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DOI: 10.4230/LIPIcs.FSCD.2022.19

Normalization Without Syntax

Authors: Willem B. Heijltjes, Dominic J. D. Hughes, and Lutz Straßburger

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

Abstract

We present normalization for intuitionistic combinatorial proofs (ICPs) and relate it to the simply-typed lambda-calculus. We prove confluence and strong normalization. Combinatorial proofs, or "proofs without syntax", form a graphical semantics of proof in various logics that is canonical yet complexity-aware: they are a polynomial-sized representation of sequent proofs that factors out exactly the non-duplicating permutations. Our approach to normalization aligns with these characteristics: it is canonical (free of permutations) and generic (readily applied to other logics). Our reduction mechanism is a canonical representation of reduction in sequent calculus with closed cuts (no abstraction is allowed below a cut), and relates to closed reduction in lambda-calculus and supercombinators. While we will use ICPs concretely, the notion of reduction is completely abstract, and can be specialized to give a reduction mechanism for any representation of typed normal forms.

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Willem B. Heijltjes, Dominic J. D. Hughes, and Lutz Straßburger. Normalization Without Syntax. In 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 228, pp. 19:1-19:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{heijltjes_et_al:LIPIcs.FSCD.2022.19,
  author =	{Heijltjes, Willem B. and Hughes, Dominic J. D. and Stra{\ss}burger, Lutz},
  title =	{{Normalization Without Syntax}},
  booktitle =	{7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022)},
  pages =	{19:1--19:19},
  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.2022.19},
  URN =		{urn:nbn:de:0030-drops-163004},
  doi =		{10.4230/LIPIcs.FSCD.2022.19},
  annote =	{Keywords: combinatorial proofs, intuitionistic logic, lambda-calculus, Curry-Howard, proof nets}
}

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DOI: 10.4230/LIPIcs.FSCD.2019.22

Proof Nets for First-Order Additive Linear Logic

Authors: Willem B. Heijltjes, Dominic J. D. Hughes, and Lutz Straßburger

Published in: LIPIcs, Volume 131, 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)

Abstract

We present canonical proof nets for first-order additive linear logic, the fragment of linear logic with sum, product, and first-order universal and existential quantification. We present two versions of our proof nets. One, witness nets, retains explicit witnessing information to existential quantification. For the other, unification nets, this information is absent but can be reconstructed through unification. Unification nets embody a central contribution of the paper: first-order witness information can be left implicit, and reconstructed as needed. Witness nets are canonical for first-order additive sequent calculus. Unification nets in addition factor out any inessential choice for existential witnesses. Both notions of proof net are defined through coalescence, an additive counterpart to multiplicative contractibility, and for witness nets an additional geometric correctness criterion is provided. Both capture sequent calculus cut-elimination as a one-step global composition operation.

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Willem B. Heijltjes, Dominic J. D. Hughes, and Lutz Straßburger. Proof Nets for First-Order Additive Linear Logic. In 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 131, pp. 22:1-22:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{heijltjes_et_al:LIPIcs.FSCD.2019.22,
  author =	{Heijltjes, Willem B. and Hughes, Dominic J. D. and Stra{\ss}burger, Lutz},
  title =	{{Proof Nets for First-Order Additive Linear Logic}},
  booktitle =	{4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)},
  pages =	{22:1--22:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-107-8},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{131},
  editor =	{Geuvers, Herman},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2019.22},
  URN =		{urn:nbn:de:0030-drops-105297},
  doi =		{10.4230/LIPIcs.FSCD.2019.22},
  annote =	{Keywords: linear logic, first-order logic, proof nets, Herbrand’s theorem}
}

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Documents authored by Hughes, Dominic J. D.

Normalization Without Syntax

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Proof Nets for First-Order Additive Linear Logic

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