan 2009
01 Jan 2016
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TL;DR: Term Rewriting and All That is a self-contained introduction to the field of term rewriting and covers all the basic material including abstract reduction systems, termination, confluence, completion, and combination problems.
Abstract: Term Rewriting and All That is a self-contained introduction to the field of term rewriting. The book starts with a simple motivating example and covers all the basic material including abstract reduction systems, termination, confluence, completion, and combination problems. Some closely connected subjects, such as universal algebra, unification theory, Grobher bases, and Buchberger's algorithm, are also covered.