Beta reduction is invariant, indeed

Proceedings of the 21st International Symposium on Principles and Practice of Declarative Programming

Coen

2019

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e λ-calculus is a handy formalism to specify the evaluation of higher-order programs. It is not very handy, however, when one interprets the speci cation as an execution mechanism, because terms can grow exponentially with the number of β-steps. is is why implementations of functional languages and proof assistants always rely on some form of sharing of subterms. ese frameworks however do not only evaluate λ-terms, they also have to compare them for equality. In presence of sharing, one is actually interested in equality of the underlying unshared λ-terms.e literature contains algorithms for such a sharing equality, that are polynomial in the sizes of the shared terms.is paper improves the bounds in the literature by presenting the rst linear time algorithm. As others before us, we are inspired by Paterson and Wegman's algorithm for rst-order uni cation, itself based on representing terms with sharing as DAGs, and sharing equality as bisimulation of DAGs. Beyond the improved complexity, a distinguishing point of our work is a dissection of the involved concepts. In particular, we show that the algorithm computes the smallest bisimulation between the given DAGs, if any.

Section: Introduction Origin and Downfall Of The Problemmentioning

confidence: 99%

Sharing Equality is Linear

Condoluci

Proceedings of the 21st International Symposium on Principles and Practice of Declarative Programming

Coen

2019

Self Cite

“…For strategies in the closed λ-calculus it is enough to use the ordinary technology for abstract machines, as first shown by Blelloch and Greiner [12], and then by Sands, Gustavsson, and Moran [24], and, with other techniques, by combining the results in Dal Lago and Martini's [15] and [14]. The case of the strong λ-calculus is subtler, and a more sophisticated form of sharing is necessary, as first shown by Accattoli and Dal Lago [7]. The topic of this paper is the study of reasonable machines for the intermediate case of Open CbV.…”

Section: Introductionmentioning

confidence: 99%

Implementing Open Call-by-Value

Fundamentals of Software Engineering

Guerrieri

2017

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The theory of the call-by-value λ-calculus relies on weak evaluation and closed terms, that are natural hypotheses in the study of programming languages. To model proof assistants, however, strong evaluation and open terms are required. Open call-by-value is the intermediate setting of weak evaluation with open terms, on top of which Grégoire and Leroy designed the abstract machine of Coq. This paper provides a theory of abstract machines for open call-by-value. The literature contains machines that are either simple but inefficient, as they have an exponential overhead, or efficient but heavy, as they rely on a labelling of environments and a technical optimization. We introduce a machine that is simple and efficient: it does not use labels and it implements open call-by-value within a bilinear overhead. Moreover, we provide a new fine understanding of how different optimizations impact on the complexity of the overhead.This work is part of a wider research effort, the COCA HOLA project https://sites.google.com/site/beniaminoaccattoli/coca-hola.

“…This paper is a longer version of the workshop paper [3]. Apart from updating the notation of [3] to that of other recent papers using the same formalism [5,9,6], it extends it with the detailed relationship between the value substitution calculus and the kernel calculus in Sect. 8.…”

Section: Introductionmentioning

confidence: 99%

Proof nets and the call-by-value λ-calculus

Theoretical Computer Science

2015

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Please cite this article in press as: B. Accattoli, Proof nets and the call-by-value λ-calculus, Theoret. Comput. Sci. (2015), http://dx. AbstractThis paper gives a detailed account of the relationship between (a variant of) the call-by-value lambda calculus and linear logic proof nets. The presentation is carefully tuned in order to realize an isomorphism between the two systems: every single rewriting step on the calculus maps to a single step on proof nets, and viceversa. In this way, we obtain an algebraic reformulation of proof nets. Moreover, we provide a simple correctness criterion for our proof nets, which employ boxes in an unusual way, and identify a subcalculus that is shown to be as expressive as the full calculus.