License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.FSCD.2022.20
URN: urn:nbn:de:0030-drops-163019
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16301/
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Das, Anupam ; De, Abhishek ; Saurin, Alexis

Decision Problems for Linear Logic with Least and Greatest Fixed Points

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LIPIcs-FSCD-2022-20.pdf (0.9 MB)

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.

BibTeX - Entry

@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/opus/volltexte/2022/16301},
  URN =		{urn:nbn:de:0030-drops-163019},
  doi =		{10.4230/LIPIcs.FSCD.2022.20},
  annote =	{Keywords: Linear logic, fixed points, decidability, vector addition systems}
}

Keywords: Linear logic, fixed points, decidability, vector addition systems
Collection: 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022)
Issue Date: 2022
Date of publication: 28.06.2022

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