Document

**Published in:** LIPIcs, Volume 279, 34th International Conference on Concurrency Theory (CONCUR 2023)

We revisit the logical characterisations of various bisimilarity relations for the finite fragment of the π-calculus. Our starting point is the early and the late bisimilarity, first defined in the seminal work of Milner, Parrow and Walker, who also proved their characterisations in fragments of a modal logic (which we refer to as the MPW logic). Two important refinements of early and late bisimilarity, called open and quasi-open bisimilarity, respectively, were subsequently proposed by Sangiorgi and Walker. Horne, et. al., showed that open and quasi-bisimilarity are characterised by intuitionistic modal logics: OM (for open bisimilarity) and FM (for quasi-open bisimilarity). In this work, we attempt to unify the logical characterisations of these bisimilarity relations, showing that they can be characterised by different sublogics of a unifying logic. A key insight to this unification derives from a reformulation of the four bisimilarity relations (early, late, open and quasi-open) that uses an explicit name context, and an observation that these relations can be distinguished by the relative scoping of names and their instantiations in the name context. This name context and name substitution then give rise to an accessibility relation in the underlying Kripke semantics of our logic, that is captured logically by an S4-like modal operator. We then show that the MPW, the OM and the FM logics can be embedded into fragments of our unifying classical modal logic. In the case of OM and FM, the embedding uses the fact that intuitionistic implication can be encoded in modal logic S4.

Tiange Liu, Alwen Tiu, and Jim de Groot. Modal Logics for Mobile Processes Revisited. In 34th International Conference on Concurrency Theory (CONCUR 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 279, pp. 34:1-34:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{liu_et_al:LIPIcs.CONCUR.2023.34, author = {Liu, Tiange and Tiu, Alwen and de Groot, Jim}, title = {{Modal Logics for Mobile Processes Revisited}}, booktitle = {34th International Conference on Concurrency Theory (CONCUR 2023)}, pages = {34:1--34:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-299-0}, ISSN = {1868-8969}, year = {2023}, volume = {279}, editor = {P\'{e}rez, Guillermo A. and Raskin, Jean-Fran\c{c}ois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2023.34}, URN = {urn:nbn:de:0030-drops-190289}, doi = {10.4230/LIPIcs.CONCUR.2023.34}, annote = {Keywords: pi-calculus, modal logic, intuitionistic logic, bisimilarity} }

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Invited Talk

**Published in:** LIPIcs, Volume 228, 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022)

In this talk I present a methodology for designing proof search calculi for a wide range of non-classical logics, such as modal and tense logics, bi-intuitionistic (linear) logics and grammar logics. Most of these logics cannot be easily formalised in the traditional Gentzen-style sequent calculus; various structural extensions to sequent calculus seem to be required. One of the more expressive extensions of sequent calculus is Belnap’s display calculus, which allows one to formalise a very wide range of logics and which provides a generic cut-elimination method for logics formalised in the calculus. The generality of display calculus derives partly from the pervasive use of structural rules to capture properties of the underlying semantics of the logic of interest, such as various frame conditions in normal modal logics, that are not easily captured by introduction rules alone. Unlike traditional sequent calculi, the subformula property in display calculi does not typically give an immediate bound on the search space (assuming contraction is absent) in proof search, as new structures may be created and their creation may not be driven by any introduction rules for logical connectives. This line of work started out as an attempt to "tame" display calculus, to make it more proof search friendly, by eliminating or restricting the use of structural rules. Two key ideas that make this possible are the adoption of deep inference, allowing inference rules to be applied inside a nested structure, and the use of propagation rules in place of structural rules. A brief survey of the applications of this methodology to a wide range of logics is presented, along with some directions for future work.

