7 Search Results for "Clouston, Ranald"

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.FSCD.2025.33
Solving Guarded Domain Equations in Presheaves over Ordinals and Mechanizing It

Authors: Sergei Stepanenko and Amin Timany

Published in: LIPIcs, Volume 337, 10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025)

Abstract
Constructing solutions to recursive domain equations is a well-known, important problem in the study of programs and programming languages. Mathematically speaking, the problem is finding a fixed point (up to isomorphism) of a suitable functor over a suitable category. A particularly useful instance, inspired by the step-indexing technique, is where the functor is over (a subcategory of) the category of presheaves over the ordinal ω and the functors are locally-contractive, also known as guarded functors. This corresponds to step-indexing over natural numbers. However, for certain problems, e.g., when dealing with infinite non-determinism, one needs to employ trans-finite step-indexing, i.e., consider presheaf categories over higher ordinals. Prior work on trans-finite step-indexing either only considers a very narrow class of functors over a particularly restricted subcategory of presheaves over higher ordinals, or treats the problem very generally working with sheaves over an arbitrary complete Heyting algebra with a well-founded basis. In this paper we present a solution to the guarded domain equations problem over all guarded functors over the category of presheaves over ordinal numbers, as well as its mechanization in the Rocq Prover. As the categories of sheaves and presheaves over ordinals are equivalent, our main contribution is simplifying prior work from the setting of the category of sheaves to the setting of the category of presheaves and mechanizing it - presheaves are more amenable to mechanization in a proof assistant.
Cite as

Sergei Stepanenko and Amin Timany. Solving Guarded Domain Equations in Presheaves over Ordinals and Mechanizing It. In 10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 337, pp. 33:1-33:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{stepanenko_et_al:LIPIcs.FSCD.2025.33,
  author =	{Stepanenko, Sergei and Timany, Amin},
  title =	{{Solving Guarded Domain Equations in Presheaves over Ordinals and Mechanizing It}},
  booktitle =	{10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025)},
  pages =	{33:1--33:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-374-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{337},
  editor =	{Fern\'{a}ndez, Maribel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2025.33},
  URN =		{urn:nbn:de:0030-drops-236486},
  doi =		{10.4230/LIPIcs.FSCD.2025.33},
  annote =	{Keywords: Domain Equations, Guarded Fixed Points, Fixed Points, Category Theory, Rocq, Presheaves, Ordinals}
}
Document
DOI: 10.4230/LIPIcs.CSL.2025.42
Unifying Sequent Systems for Gödel-Löb Provability Logic via Syntactic Transformations

Authors: Tim S. Lyon

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

Abstract
We demonstrate the inter-translatability of proofs between the most prominent sequent-based formalisms for Gödel-Löb provability logic. In particular, we consider Sambin and Valentini’s sequent system GL_{seq}, Shamkanov’s non-wellfounded and cyclic sequent systems GL_∞ and GL_{circ}, Poggiolesi’s tree-hypersequent system CSGL, and Negri’s labeled sequent system G3GL. Shamkanov provided proof-theoretic correspondences between GL_{seq}, GL_∞, and GL_{circ}, and Goré and Ramanayake showed how to transform proofs between CSGL and G3GL, however, the exact nature of proof transformations between the former three systems and the latter two systems has remained an open problem. We solve this open problem by showing how to restructure tree-hypersequent proofs into an end-active form and introduce a novel linearization technique that transforms such proofs into linear nested sequent proofs. As a result, we obtain a new proof-theoretic tool for extracting linear nested sequent systems from tree-hypersequent systems, which yields the first cut-free linear nested sequent calculus LNGL for Gödel-Löb provability logic. We show how to transform proofs in LNGL into a certain normal form, where proofs repeat in stages of modal and local rule applications, and which are translatable into GL_{seq} and G3GL proofs. These new syntactic transformations, together with those mentioned above, establish full proof-theoretic correspondences between GL_{seq}, GL_∞, GL_{circ}, CSGL, G3GL, and LNGL while also giving (to the best of the author’s knowledge) the first constructive proof mappings between structural (viz. labeled, tree-hypersequent, and linear nested sequent) systems and a cyclic sequent system.
Cite as

Tim S. Lyon. Unifying Sequent Systems for Gödel-Löb Provability Logic via Syntactic Transformations. In 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 326, pp. 42:1-42:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{lyon:LIPIcs.CSL.2025.42,
  author =	{Lyon, Tim S.},
  title =	{{Unifying Sequent Systems for G\"{o}del-L\"{o}b Provability Logic via Syntactic Transformations}},
  booktitle =	{33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)},
  pages =	{42:1--42:23},
  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.2025.42},
  URN =		{urn:nbn:de:0030-drops-227992},
  doi =		{10.4230/LIPIcs.CSL.2025.42},
  annote =	{Keywords: Cyclic proof, G\"{o}del-L\"{o}b logic, Labeled sequent, Linear nested sequent, Modal logic, Non-wellfounded proof, Proof theory, Proof transformation, Tree-hypersequent}
}
Document
DOI: 10.4230/LIPIcs.CSL.2025.41
Taking Bi-Intuitionistic Logic First-Order: A Proof-Theoretic Investigation via Polytree Sequents

