DROPS

Document

DOI: 10.4230/LIPIcs.FSCD.2026.14

How Term Rewriting Structures Shape the Decidability of Knowledge Problems

Authors: Raja O. P. Damanik and Alwen Tiu

Published in: LIPIcs, Volume 378, 11th International Conference on Formal Structures for Computation and Deduction (FSCD 2026)

Abstract

Deduction and static equivalence are central knowledge problems in the formal analysis of security protocols and are known to be undecidable for general equational theories. Several decidable classes have been identified through structural restrictions, including subterm convergent theories, shallow permutative theories, contracting theories and some are implemented in tools such as ProVerif and DeepSec. We identify two recurring themes: symbol preservation, where symbols are maintained across axioms, and symbol contraction, where symbols decrease in depth or number from left to right. For symbol-preserving systems, we introduce measure-invariant (MI) and separate measure-invariant (SMI) theories, generalizing permutative classes and providing new decidable fragments for deduction and static equivalence. Depth-sensitive refinements, including depth-preserving permutative (DPP) and depth-preserving variable-permuting (DPVP) theories, are case studies to understand whether the depth of occurrences of symbols matter. For symbol-contraction systems, we define depth-decreasing (DD) and variable-preserving function-decreasing (VBFD) theories, capturing some simple relaxations of term contraction; while deduction is undecidable in general, these restrictions highlight potential decidable fragments. Overall, our results show that controlling symbol dynamics in rewrite rules provides a unifying perspective on the decidability of knowledge problems, offering conceptual clarity on what makes these problems hard for different equational theories.

Cite as

Raja O. P. Damanik and Alwen Tiu. How Term Rewriting Structures Shape the Decidability of Knowledge Problems. In 11th International Conference on Formal Structures for Computation and Deduction (FSCD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 378, pp. 14:1-14:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{damanik_et_al:LIPIcs.FSCD.2026.14,
  author =	{Damanik, Raja O. P. and Tiu, Alwen},
  title =	{{How Term Rewriting Structures Shape the Decidability of Knowledge Problems}},
  booktitle =	{11th International Conference on Formal Structures for Computation and Deduction (FSCD 2026)},
  pages =	{14:1--14:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-433-8},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{378},
  editor =	{Pfenning, Frank},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2026.14},
  URN =		{urn:nbn:de:0030-drops-263649},
  doi =		{10.4230/LIPIcs.FSCD.2026.14},
  annote =	{Keywords: Security Protocol, Intruder Deduction, Static Equivalence, Term Rewriting}
}

Document

Invited Talk

DOI: 10.4230/LIPIcs.ITP.2025.40

Autosubst: On Mechanising Binders in a General-Purpose Proof Assistant (Invited Talk)

Authors: Kathrin Stark

Published in: LIPIcs, Volume 352, 16th International Conference on Interactive Theorem Proving (ITP 2025)

Abstract

The question of handling binders effectively regularly comes up when mechanising results in a proof assistant. Since the beginning of proof assistants, many solutions have been suggested: these include de Bruijn syntax, locally nameless binders, nominal syntax or HOAS. But even today - 20 years after the POPLMark Challenge [Brian E. Aydemir et al., 2005] - new tools and approaches to binding are still being published. Binders are still mentioned as being a complication of mechanised proofs over pen-and-paper proofs and the source of uninteresting technicalities and heaps of boilerplate - particularly, when working in a proof assistant without native support for binders or quotients. Autosubst [Steven Schäfer et al., 2015; Steven Schäfer et al., 2015], initially developed a decade ago as a teaching tool, addresses this problem by automating boilerplate code generation for a de Bruijn representation in the Rocq prover. Autosubst translates a specification into the corresponding pure or scoped de Bruijn algebra: It hence generates a corresponding instantiation operation for parallel substitutions, and several equational substitution lemmas. Central to its usability is a rewriting tactic that automatically decides equality up to substitutions, requiring minimal input and knowledge from the user’s side. This greatly simplifies using the otherwise notoriously difficult de Bruijn representation. Since then, Autosubst has been successfully used in several projects. In this talk, I will give an overview of the background and history of Autosubst and give an overview on the work, tools, and formalisations around it [Kathrin Stark et al., 2019; Kathrin Stark, 2020; Andreas Abel et al., 2019]. Moreover, this talk will highlight some of the more recent extensions [Yannick Forster and Kathrin Stark, 2020] as well as outline future challenges and directions.

