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DOI: 10.4230/LIPIcs.FSCD.2022.32

Galois Connecting Call-by-Value and Call-by-Name

Authors: Dylan McDermott and Alan Mycroft

Published in: LIPIcs, Volume 228, 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022)

Abstract

We establish a general framework for reasoning about the relationship between call-by-value and call-by-name. In languages with side-effects, call-by-value and call-by-name executions of programs often have different, but related, observable behaviours. For example, if a program might diverge but otherwise has no side-effects, then whenever it terminates under call-by-value, it terminates with the same result under call-by-name. We propose a technique for stating and proving these properties. The key ingredient is Levy’s call-by-push-value calculus, which we use as a framework for reasoning about evaluation orders. We construct maps between the call-by-value and call-by-name interpretations of types. We then identify properties of side-effects that imply these maps form a Galois connection. These properties hold for some side-effects (such as divergence), but not others (such as mutable state). This gives rise to a general reasoning principle that relates call-by-value and call-by-name. We apply the reasoning principle to example side-effects including divergence and nondeterminism.

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Dylan McDermott and Alan Mycroft. Galois Connecting Call-by-Value and Call-by-Name. In 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 228, pp. 32:1-32:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{mcdermott_et_al:LIPIcs.FSCD.2022.32,
  author =	{McDermott, Dylan and Mycroft, Alan},
  title =	{{Galois Connecting Call-by-Value and Call-by-Name}},
  booktitle =	{7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022)},
  pages =	{32:1--32:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-233-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{228},
  editor =	{Felty, Amy P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2022.32},
  URN =		{urn:nbn:de:0030-drops-163138},
  doi =		{10.4230/LIPIcs.FSCD.2022.32},
  annote =	{Keywords: computational effect, evaluation order, call-by-push-value, categorical semantics}
}

Document

DOI: 10.4230/LIPIcs.FSCD.2021.30

Abstract Clones for Abstract Syntax

Authors: Nathanael Arkor and Dylan McDermott

Published in: LIPIcs, Volume 195, 6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021)

Abstract

We give a formal treatment of simple type theories, such as the simply-typed λ-calculus, using the framework of abstract clones. Abstract clones traditionally describe first-order structures, but by equipping them with additional algebraic structure, one can further axiomatize second-order, variable-binding operators. This provides a syntax-independent representation of simple type theories. We describe multisorted second-order presentations, such as the presentation of the simply-typed λ-calculus, and their clone-theoretic algebras; free algebras on clones abstractly describe the syntax of simple type theories quotiented by equations such as β- and η-equality. We give a construction of free algebras and derive a corresponding induction principle, which facilitates syntax-independent proofs of properties such as adequacy and normalization for simple type theories. Working only with clones avoids some of the complexities inherent in presheaf-based frameworks for abstract syntax.

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Nathanael Arkor and Dylan McDermott. Abstract Clones for Abstract Syntax. In 6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 195, pp. 30:1-30:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{arkor_et_al:LIPIcs.FSCD.2021.30,
  author =	{Arkor, Nathanael and McDermott, Dylan},
  title =	{{Abstract Clones for Abstract Syntax}},
  booktitle =	{6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021)},
  pages =	{30:1--30:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-191-7},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{195},
  editor =	{Kobayashi, Naoki},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2021.30},
  URN =		{urn:nbn:de:0030-drops-142686},
  doi =		{10.4230/LIPIcs.FSCD.2021.30},
  annote =	{Keywords: simple type theories, abstract clones, second-order abstract syntax, substitution, variable binding, presentations, free algebras, induction, logical relations}
}

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Galois Connecting Call-by-Value and Call-by-Name

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Abstract Clones for Abstract Syntax

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