6 Search Results for "Enqvist, Sebastian"

Document
DOI: 10.4230/LIPIcs.CSL.2026.18
A Modular Framework for Proof-Search via Formalised Modal Completeness in HOL Light

Authors: Antonella Bilotta, Marco Maggesi, and Cosimo Perini Brogi

Published in: LIPIcs, Volume 363, 34th EACSL Annual Conference on Computer Science Logic (CSL 2026)

Abstract
We extend the existing HOL Light Library for Modal Systems (HOLMS) to support a modular implementation of modal reasoning within the HOL Light proof assistant. We deeply embed axiomatic calculi and relational semantics for seven normal modal logics (K, T, B, K4, S4, S5, GL) and formalise modal adequacy theorems for these systems. We then leverage those formalisations to implement a mechanism for automated reasoning via proof-search in the associated labelled sequent calculi, which we shallowly embed in HOL Light’s goal-stack mechanism. This way, we equip the general-purpose proof assistant with (semi)decision procedures for these logics that, in case of failure to construct a proof for the input formula, return a certified countermodel within the appropriate class for the logic under consideration. On the methodological side, we propose a precise measure of the modularity of our approach by systematically adopting Christopher Strachey’s distinction between ad hoc and parametric polymorphism throughout the library.
Cite as

Antonella Bilotta, Marco Maggesi, and Cosimo Perini Brogi. A Modular Framework for Proof-Search via Formalised Modal Completeness in HOL Light. In 34th EACSL Annual Conference on Computer Science Logic (CSL 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 363, pp. 18:1-18:29, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{bilotta_et_al:LIPIcs.CSL.2026.18,
  author =	{Bilotta, Antonella and Maggesi, Marco and Perini Brogi, Cosimo},
  title =	{{A Modular Framework for Proof-Search via Formalised Modal Completeness in HOL Light}},
  booktitle =	{34th EACSL Annual Conference on Computer Science Logic (CSL 2026)},
  pages =	{18:1--18:29},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-411-6},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{363},
  editor =	{Guerrini, Stefano and K\"{o}nig, Barbara},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2026.18},
  URN =		{urn:nbn:de:0030-drops-254427},
  doi =		{10.4230/LIPIcs.CSL.2026.18},
  annote =	{Keywords: Modal logic, HOL Light, Labelled sequent calculi, Logical verification, Interactive theorem proving, Automated proof-search}
}
Document
DOI: 10.4230/LIPIcs.CSL.2026.15
Cyclic Proof Theory of Generalised Inductive Definitions

Authors: Gianluca Curzi and Lukas Melgaard

Published in: LIPIcs, Volume 363, 34th EACSL Annual Conference on Computer Science Logic (CSL 2026)

Abstract
We study cyclic proof systems for μPA, an extension of Peano arithmetic by generalised inductive definitions that is arithmetically equivalent to the (impredicative) subsystem of second-order arithmetic Π^1_2-CA₀ by Möllerfeld. The main result of this paper is that cyclic and inductive μPA have the same proof-theoretic strength. First, we translate cyclic proofs into an annotated variant based on Sprenger and Dam’s systems for first-order μ-calculus, whose stronger validity condition allows for a simpler proof of soundness. We then formalise this argument within Π^1_2-CA₀, leveraging Möllerfeld’s conservativity properties. To this end, we build on prior work by Curzi and Das on the reverse mathematics of the Knaster-Tarski theorem. As a byproduct of our proof methods we show that, despite the stronger validity condition, annotated and "plain" cyclic proofs for μPA prove the same theorems. This work represents a further step in the non-wellfounded proof-theoretic analysis of theories of arithmetic via impredicative fragments of second-order arithmetic, an approach initiated by Simpson’s Cyclic Arithmetic, and continued by Das and Melgaard in the context of arithmetical inductive definitions.
Cite as

Gianluca Curzi and Lukas Melgaard. Cyclic Proof Theory of Generalised Inductive Definitions. In 34th EACSL Annual Conference on Computer Science Logic (CSL 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 363, pp. 15:1-15:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{curzi_et_al:LIPIcs.CSL.2026.15,
  author =	{Curzi, Gianluca and Melgaard, Lukas},
  title =	{{Cyclic Proof Theory of Generalised Inductive Definitions}},
  booktitle =	{34th EACSL Annual Conference on Computer Science Logic (CSL 2026)},
  pages =	{15:1--15:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-411-6},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{363},
  editor =	{Guerrini, Stefano and K\"{o}nig, Barbara},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2026.15},
  URN =		{urn:nbn:de:0030-drops-254399},
  doi =		{10.4230/LIPIcs.CSL.2026.15},
  annote =	{Keywords: cyclic proofs, positive inductive definitions, arithmetic, fixed points, proof theory, reset proof systems}
}
Document
DOI: 10.4230/LIPIcs.CALCO.2025.9
Safety and Strong Completeness via Reducibility for Many-Valued Coalgebraic Dynamic Logics