Cite as

Kathrin Stark. Autosubst: On Mechanising Binders in a General-Purpose Proof Assistant (Invited Talk). In 16th International Conference on Interactive Theorem Proving (ITP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 352, pp. 40:1-40:2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{stark:LIPIcs.ITP.2025.40,
  author =	{Stark, Kathrin},
  title =	{{Autosubst: On Mechanising Binders in a General-Purpose Proof Assistant}},
  booktitle =	{16th International Conference on Interactive Theorem Proving (ITP 2025)},
  pages =	{40:1--40:2},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-396-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{352},
  editor =	{Forster, Yannick and Keller, Chantal},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2025.40},
  URN =		{urn:nbn:de:0030-drops-246385},
  doi =		{10.4230/LIPIcs.ITP.2025.40},
  annote =	{Keywords: Syntax, binders, Rocq}
}

Document

DOI: 10.4230/LIPIcs.ITP.2025.13

Barendregt’s Theory of the λ-Calculus, Refreshed and Formalized

Authors: Adrienne Lancelot, Beniamino Accattoli, and Maxime Vemclefs

Published in: LIPIcs, Volume 352, 16th International Conference on Interactive Theorem Proving (ITP 2025)

Abstract

Barendregt’s book on the untyped λ-calculus refines the inconsistent view of β-divergence as representation of the undefined via the key concept of head reduction. In this paper, we put together recent revisitations of some key theorems laid out in Barendregt’s book, and we formalize them in the Abella proof assistant. Our work provides a compact and refreshed presentation of the core of the book. The formalization faithfully mimics pen-and-paper proofs. Two interesting aspects are the manipulation of contexts for the study of contextual equivalence and a formal alternative to the informal trick at work in Takahashi’s proof of the genericity lemma. As a by-product, we obtain an alternative definition of contextual equivalence that does not mention contexts.

Cite as

Adrienne Lancelot, Beniamino Accattoli, and Maxime Vemclefs. Barendregt’s Theory of the λ-Calculus, Refreshed and Formalized. In 16th International Conference on Interactive Theorem Proving (ITP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 352, pp. 13:1-13:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{lancelot_et_al:LIPIcs.ITP.2025.13,
  author =	{Lancelot, Adrienne and Accattoli, Beniamino and Vemclefs, Maxime},
  title =	{{Barendregt’s Theory of the \lambda-Calculus, Refreshed and Formalized}},
  booktitle =	{16th International Conference on Interactive Theorem Proving (ITP 2025)},
  pages =	{13:1--13:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-396-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{352},
  editor =	{Forster, Yannick and Keller, Chantal},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2025.13},
  URN =		{urn:nbn:de:0030-drops-246114},
  doi =		{10.4230/LIPIcs.ITP.2025.13},
  annote =	{Keywords: lambda-calculus, head reduction, equational theory}
}

Document

DOI: 10.4230/LIPIcs.ITP.2025.11

Animating MRBNFs: Truly Modular Binding-Aware Datatypes in Isabelle/HOL

Authors: Jan van Brügge, Andrei Popescu, and Dmitriy Traytel

Published in: LIPIcs, Volume 352, 16th International Conference on Interactive Theorem Proving (ITP 2025)

Abstract

Nominal Isabelle provides powerful tools for meta-theoretic reasoning about syntax of logics or programming languages, in which variables are bound. It has been instrumental to major verification successes, such as Gödel’s incompleteness theorems. However, the existing tooling is not compositional. In particular, it does not support nested recursion, linear binding patterns, or infinitely branching syntax. These limitations are fundamental in the way nominal datatypes and functions on them are constructed within Nominal Isabelle. Taking advantage of recent theoretical advancements that overcome these limitations through a modular approach using the concept of map-restricted bounded natural functor (MRBNF), we develop and implement a new definitional package for binding-aware datatypes in Isabelle/HOL, called MrBNF. We describe the journey from the user specification to the end-product types, constants and theorems the tool generates. We validate MrBNF in two formalization case studies that so far were out of reach of nominal approaches: (1) Mazza’s isomorphism between the finitary and the infinitary affine λ-calculus, and (2) the POPLmark 2B challenge, which involves non-free binders for linear pattern matching.

Cite as

Jan van Brügge, Andrei Popescu, and Dmitriy Traytel. Animating MRBNFs: Truly Modular Binding-Aware Datatypes in Isabelle/HOL. In 16th International Conference on Interactive Theorem Proving (ITP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 352, pp. 11:1-11:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{vanbrugge_et_al:LIPIcs.ITP.2025.11,
  author =	{van Br\"{u}gge, Jan and Popescu, Andrei and Traytel, Dmitriy},
  title =	{{Animating MRBNFs: Truly Modular Binding-Aware Datatypes in Isabelle/HOL}},
  booktitle =	{16th International Conference on Interactive Theorem Proving (ITP 2025)},
  pages =	{11:1--11:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-396-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{352},
  editor =	{Forster, Yannick and Keller, Chantal},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2025.11},
  URN =		{urn:nbn:de:0030-drops-246091},
  doi =		{10.4230/LIPIcs.ITP.2025.11},
  annote =	{Keywords: syntax with bindings, datatypes, inductive predicates, Isabelle/HOL}
}

Document

DOI: 10.4230/LIPIcs.CONCUR.2025.30

Open Bisimilarity for the π-Calculus with Mismatch

Authors: Tiange Liu, Alwen Tiu, and Ross Horne

Published in: LIPIcs, Volume 348, 36th International Conference on Concurrency Theory (CONCUR 2025)

