Intuitionistic proofs without syntax

Willem Heijltjes; Dominic Hughes; Lutz Strassburger

doi:10.1109/LICS.2019.8785827

Intuitionistic proofs without syntax

Willem Heijltjes, Dominic Hughes, Lutz Strassburger

Research output: Chapter or section in a book/report/conference proceeding › Chapter in a published conference proceeding

7 Citations (SciVal)

94 Downloads (Pure)

Abstract

We present Intuitionistic Combinatorial Proofs (ICPs), a concrete geometric semantics of intuitionistic logic based on the principles of the second author’s classical Com- binatorial Proofs.
An ICP naturally factorizes into a linear fragment, a graphical abstraction of an IMLL proof net (an arena net), and a parallel contraction-weakening fragment (a skew fibration). ICPs relate to game semantics, and can be seen as a strategy in a Hyland-Ong arena, generalized from a tree-like to a dag-like strategy.

Our first main result, Polynomial Full Completeness, is that ICPs as a semantics are complexity-aware: the translations to and from sequent calculus are size-preserving (up to a polynomial). By contrast, lambda-calculus and game semantics incur an exponential blowup. Our second main result, Local Canonicity, is that ICPs abstract fully and faithfully over the non-duplicating permutations of the sequent calculus, analogously to the first and second authors’ recent result for MALL.

Original language	English
Title of host publication	2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2019
Publisher	IEEE
Pages	1-13
Number of pages	13
ISBN (Electronic)	9781728136080
ISBN (Print)	9781728136097
DOIs	https://doi.org/10.1109/LICS.2019.8785827
Publication status	Published - 6 Aug 2019
Event	Thirty-Fourth Annual ACM/IEEE Symposium on Logic in Computer Science - Vancouver, Canada Duration: 24 Jun 2019 → 27 Jun 2019

Publication series

Name	Annual Symposium on Logic in Computer Science
Volume	2019-June
ISSN (Print)	1043-6871

Conference

Conference	Thirty-Fourth Annual ACM/IEEE Symposium on Logic in Computer Science
Abbreviated title	LICS
Country/Territory	Canada
City	Vancouver
Period	24/06/19 → 27/06/19

Keywords

intuitionistic logic
proof nets
game semantics

ASJC Scopus subject areas

Software
General Mathematics

Access to Document

10.1109/LICS.2019.8785827

icp.PDF
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Accepted author manuscript, 377 KB

Cite this

Intuitionistic proofs without syntax. / Heijltjes, Willem; Hughes, Dominic; Strassburger, Lutz.
2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2019. IEEE, 2019. p. 1-13 (Annual Symposium on Logic in Computer Science; Vol. 2019-June).

Research output: Chapter or section in a book/report/conference proceeding › Chapter in a published conference proceeding

Heijltjes, W, Hughes, D & Strassburger, L 2019, Intuitionistic proofs without syntax. in 2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2019. Annual Symposium on Logic in Computer Science, vol. 2019-June, IEEE, pp. 1-13, Thirty-Fourth Annual ACM/IEEE Symposium on Logic in Computer Science, Vancouver, Canada, 24/06/19. https://doi.org/10.1109/LICS.2019.8785827

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abstract = "We present Intuitionistic Combinatorial Proofs (ICPs), a concrete geometric semantics of intuitionistic logic based on the principles of the second author{\textquoteright}s classical Com- binatorial Proofs.An ICP naturally factorizes into a linear fragment, a graphical abstraction of an IMLL proof net (an arena net), and a parallel contraction-weakening fragment (a skew fibration). ICPs relate to game semantics, and can be seen as a strategy in a Hyland-Ong arena, generalized from a tree-like to a dag-like strategy.Our first main result, Polynomial Full Completeness, is that ICPs as a semantics are complexity-aware: the translations to and from sequent calculus are size-preserving (up to a polynomial). By contrast, lambda-calculus and game semantics incur an exponential blowup. Our second main result, Local Canonicity, is that ICPs abstract fully and faithfully over the non-duplicating permutations of the sequent calculus, analogously to the first and second authors{\textquoteright} recent result for MALL.",

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AB - We present Intuitionistic Combinatorial Proofs (ICPs), a concrete geometric semantics of intuitionistic logic based on the principles of the second author’s classical Com- binatorial Proofs.An ICP naturally factorizes into a linear fragment, a graphical abstraction of an IMLL proof net (an arena net), and a parallel contraction-weakening fragment (a skew fibration). ICPs relate to game semantics, and can be seen as a strategy in a Hyland-Ong arena, generalized from a tree-like to a dag-like strategy.Our first main result, Polynomial Full Completeness, is that ICPs as a semantics are complexity-aware: the translations to and from sequent calculus are size-preserving (up to a polynomial). By contrast, lambda-calculus and game semantics incur an exponential blowup. Our second main result, Local Canonicity, is that ICPs abstract fully and faithfully over the non-duplicating permutations of the sequent calculus, analogously to the first and second authors’ recent result for MALL.

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Intuitionistic proofs without syntax

Abstract

Publication series

Conference

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Typed Lambda-Calculi with Sharing and Unsharing

Willem Heijltjes

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Intuitionistic proofs without syntax

Abstract

Publication series

Conference

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Projects

Typed Lambda-Calculi with Sharing and Unsharing

Profiles

Willem Heijltjes

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