Which simple types have a unique inhabitant?

Scherer, Gabriel; Rémy, Didier

doi:10.1145/2858949.2784757

Cited by 4 publications

(3 citation statements)

References 41 publications

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“…Focusing has traditionally been applied in the context of logics [Andreoli 1992] and pure programming languages [Scherer 2017;Scherer and Rémy 2015]. Thus, much work remains to be done in applying computation focusing to languages with more complex types and effects.…”

Section: Future Workmentioning

confidence: 99%

Computation focusing

Rioux

Zdancewic

2020

Proc. ACM Program. Lang.

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Focusing is a technique from proof theory that exploits type information to prune inessential nondeterminism from proof search procedures. Viewed through the lens of the Curry-Howard correspondence, a focused typing derivation yields terms in normal form. This paper explores how to exploit focusing for reasoning about contextual equivalences and full abstraction. We present a focused polymorphic call-by-push-value calculus and prove a computational completeness result: for every well-typed term, there exists a focused term that is-equivalent to it. This completeness result yields a powerful way to refine the context lemmas for establishing contextual equivalences, cutting down the set that must be considered to just focused contexts. The paper demonstrates the application of focusing to establish program equivalences, including free theorems. It also uses focusing to prove full abstraction of a translation of the pure, total call-by-push-value language into a language with divergence and simple effect types, yielding a novel solution to a simple-to-state, but hitherto difficult to solve problem.

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Section: Future Workmentioning

confidence: 99%

Computation focusing

Rioux

Zdancewic

2020

Proc. ACM Program. Lang.

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“…There are many other recent results showing how to synthesize functions from type-based specifications [Augustsson 2004;Feser et al 2015;Frankle et al 2015;Polikarpova et al 2016;Scherer and Rèmy 2015]. These systems enumerate programs of their target language, orienting their search procedures to process only terms that are well-typed.…”

Section: Related Workmentioning

confidence: 99%

Synthesizing bijective lenses

Miltner

Fisher

Pierce

et al. 2017

Proc. ACM Program. Lang.

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Bidirectional transformations between different data representations occur frequently in modern software systems. They appear as serializers and deserializers, as parsers and pretty printers, as database views and view updaters, and as a multitude of different kinds of ad hoc data converters. Manually building bidirectional transformationsÐby writing two separate functions that are intended to be inversesÐis tedious and error prone. A better approach is to use a domain-specific language in which both directions can be written as a single expression. However, these domain-specific languages can be difficult to program in, requiring programmers to manage fiddly details while working in a complex type system.We present an alternative approach. Instead of coding transformations manually, we synthesize them from declarative format descriptions and examples. Specifically, we present Optician, a tool for type-directed synthesis of bijective string transformers. The inputs to Optician are a pair of ordinary regular expressions representing two data formats and a few concrete examples for disambiguation. The output is a well-typed program in Boomerang (a bidirectional language based on the theory of lenses). The main technical challenge involves navigating the vast program search space efficiently. In particular, and unlike most prior work on type-directed synthesis, our system operates in the context of a language with a rich equivalence relation on types (the theory of regular expressions). Consequently, program synthesis requires search in two dimensions: First, our synthesis algorithm must find a pair of łsyntactically compatible types,ž and second, using the structure of those types, it must find a type-and example-compliant term. Our key insight is that it is possible to reduce the size of this search space without losing any computational power by defining a new language of lenses designed specifically for synthesis. The new language is free from arbitrary function composition and operates only over types and terms in a new disjunctive normal form. We prove (1) our new language is just as powerful as a more natural, compositional, and declarative language and (2) our synthesis algorithm is sound and complete with respect to the new language. We also demonstrate empirically that our new language changes the synthesis problem from one that admits intractable solutions to one that admits highly efficient solutions, able to synthesize intricate lenses between complex file formats in seconds. We evaluate Optician on a benchmark suite of 39 examples that includes both microbenchmarks and realistic examples derived from other data management systems including Flash Fill, a tool for synthesizing string transformations in spreadsheets, and Augeas, a tool for bidirectional processing of Linux system configuration files.

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“…In earlier work (Liang and Miller, 2009), we have presented the focused proof system LJF as an improved version of Gentzen's sequent system LJ for intuitionistic logic. Such focused proof systems have been used to give a foundation to logic programming (Miller, 1989;Miller, Nadathur, Pfenning, and Scedrov, 1991), model checking (Heath and Miller, 2019), and term representation (Herbelin, 1995;Scherer, 2016).…”

Section: Introductionmentioning

confidence: 99%

Focusing Gentzen’s LK Proof System

Liang,

Miller

2024

Outstanding Contributions to Logic

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Gentzen's sequent calculi LK and LJ are landmark proof systems. They identify the structural rules of weakening and contraction as notable inference rules, and they allow for an elegant statement and proof of both cut elimination and consistency for classical and intuitionistic logics. Among the undesirable features of those sequent calculi is that their inferences rules are low-level and frequently permute over each other. As a result, large-scale structures within sequent calculus proofs are hard to identify. In this paper, we present a different approach to designing a sequent calculus for classical logic. Starting with Gentzen's LK proof system, we examine the proof search meaning of his inference rules and classify those rules as involving either don't care nondeterminism or don't know nondeterminism. Based on that classification, we design the focused proof system LKF in which inference rules belong to one of two phases of proof construction depending on which flavor of nondeterminism they involve. We then prove that the cut rule and the general form of the initial rule are admissible in LKF. Finally, by showing that the inference rules for LK are all admissible in LKF, we can give a relative completeness proof for LKF provability with respect to LK provability. We shall also apply these properties of the LKF proof system to establish other meta-theoretic properties of classical logic, including Herbrand's theorem.

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Which simple types have a unique inhabitant?

Cited by 4 publications

References 41 publications

Computation focusing

Computation focusing

Synthesizing bijective lenses

Focusing Gentzen’s LK Proof System

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