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DOI: 10.4230/LIPIcs.CONCUR.2021.21

Fibrational Initial Algebra-Final Coalgebra Coincidence over Initial Algebras: Turning Verification Witnesses Upside Down

Authors: Mayuko Kori, Ichiro Hasuo, and Shin-ya Katsumata

Published in: LIPIcs, Volume 203, 32nd International Conference on Concurrency Theory (CONCUR 2021)

Abstract

The coincidence between initial algebras (IAs) and final coalgebras (FCs) is a phenomenon that underpins various important results in theoretical computer science. In this paper, we identify a general fibrational condition for the IA-FC coincidence, namely in the fiber over an initial algebra in the base category. Identifying (co)algebras in a fiber as (co)inductive predicates, our fibrational IA-FC coincidence allows one to use coinductive witnesses (such as invariants) for verifying inductive properties (such as liveness). Our general fibrational theory features the technical condition of stability of chain colimits; we extend the framework to the presence of a monadic effect, too, restricting to fibrations of complete lattice-valued predicates. Practical benefits of our categorical theory are exemplified by new "upside-down" witness notions for three verification problems: probabilistic liveness, and acceptance and model-checking with respect to bottom-up tree automata.

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Mayuko Kori, Ichiro Hasuo, and Shin-ya Katsumata. Fibrational Initial Algebra-Final Coalgebra Coincidence over Initial Algebras: Turning Verification Witnesses Upside Down. In 32nd International Conference on Concurrency Theory (CONCUR 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 203, pp. 21:1-21:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{kori_et_al:LIPIcs.CONCUR.2021.21,
  author =	{Kori, Mayuko and Hasuo, Ichiro and Katsumata, Shin-ya},
  title =	{{Fibrational Initial Algebra-Final Coalgebra Coincidence over Initial Algebras: Turning Verification Witnesses Upside Down}},
  booktitle =	{32nd International Conference on Concurrency Theory (CONCUR 2021)},
  pages =	{21:1--21:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-203-7},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{203},
  editor =	{Haddad, Serge and Varacca, Daniele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2021.21},
  URN =		{urn:nbn:de:0030-drops-143982},
  doi =		{10.4230/LIPIcs.CONCUR.2021.21},
  annote =	{Keywords: initial algebra, final coalgebra, fibration, category theory}
}

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DOI: 10.4230/LIPIcs.CSL.2021.29

A Cyclic Proof System for HFL_ℕ

Authors: Mayuko Kori, Takeshi Tsukada, and Naoki Kobayashi

Published in: LIPIcs, Volume 183, 29th EACSL Annual Conference on Computer Science Logic (CSL 2021)

Abstract

A cyclic proof system allows us to perform inductive reasoning without explicit inductions. We propose a cyclic proof system for HFL_ℕ, which is a higher-order predicate logic with natural numbers and alternating fixed-points. Ours is the first cyclic proof system for a higher-order logic, to our knowledge. Due to the presence of higher-order predicates and alternating fixed-points, our cyclic proof system requires a more delicate global condition on cyclic proofs than the original system of Brotherston and Simpson. We prove the decidability of checking the global condition and soundness of this system, and also prove a restricted form of standard completeness for an infinitary variant of our cyclic proof system. A potential application of our cyclic proof system is semi-automated verification of higher-order programs, based on Kobayashi et al.’s recent work on reductions from program verification to HFL_ℕ validity checking.

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Mayuko Kori, Takeshi Tsukada, and Naoki Kobayashi. A Cyclic Proof System for HFL_ℕ. In 29th EACSL Annual Conference on Computer Science Logic (CSL 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 183, pp. 29:1-29:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{kori_et_al:LIPIcs.CSL.2021.29,
  author =	{Kori, Mayuko and Tsukada, Takeshi and Kobayashi, Naoki},
  title =	{{A Cyclic Proof System for HFL\underline\mathbb{N}}},
  booktitle =	{29th EACSL Annual Conference on Computer Science Logic (CSL 2021)},
  pages =	{29:1--29:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-175-7},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{183},
  editor =	{Baier, Christel and Goubault-Larrecq, Jean},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2021.29},
  URN =		{urn:nbn:de:0030-drops-134632},
  doi =		{10.4230/LIPIcs.CSL.2021.29},
  annote =	{Keywords: Cyclic proof, higher-order logic, fixed-point logic, sequent calculus}
}

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Documents authored by Kori, Mayuko

Fibrational Initial Algebra-Final Coalgebra Coincidence over Initial Algebras: Turning Verification Witnesses Upside Down

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A Cyclic Proof System for HFL_ℕ

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