Authors: Tim S. Lyon, Ian Shillito, and Alwen Tiu

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

Abstract
It is well-known that extending the Hilbert axiomatic system for first-order intuitionistic logic with an exclusion operator, that is dual to implication, collapses the domains of models into a constant domain. This makes it an interesting problem to find a sound and complete proof system for first-order bi-intuitionistic logic with non-constant domains that is also conservative over first-order intuitionistic logic. We solve this problem by presenting the first sound and complete proof system for first-order bi-intuitionistic logic with increasing domains. We formalize our proof system as a polytree sequent calculus (a notational variant of nested sequents), and prove that it enjoys cut-elimination and is conservative over first-order intuitionistic logic. A key feature of our calculus is an explicit eigenvariable context, which allows us to control precisely the scope of free variables in a polytree structure. Semantically this context can be seen as encoding a notion of Scott’s existence predicate for intuitionistic logic. This turns out to be crucial to avoid the collapse of domains and to prove the completeness of our proof system. The explicit consideration of the variable context in a formula sheds light on a previously overlooked dependency between the residuation principle and the existence predicate in the first-order setting, which may help to explain the difficulty in designing a sound and complete proof system for first-order bi-intuitionistic logic.
Cite as

Tim S. Lyon, Ian Shillito, and Alwen Tiu. Taking Bi-Intuitionistic Logic First-Order: A Proof-Theoretic Investigation via Polytree Sequents. In 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 326, pp. 41:1-41:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{lyon_et_al:LIPIcs.CSL.2025.41,
  author =	{Lyon, Tim S. and Shillito, Ian and Tiu, Alwen},
  title =	{{Taking Bi-Intuitionistic Logic First-Order: A Proof-Theoretic Investigation via Polytree Sequents}},
  booktitle =	{33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)},
  pages =	{41:1--41:23},
  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.2025.41},
  URN =		{urn:nbn:de:0030-drops-227987},
  doi =		{10.4230/LIPIcs.CSL.2025.41},
  annote =	{Keywords: Bi-intuitionistic, Cut-elimination, Conservativity, Domain, First-order, Polytree, Proof theory, Reachability, Sequent}
}
Document
DOI: 10.4230/LIPIcs.CSL.2020.28
Syntactic Interpolation for Tense Logics and Bi-Intuitionistic Logic via Nested Sequents

Authors: Tim Lyon, Alwen Tiu, Rajeev Goré, and Ranald Clouston

Published in: LIPIcs, Volume 152, 28th EACSL Annual Conference on Computer Science Logic (CSL 2020)

Abstract
We provide a direct method for proving Craig interpolation for a range of modal and intuitionistic logics, including those containing a "converse" modality. We demonstrate this method for classical tense logic, its extensions with path axioms, and for bi-intuitionistic logic. These logics do not have straightforward formalisations in the traditional Gentzen-style sequent calculus, but have all been shown to have cut-free nested sequent calculi. The proof of the interpolation theorem uses these calculi and is purely syntactic, without resorting to embeddings, semantic arguments, or interpreted connectives external to the underlying logical language. A novel feature of our proof includes an orthogonality condition for defining duality between interpolants.
Cite as

Tim Lyon, Alwen Tiu, Rajeev Goré, and Ranald Clouston. Syntactic Interpolation for Tense Logics and Bi-Intuitionistic Logic via Nested Sequents. In 28th EACSL Annual Conference on Computer Science Logic (CSL 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 152, pp. 28:1-28:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{lyon_et_al:LIPIcs.CSL.2020.28,
  author =	{Lyon, Tim and Tiu, Alwen and Gor\'{e}, Rajeev and Clouston, Ranald},
  title =	{{Syntactic Interpolation for Tense Logics and Bi-Intuitionistic Logic via Nested Sequents}},
  booktitle =	{28th EACSL Annual Conference on Computer Science Logic (CSL 2020)},
  pages =	{28:1--28:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-132-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{152},
  editor =	{Fern\'{a}ndez, Maribel and Muscholl, Anca},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2020.28},
  URN =		{urn:nbn:de:0030-drops-116713},
  doi =		{10.4230/LIPIcs.CSL.2020.28},
  annote =	{Keywords: Bi-intuitionistic logic, Interpolation, Nested calculi, Proof theory, Sequents, Tense logics}
}
Document
DOI: 10.4230/LIPIcs.CSL.2016.23
Guarded Cubical Type Theory: Path Equality for Guarded Recursion