Authors: Helle Hvid Hansen and Wolfgang Poiger

Published in: LIPIcs, Volume 342, 11th Conference on Algebra and Coalgebra in Computer Science (CALCO 2025)

Abstract
We present a coalgebraic framework for studying generalisations of dynamic modal logics such as PDL and game logic in which both the propositions and the semantic structures can take values in an algebra 𝐀 of truth-degrees. More precisely, we work with coalgebraic modal logic via 𝐀-valued predicate liftings where 𝐀 is an FLew-algebra, and interpret actions (abstracting programs and games) as 𝖥-coalgebras where the functor 𝖥 represents some type of 𝐀-weighted system. We also allow combinations of crisp propositions with 𝐀-weighted systems and vice versa. We introduce coalgebra operations and tests, with a focus on operations which are reducible in the sense that modalities for composed actions can be reduced to compositions of modalities for the constituent actions. We prove that reducible operations are safe for bisimulation and behavioural equivalence, and prove a general strong completeness result, from which we obtain new strong completeness results for 𝟐-valued iteration-free PDL with 𝐀-valued accessibility relations when 𝐀 is a finite chain, and for many-valued iteration-free game logic with many-valued strategies based on finite Lukasiewicz logic.
Cite as

Helle Hvid Hansen and Wolfgang Poiger. Safety and Strong Completeness via Reducibility for Many-Valued Coalgebraic Dynamic Logics. In 11th Conference on Algebra and Coalgebra in Computer Science (CALCO 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 342, pp. 9:1-9:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{hansen_et_al:LIPIcs.CALCO.2025.9,
  author =	{Hansen, Helle Hvid and Poiger, Wolfgang},
  title =	{{Safety and Strong Completeness via Reducibility for Many-Valued Coalgebraic Dynamic Logics}},
  booktitle =	{11th Conference on Algebra and Coalgebra in Computer Science (CALCO 2025)},
  pages =	{9:1--9:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-383-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{342},
  editor =	{C\^{i}rstea, Corina and Knapp, Alexander},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2025.9},
  URN =		{urn:nbn:de:0030-drops-235681},
  doi =		{10.4230/LIPIcs.CALCO.2025.9},
  annote =	{Keywords: dynamic logic, many-valued coalgebraic logic, safety, strong completeness}
}
Document
Invited Talk
DOI: 10.4230/LIPIcs.CSL.2025.5
Modal Automata: Analysing Modal Fixpoint Logics, One Step at a Time (Invited Talk)

Authors: Yde Venema

Published in: LIPIcs, Volume 326, 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)

Abstract
We present and investigate a general framework for studying modal fixpoint logics and some related versions of monadic second-order logic, by means of certain finite automata that operate on Kripke structures. Characteristic of these modal automata is that the co-domain of their transition function is a set of formulas of a so-called one-step logic. The motivation for taking this perspective is that if a logic is characterised by a class of modal automata, many of its properties are already determined at the level of the much simpler one-step logic.
Cite as

Yde Venema. Modal Automata: Analysing Modal Fixpoint Logics, One Step at a Time (Invited Talk). In 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 326, pp. 5:1-5:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{venema:LIPIcs.CSL.2025.5,
  author =	{Venema, Yde},
  title =	{{Modal Automata: Analysing Modal Fixpoint Logics, One Step at a Time}},
  booktitle =	{33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)},
  pages =	{5:1--5:5},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-362-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{326},
  editor =	{Endrullis, J\"{o}rg and Schmitz, Sylvain},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2025.5},
  URN =		{urn:nbn:de:0030-drops-227627},
  doi =		{10.4230/LIPIcs.CSL.2025.5},
  annote =	{Keywords: modal logic, parity automata, fixpoint logic, one-step logic}
}
Document
DOI: 10.4230/LIPIcs.CALCO.2017.11
Disjunctive Bases: Normal Forms for Modal Logics

Authors: Sebastian Enqvist and Yde Venema

Published in: LIPIcs, Volume 72, 7th Conference on Algebra and Coalgebra in Computer Science (CALCO 2017)

