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Documents authored by De, Abhishek

Document
DOI: 10.4230/LIPIcs.FSTTCS.2023.40
Comparing Infinitary Systems for Linear Logic with Fixed Points

Authors: Anupam Das, Abhishek De, and Alexis Saurin

Published in: LIPIcs, Volume 284, 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)

Abstract
Extensions of Girard’s linear logic by least and greatest fixed point operators (μMALL) have been an active field of research for almost two decades. Various proof systems are known viz. finitary and non-wellfounded, based on explicit and implicit (co)induction respectively. In this paper, we compare the relative expressivity, at the level of provability, of two complementary infinitary proof systems: finitely branching non-wellfounded proofs (μMALL^∞) vs. infinitely branching well-founded proofs (μMALL_{ω,∞}). Our main result is that μMALL^∞ is strictly contained in μMALL_{ω,∞}. For inclusion, we devise a novel technique involving infinitary rewriting of non-wellfounded proofs that yields a wellfounded proof in the limit. For strictness of the inclusion, we improve previously known lower bounds on μMALL^∞ provability from Π⁰₁-hard to Σ¹₁-hard, by encoding a sort of Büchi condition for Minsky machines.
Cite as

Anupam Das, Abhishek De, and Alexis Saurin. Comparing Infinitary Systems for Linear Logic with Fixed Points. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 40:1-40:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{das_et_al:LIPIcs.FSTTCS.2023.40,
  author =	{Das, Anupam and De, Abhishek and Saurin, Alexis},
  title =	{{Comparing Infinitary Systems for Linear Logic with Fixed Points}},
  booktitle =	{43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)},
  pages =	{40:1--40:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-304-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{284},
  editor =	{Bouyer, Patricia and Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2023.40},
  URN =		{urn:nbn:de:0030-drops-194131},
  doi =		{10.4230/LIPIcs.FSTTCS.2023.40},
  annote =	{Keywords: linear logic, fixed points, non-wellfounded proofs, omega-branching proofs, analytical hierarchy}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2022.35
Phase Semantics for Linear Logic with Least and Greatest Fixed Points

Authors: Abhishek De, Farzad Jafarrahmani, and Alexis Saurin

Published in: LIPIcs, Volume 250, 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)

Abstract
The truth semantics of linear logic (i.e. phase semantics) is often overlooked despite having a wide range of applications and deep connections with several denotational semantics. In phase semantics, one is concerned about the provability of formulas rather than the contents of their proofs (or refutations). Linear logic equipped with the least and greatest fixpoint operators (μMALL) has been an active field of research for the past one and a half decades. Various proof systems are known viz. finitary and non-wellfounded, based on explicit and implicit (co)induction respectively. In this paper, we extend the phase semantics of multiplicative additive linear logic (a.k.a. MALL) to μMALL with explicit (co)induction (i.e. μMALL^{ind}). We introduce a Tait-style system for μMALL called μMALL_ω where proofs are wellfounded but potentially infinitely branching. We study its phase semantics and prove that it does not have the finite model property.
Cite as

Abhishek De, Farzad Jafarrahmani, and Alexis Saurin. Phase Semantics for Linear Logic with Least and Greatest Fixed Points. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 35:1-35:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{de_et_al:LIPIcs.FSTTCS.2022.35,
  author =	{De, Abhishek and Jafarrahmani, Farzad and Saurin, Alexis},
  title =	{{Phase Semantics for Linear Logic with Least and Greatest Fixed Points}},
  booktitle =	{42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
  pages =	{35:1--35:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-261-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{250},
  editor =	{Dawar, Anuj and Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.35},
  URN =		{urn:nbn:de:0030-drops-174272},
  doi =		{10.4230/LIPIcs.FSTTCS.2022.35},
  annote =	{Keywords: Linear logic, fixed points, phase semantics, closure ordinals, cut elimination}
}
Document
DOI: 10.4230/LIPIcs.FSCD.2022.20
Decision Problems for Linear Logic with Least and Greatest Fixed Points

Authors: Anupam Das, Abhishek De, and Alexis Saurin

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

Abstract
Linear logic is an important logic for modelling resources and decomposing computational interpretations of proofs. Decision problems for fragments of linear logic exhibiting "infinitary" behaviour (such as exponentials) are notoriously complicated. In this work, we address the decision problems for variations of linear logic with fixed points (μMALL), in particular, recent systems based on "circular" and "non-wellfounded" reasoning. In this paper, we show that μMALL is undecidable. More explicitly, we show that the general non-wellfounded system is Π⁰₁-hard via a reduction to the non-halting of Minsky machines, and thus is strictly stronger than its circular counterpart (which is in Σ⁰₁). Moreover, we show that the restriction of these systems to theorems with only the least fixed points is already Σ⁰₁-complete via a reduction to the reachability problem of alternating vector addition systems with states. This implies that both the circular system and the finitary system (with explicit (co)induction) are Σ⁰₁-complete.
Cite as

Anupam Das, Abhishek De, and Alexis Saurin. Decision Problems for Linear Logic with Least and Greatest Fixed Points. In 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 228, pp. 20:1-20:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{das_et_al:LIPIcs.FSCD.2022.20,
  author =	{Das, Anupam and De, Abhishek and Saurin, Alexis},
  title =	{{Decision Problems for Linear Logic with Least and Greatest Fixed Points}},
  booktitle =	{7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022)},
  pages =	{20:1--20:20},
  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.2022.20},
  URN =		{urn:nbn:de:0030-drops-163019},
  doi =		{10.4230/LIPIcs.FSCD.2022.20},
  annote =	{Keywords: Linear logic, fixed points, decidability, vector addition systems}
}
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