Alwen Tiu. A Methodology for Designing Proof Search Calculi for Non-Classical Logics (Invited Talk). In 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 228, pp. 2:1-2:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{tiu:LIPIcs.FSCD.2022.2, author = {Tiu, Alwen}, title = {{A Methodology for Designing Proof Search Calculi for Non-Classical Logics}}, booktitle = {7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022)}, pages = {2:1--2:4}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2022.2}, URN = {urn:nbn:de:0030-drops-162834}, doi = {10.4230/LIPIcs.FSCD.2022.2}, annote = {Keywords: Proof theory, Sequent calculus, Display calculus, Nested sequent calculus, Deep inference} }

Document

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

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.

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-dev.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} }

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**Published in:** LIPIcs, Volume 85, 28th International Conference on Concurrency Theory (CONCUR 2017)

Open bisimilarity is a strong bisimulation congruence for the pi-calculus. In open bisimilarity, free names in processes are treated as variables that may be instantiated; in contrast to late bisimilarity where free names are constants. An established modal logic due to Milner, Parrow, and Walker characterises late bisimilarity, that is, two processes satisfy the same set of formulae if and only if they are bisimilar. We propose an intuitionistic variation of this modal logic and prove that it characterises open bisimilarity. The soundness proof is mechanised in Abella. The completeness proof provides an algorithm for generating distinguishing formulae, useful for explaining and certifying whenever processes are non-bisimilar.

Ki Yung Ahn, Ross Horne, and Alwen Tiu. A Characterisation of Open Bisimilarity using an Intuitionistic Modal Logic. In 28th International Conference on Concurrency Theory (CONCUR 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 85, pp. 7:1-7:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{ahn_et_al:LIPIcs.CONCUR.2017.7, author = {Ahn, Ki Yung and Horne, Ross and Tiu, Alwen}, title = {{A Characterisation of Open Bisimilarity using an Intuitionistic Modal Logic}}, booktitle = {28th International Conference on Concurrency Theory (CONCUR 2017)}, pages = {7:1--7:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-048-4}, ISSN = {1868-8969}, year = {2017}, volume = {85}, editor = {Meyer, Roland and Nestmann, Uwe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2017.7}, URN = {urn:nbn:de:0030-drops-77899}, doi = {10.4230/LIPIcs.CONCUR.2017.7}, annote = {Keywords: bisimulation, modal logic, intuitionistic logic} }

Document

**Published in:** LIPIcs, Volume 59, 27th International Conference on Concurrency Theory (CONCUR 2016)

We present an expressive but decidable first-order system (named MAV1) defined by using the calculus of structures, a generalisation of the sequent calculus. In addition to first-order universal and existential quantifiers the system incorporates a de Morgan dual pair of nominal quantifiers called `new' and `wen', distinct from the self-dual Gabbay-Pitts and Miller-Tiu nominal quantifiers. The novelty of the operators `new' and `wen' is they are polarised in the sense that `new' distributes over positive operators while `wen' distributes over negative operators. This greater control of bookkeeping enables private names to be modelled in processes embedded as predicates in MAV1. Modelling processes as predicates in MAV1 has the advantage that linear implication defines a precongruence over processes that fully respects causality and branching. The transitivity of this precongruence is established by novel techniques for handling first-order quantifiers in the cut elimination proof.

Ross Horne, Alwen Tiu, Bogdan Aman, and Gabriel Ciobanu. Private Names in Non-Commutative Logic. In 27th International Conference on Concurrency Theory (CONCUR 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 59, pp. 31:1-31:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{horne_et_al:LIPIcs.CONCUR.2016.31, author = {Horne, Ross and Tiu, Alwen and Aman, Bogdan and Ciobanu, Gabriel}, title = {{Private Names in Non-Commutative Logic}}, booktitle = {27th International Conference on Concurrency Theory (CONCUR 2016)}, pages = {31:1--31:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-017-0}, ISSN = {1868-8969}, year = {2016}, volume = {59}, editor = {Desharnais, Jos\'{e}e and Jagadeesan, Radha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2016.31}, URN = {urn:nbn:de:0030-drops-61759}, doi = {10.4230/LIPIcs.CONCUR.2016.31}, annote = {Keywords: process calculi, calculus of structures, nominal logic, causal consistency} }

Document

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

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.

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-dev.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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