Abstract
We present the concept of a disjunctive basis as a generic framework for normal forms in modal logic based on coalgebra. Disjunctive bases were defined in previous work on completeness for modal fixpoint logics, where they played a central role in the proof of a generic completeness theorem for coalgebraic mu-calculi. Believing the concept has a much wider significance, here we investigate it more thoroughly in its own right. We show that the presence of a disjunctive basis at the "one-step" level entails a number of good properties for a coalgebraic mu-calculus, in particular, a simulation theorem showing that every alternating automaton can be transformed into an equivalent nondeterministic one. Based on this, we prove a Lyndon theorem for the full fixpoint logic, its fixpoint-free fragment and its one-step fragment, and a Uniform Interpolation result, for both the full mu-calculus and its fixpoint-free fragment. We also raise the questions, when a disjunctive basis exists, and how disjunctive bases are related to Moss' coalgebraic "nabla" modalities. Nabla formulas provide disjunctive bases for many coalgebraic modal logics, but there are cases where disjunctive bases give useful normal forms even when nabla formulas fail to do so, our prime example being graded modal logic. Finally, we consider the problem of giving a category-theoretic formulation of disjunctive bases, and provide a partial solution.
Cite as

Sebastian Enqvist and Yde Venema. Disjunctive Bases: Normal Forms for Modal Logics. In 7th Conference on Algebra and Coalgebra in Computer Science (CALCO 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 72, pp. 11:1-11:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{enqvist_et_al:LIPIcs.CALCO.2017.11,
  author =	{Enqvist, Sebastian and Venema, Yde},
  title =	{{Disjunctive Bases: Normal Forms for Modal Logics}},
  booktitle =	{7th Conference on Algebra and Coalgebra in Computer Science (CALCO 2017)},
  pages =	{11:1--11:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-033-0},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{72},
  editor =	{Bonchi, Filippo and K\"{o}nig, Barbara},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2017.11},
  URN =		{urn:nbn:de:0030-drops-80357},
  doi =		{10.4230/LIPIcs.CALCO.2017.11},
  annote =	{Keywords: Modal logic, fixpoint logic, automata, coalgebra, graded modal logic, Lyndon theorem, uniform interpolation}
}
Document
DOI: 10.4230/LIPIcs.CSL.2016.7
Completeness for Coalgebraic Fixpoint Logic

Authors: Sebastian Enqvist, Fatemeh Seifan, and Yde Venema

Published in: LIPIcs, Volume 62, 25th EACSL Annual Conference on Computer Science Logic (CSL 2016)

Abstract
We introduce an axiomatization for the coalgebraic fixed point logic which was introduced by Venema as a generalization, based on Moss' coalgebraic modality, of the well-known modal mu-calculus. Our axiomatization can be seen as a generalization of Kozen's proof system for the modal mu-calculus to the coalgebraic level of generality. It consists of a complete axiomatization for Moss'modality, extended with Kozen's axiom and rule for the fixpoint operators. Our main result is a completeness theorem stating that, for functors that preserve weak pullbacks and restrict to finite sets, our axiomatization is sound and complete for the standard interpretation of the language in coalgebraic models. Our proof is based on automata-theoretic ideas: in particular, we introduce the notion of consequence game for modal automata, which plays a crucial role in the proof of our main result. The result generalizes the celebrated Kozen-Walukiewicz completeness theorem for the modal mu-calculus, and our automata-theoretic methods simplify parts of Walukiewicz' proof.
Cite as

Sebastian Enqvist, Fatemeh Seifan, and Yde Venema. Completeness for Coalgebraic Fixpoint Logic. In 25th EACSL Annual Conference on Computer Science Logic (CSL 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 62, pp. 7:1-7:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{enqvist_et_al:LIPIcs.CSL.2016.7,
  author =	{Enqvist, Sebastian and Seifan, Fatemeh and Venema, Yde},
  title =	{{Completeness for Coalgebraic Fixpoint Logic}},
  booktitle =	{25th EACSL Annual Conference on Computer Science Logic (CSL 2016)},
  pages =	{7:1--7:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-022-4},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{62},
  editor =	{Talbot, Jean-Marc and Regnier, Laurent},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2016.7},
  URN =		{urn:nbn:de:0030-drops-65470},
  doi =		{10.4230/LIPIcs.CSL.2016.7},
  annote =	{Keywords: mu-calculus, coalgebra, coalgebraic modal logic, automata, completeness}
}
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