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DOI: 10.4230/LIPIcs.MFCS.2022.70

Generalized Bundled Fragments for First-Order Modal Logic

Authors: Mo Liu, Anantha Padmanabha, R. Ramanujam, and Yanjing Wang

Published in: LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

Abstract

When we bundle quantifiers and modalities together (as in ∃x□, ◇∀x etc.) in first-order modal logic (FOML), we get new logical operators whose combinations produce interesting bundled fragments of FOML. It is well-known that finding decidable fragments of FOML is hard, but existing work shows that certain bundled fragments are decidable [Anantha Padmanabha et al., 2018], without any restriction on the arity of predicates, the number of variables, or the modal scope. In this paper, we explore generalized bundles such as ∀x∀y□, ∀x∃y◇ etc., and map the terrain with regard to decidability, presenting both decidability and undecidability results. In particular, we propose the loosely bundled fragment, which is decidable over increasing domains and encompasses all known decidable bundled fragments.

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Mo Liu, Anantha Padmanabha, R. Ramanujam, and Yanjing Wang. Generalized Bundled Fragments for First-Order Modal Logic. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 70:1-70:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{liu_et_al:LIPIcs.MFCS.2022.70,
  author =	{Liu, Mo and Padmanabha, Anantha and Ramanujam, R. and Wang, Yanjing},
  title =	{{Generalized Bundled Fragments for First-Order Modal Logic}},
  booktitle =	{47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
  pages =	{70:1--70:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-256-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{241},
  editor =	{Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.70},
  URN =		{urn:nbn:de:0030-drops-168684},
  doi =		{10.4230/LIPIcs.MFCS.2022.70},
  annote =	{Keywords: bundled fragments, first-order modal logic, decidability, tableaux}
}

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DOI: 10.4230/LIPIcs.FSTTCS.2018.43

Bundled Fragments of First-Order Modal Logic: (Un)Decidability

Authors: Anantha Padmanabha, R Ramanujam, and Yanjing Wang

Published in: LIPIcs, Volume 122, 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)

Abstract

Quantified modal logic is notorious for being undecidable, with very few known decidable fragments such as the monodic ones. For instance, even the two-variable fragment over unary predicates is undecidable. In this paper, we study a particular fragment, namely the bundled fragment, where a first-order quantifier is always followed by a modality when occurring in the formula, inspired by the proposal of [Yanjing Wang, 2017] in the context of non-standard epistemic logics of know-what, know-how, know-why, and so on. As always with quantified modal logics, it makes a significant difference whether the domain stays the same across possible worlds. In particular, we show that the predicate logic with the bundle "forall Box" alone is undecidable over constant domain interpretations, even with only monadic predicates, whereas having the "exists Box" bundle instead gives us a decidable logic. On the other hand, over increasing domain interpretations, we get decidability with both "forall Box" and "exists Box" bundles with unrestricted predicates, where we obtain tableau based procedures that run in PSPACE. We further show that the "exists Box" bundle cannot distinguish between constant domain and variable domain interpretations.

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Anantha Padmanabha, R Ramanujam, and Yanjing Wang. Bundled Fragments of First-Order Modal Logic: (Un)Decidability. In 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 122, pp. 43:1-43:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{padmanabha_et_al:LIPIcs.FSTTCS.2018.43,
  author =	{Padmanabha, Anantha and Ramanujam, R and Wang, Yanjing},
  title =	{{Bundled Fragments of First-Order Modal Logic: (Un)Decidability}},
  booktitle =	{38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)},
  pages =	{43:1--43:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-093-4},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{122},
  editor =	{Ganguly, Sumit and Pandya, Paritosh},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2018.43},
  URN =		{urn:nbn:de:0030-drops-99424},
  doi =		{10.4230/LIPIcs.FSTTCS.2018.43},
  annote =	{Keywords: First-order modal logic, decidability, bundled fragments}
}

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Generalized Bundled Fragments for First-Order Modal Logic

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Bundled Fragments of First-Order Modal Logic: (Un)Decidability

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