Authors: Lars Birkedal, Aleš Bizjak, Ranald Clouston, Hans Bugge Grathwohl, Bas Spitters, and Andrea Vezzosi

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

Abstract
This paper improves the treatment of equality in guarded dependent type theory (GDTT), by combining it with cubical type theory (CTT). GDTT is an extensional type theory with guarded recursive types, which are useful for building models of program logics, and for programming and reasoning with coinductive types. We wish to implement GDTT with decidable type checking, while still supporting non-trivial equality proofs that reason about the extensions of guarded recursive constructions. CTT is a variation of Martin-Löf type theory in which the identity type is replaced by abstract paths between terms. CTT provides a computational interpretation of functional extensionality, is conjectured to have decidable type checking, and has an implemented type checker. Our new type theory, called guarded cubical type theory, provides a computational interpretation of extensionality for guarded recursive types. This further expands the foundations of CTT as a basis for formalisation in mathematics and computer science. We present examples to demonstrate the expressivity of our type theory, all of which have been checked using a prototype type-checker implementation, and present semantics in a presheaf category.
Cite as

Lars Birkedal, Aleš Bizjak, Ranald Clouston, Hans Bugge Grathwohl, Bas Spitters, and Andrea Vezzosi. Guarded Cubical Type Theory: Path Equality for Guarded Recursion. In 25th EACSL Annual Conference on Computer Science Logic (CSL 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 62, pp. 23:1-23:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{birkedal_et_al:LIPIcs.CSL.2016.23,
  author =	{Birkedal, Lars and Bizjak, Ale\v{s} and Clouston, Ranald and Grathwohl, Hans Bugge and Spitters, Bas and Vezzosi, Andrea},
  title =	{{Guarded Cubical Type Theory: Path Equality for Guarded Recursion}},
  booktitle =	{25th EACSL Annual Conference on Computer Science Logic (CSL 2016)},
  pages =	{23:1--23:17},
  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.2016.23},
  URN =		{urn:nbn:de:0030-drops-65638},
  doi =		{10.4230/LIPIcs.CSL.2016.23},
  annote =	{Keywords: Guarded Recursion, Dependent Type Theory, Cubical Type Theory, Denotational Semantics, Homotopy Type Theory}
}
Document
DOI: 10.4230/LIPIcs.CSL.2013.197
Annotation-Free Sequent Calculi for Full Intuitionistic Linear Logic

Authors: Ranald Clouston, Jeremy Dawson, Rajeev Goré, and Alwen Tiu

Published in: LIPIcs, Volume 23, Computer Science Logic 2013 (CSL 2013)

Abstract
Full Intuitionistic Linear Logic (FILL) is multiplicative intuitionistic linear logic extended with par. Its proof theory has been notoriously difficult to get right, and existing sequent calculi all involve inference rules with complex annotations to guarantee soundness and cut-elimination. We give a simple and annotation-free display calculus for FILL which satisfies Belnap’s generic cut-elimination theorem. To do so, our display calculus actually handles an extension of FILL, called Bi-Intuitionistic Linear Logic (BiILL), with an ‘exclusion’ connective defined via an adjunction with par. We refine our display calculus for BiILL into a cut-free nested sequent calculus with deep inference in which the explicit structural rules of the display calculus become admissible. A separation property guarantees that proofs of FILL formulae in the deep inference calculus contain no trace of exclusion. Each such rule is sound for the semantics of FILL, thus our deep inference calculus and display calculus are conservative over FILL. The deep inference calculus also enjoys the subformula property and terminating backward proof search, which gives the NP-completeness of BiILL and FILL.
Cite as

Ranald Clouston, Jeremy Dawson, Rajeev Goré, and Alwen Tiu. Annotation-Free Sequent Calculi for Full Intuitionistic Linear Logic. In Computer Science Logic 2013 (CSL 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 23, pp. 197-214, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{clouston_et_al:LIPIcs.CSL.2013.197,
  author =	{Clouston, Ranald and Dawson, Jeremy and Gor\'{e}, Rajeev and Tiu, Alwen},
  title =	{{Annotation-Free Sequent Calculi for Full Intuitionistic Linear Logic}},
  booktitle =	{Computer Science Logic 2013 (CSL 2013)},
  pages =	{197--214},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-60-6},
  ISSN =	{1868-8969},
  year =	{2013},
  volume =	{23},
  editor =	{Ronchi Della Rocca, Simona},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2013.197},
  URN =		{urn:nbn:de:0030-drops-41981},
  doi =		{10.4230/LIPIcs.CSL.2013.197},
  annote =	{Keywords: Linear logic, display calculus, nested sequent calculus, deep inference}
}
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