Abstract

Open bisimilarity is an equivalence relation for the π-calculus that is also congruence, making it suitable to use in compositional reasoning for mobile processes and communication protocols. The original definition of open bisimilarity, due to Sangiorgi, does not account for the mismatch operator, that is crucial in modelling real-world protocols. When mismatch is present, the congruence property no longer holds for open bisimilarity. In a LICS 2018 paper, Horne et al. proposed an extension of open bisimilarity, using a history-indexed class of relations, to address this problem. That definition, however, turns out to be non-compositional as we shall demonstrate in this paper. This paper presents a new definition of open bisimilarity in the π-calculus that incorporates mismatch. This is achieved by augmenting the transition semantics of the π-calculus with an explicit assumption about name distinctions, and by requiring that open bisimulation to be closed under an arbitary extension of the name distinctions assumption. We then prove that the resulting open bisimilarity is both an equivalence relation and a congruence.

Cite as

Tiange Liu, Alwen Tiu, and Ross Horne. Open Bisimilarity for the π-Calculus with Mismatch. In 36th International Conference on Concurrency Theory (CONCUR 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 348, pp. 30:1-30:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{liu_et_al:LIPIcs.CONCUR.2025.30,
  author =	{Liu, Tiange and Tiu, Alwen and Horne, Ross},
  title =	{{Open Bisimilarity for the \pi-Calculus with Mismatch}},
  booktitle =	{36th International Conference on Concurrency Theory (CONCUR 2025)},
  pages =	{30:1--30:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-389-8},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{348},
  editor =	{Bouyer, Patricia and van de Pol, Jaco},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2025.30},
  URN =		{urn:nbn:de:0030-drops-239805},
  doi =		{10.4230/LIPIcs.CONCUR.2025.30},
  annote =	{Keywords: mismatch, open bisimilarity, pi calculus}
}

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.CONCUR.2023.34

Modal Logics for Mobile Processes Revisited

Authors: Tiange Liu, Alwen Tiu, and Jim de Groot

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

Abstract

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.

Cite as

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.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}
}

Document

Invited Talk

DOI: 10.4230/LIPIcs.FSCD.2022.2

A Methodology for Designing Proof Search Calculi for Non-Classical Logics (Invited Talk)

Authors: Alwen Tiu

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

Abstract

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.

Cite as

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.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

DOI: 10.4230/LIPIcs.CONCUR.2020.31

Decidable Inductive Invariants for Verification of Cryptographic Protocols with Unbounded Sessions

Authors: Emanuele D'Osualdo and Felix Stutz

Published in: LIPIcs, Volume 171, 31st International Conference on Concurrency Theory (CONCUR 2020)

Abstract

We develop a theory of decidable inductive invariants for an infinite-state variant of the Applied πcalc, with applications to automatic verification of stateful cryptographic protocols with unbounded sessions/nonces. Since the problem is undecidable in general, we introduce depth-bounded protocols, a strict generalisation of a class from the literature, for which our decidable analysis is sound and complete. Our core contribution is a procedure to check that an invariant is inductive, which implies that every reachable configuration satisfies it. Our invariants can capture security properties like secrecy, can be inferred automatically, and represent an independently checkable certificate of correctness. We provide a prototype implementation and we report on its performance on some textbook examples.

Cite as

Emanuele D'Osualdo and Felix Stutz. Decidable Inductive Invariants for Verification of Cryptographic Protocols with Unbounded Sessions. In 31st International Conference on Concurrency Theory (CONCUR 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 171, pp. 31:1-31:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{dosualdo_et_al:LIPIcs.CONCUR.2020.31,
  author =	{D'Osualdo, Emanuele and Stutz, Felix},
  title =	{{Decidable Inductive Invariants for Verification of Cryptographic Protocols with Unbounded Sessions}},
  booktitle =	{31st International Conference on Concurrency Theory (CONCUR 2020)},
  pages =	{31:1--31:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-160-3},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{171},
  editor =	{Konnov, Igor and Kov\'{a}cs, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2020.31},
  URN =		{urn:nbn:de:0030-drops-128433},
  doi =		{10.4230/LIPIcs.CONCUR.2020.31},
  annote =	{Keywords: Security Protocols, Infinite-State Verification, Ideal Completions for WSTS}
}

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.CONCUR.2017.7

A Characterisation of Open Bisimilarity using an Intuitionistic Modal Logic

Authors: Ki Yung Ahn, Ross Horne, and Alwen Tiu

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

Abstract

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.

Cite as

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.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

DOI: 10.4230/LIPIcs.CONCUR.2016.31

Private Names in Non-Commutative Logic

Authors: Ross Horne, Alwen Tiu, Bogdan Aman, and Gabriel Ciobanu

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

Abstract

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.

Cite as

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.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

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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