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Jim de Groot

Publications and source records attributed to Jim de Groot.

At least 19 recordsLinked to original sources
arXiv2026

Intuitionistic Monotone Modal Logic: Proof Theory and Semantics

We study the recently introduced intuitionistic monotone modal logic IM. We first provide a semantic characterisation for a family of natural extensions of IM in terms of constructive neighbourhood models. We then present a calculus for IM and its extensions, obtained by adapting a structured calculus for the classical monotone modal logic M. Based on the calculus, we prove some preliminary results for IM, including its decidability. Our calculus also reveals an interesting analogy between constructive and intuitionistic variants of M and the corresponding variants of K, thereby further justifying IM as a faithful intuitionistic variant of M.

cs.LO↗
arXiv2026

Intuitionistic K is a Bisimulation-Invariant Fragment of Intuitionistic First-Order Logic

We define the notion of IK-bisimulation between the relational semantics for the intuitionistic modal logic IK, and prove that IK arises as the IK-bisimulation-invariant fragment of intuitionistic first-order logic. En route, we provide an intrinsic characterisation result of this logic by way of a Hennessy-Milner-style theorem and develop some intuitionistic first-order model theory, including intuitionistic analogues of Los's Theorem, elementary embeddings and countable saturation.

math.LO↗
arXiv2026

Modal Measurable Logics via a Modal Loomis-Sikorski Representation Theorem

We investigate a modal extension of the infinitary classical logic with countable meets and joins, formulated with an eye toward measure-theoretic work in dynamical systems and in point-free ergodic theory. We define a modal formalism in this language, which we call modal measurable logics. We also introduce a Kripke-like semantics for these logics in measurable spaces taking a designated modal sigma-ideal into consideration. Using a restriction of Jonsson-Tarski duality and a modal extension of the Loomis-Sikorski theorem, we prove completeness of modal measurable logics with respect to this new semantics.

math.LO↗
arXiv2026

Relational Semantics for Flat Heyting-Lewis Logic

We introduce relational semantics for "flat Heyting-Lewis logic" HLC-flat. This logic arises as the extension of intuitionistic logic with a Lewis-style strict implication modality that, contrary to its "sharp" counterpart HLC-sharp, does not turn meets into joins in its first argument. We prove completeness and the finite model property for HLC-flat and for several extensions with additional axioms.

math.LO↗
arXiv2026

Duality for Constructive Modal Logics: from Sahqlvist to Goldblatt-Thomason

We carry out a semantic study of the constructive modal logic CK. We provide a categorical duality linking the algebraic and birelational semantics of the logic. We then use this to prove Sahlqvist style correspondence and completeness results, as well as a Goldblatt-Thomason style theorem on definability of classes of frames.

math.LO↗
arXiv2025

Intrinsic and relative characterization results for logics with negative modalities

We introduce simulations for modal logics with subclassical negations and restoration modalities, establish an adequacy theorem, and prove intrinsic (Hennessy-Milner-type) and relative (Van Benthem-type) characterization results. These results identify each restorative language with the fragment of first-order logic invariant under its simulations and delineate the expressive profile of modal logics with non-classical negations.

math.LO↗
arXiv2025

Filling in the semantics for intuitionistic conditional logic

We prove completeness results for a wide variety of intuitionistic conditional logics. We do so by first using a canonical model construction obtain completeness with respect to descriptive conditional frames, and then introducing the fill-in method to transfer this to classes of conditional frames without extra structure. The fill-in method closes the gap between descriptive conditional frames, which do not have a canonical underlying frame, and conditional frames.

math.LO↗
arXiv2025

Intuitionistic monotone modal logic via translation

We introduce a monotone modal analogue of the intuitionistic (normal) modal logic IK using a translation into a suitable (intuitionistic) first-order logic. We axiomatise the logic and give a semantics by means of intuitionistic neighbourhood models, which contain neighbourhoods whose value can change when moving along the intuitionistic accessibility relation. We compare the resulting logic with other intuitionistic monotone modal logics and show how it can be embedded into a multimodal version of IK.

math.LO↗
arXiv2024

Tamgram: A Frontend for Large-scale Protocol Modeling in Tamarin

Automated security protocol verifiers such as ProVerif and Tamarin have been increasingly applied to verify large scale complex real-world protocols. While their ability to automate difficult reasoning processes required to handle protocols at that scale is impressive, there remains a gap in the modeling languages used. In particular, providing support for writing and maintaining large protocol specifications. This work attempts to fill this gap by introducing a high-level protocol modeling language, called Tamgram, with a formal semantics that can be translated to the multiset rewriting semantics of Tamarin. Tamgram supports writing native Tamarin code directly, but also allows for easier structuring of large specifications through various high-level constructs, in particular those needed to manipulate states in protocols. We prove the soundness and the completeness of Tamgram with respect to the trace semantics of Tamarin, discuss different translation strategies, and identify an optimal strategy that yields performance comparable to manually coded Tamarin specifications. Finally we show the practicality of Tamgram with a set of small case studies and one large scale case study.

cs.CR↗
arXiv2024

Sub-sub-intuitionistic logic

Sub-sub-intuitionistic logic is obtained from intuitionistic logic by weakening the implication and removing distributivity. It can alternatively be viewed as conditional weak positive logic. We provide semantics for sub-sub-intuitionistic logic by means of semilattices with a selection function, prove a categorical duality for the algebraic semantics of the logic, and use this to derive completeness. We then consider the extension of sub-sub-intuitionistic logic with a variety of axioms.

math.LO↗
arXiv2024

Semantical Analysis of Intuitionistic Modal Logics between CK and IK

The intuitionistic modal logics considered between Constructive K (CK) and Intuitionistic K (IK) differ in their treatment of the possibility (diamond) connective. It was recently rediscovered that some logics between CK and IK also disagree on their diamond-free fragments, with only some remaining conservative over the standard axiomatisation of intuitionistic modal logic with necessity (box) alone. We show that relational Kripke semantics for CK can be extended with frame conditions for all axioms in the standard axiomatisation of IK, as well as other axioms previously studied. This allows us to answer open questions about the (non-)conservativity of such logics over intuitionistic modal logic without diamond. Our results are formalised using the Coq Proof Assistant.

cs.LO↗
arXiv2023

Positive Modal Logic Beyond Distributivity

We develop a duality for (modal) lattices that need not be distributive, and use it to study positive (modal) logic beyond distributivity, which we call weak positive (modal) logic. This duality builds on the Hofmann, Mislove and Stralka duality for meet-semilattices. We introduce the notion of $Π_1$-persistence and show that every weak positive modal logic is $Π_1$-persistent. This approach leads to a new relational semantics for weak positive modal logic, for which we prove an analogue of Sahlqvist correspondence result.

math.LO↗
arXiv2023

Non-distributive positive logic as a fragment of first-order logic over semilattices

We characterise non-distributive positive logic as the fragment of a single-sorted first-order language that is preserved by a new notion of simulation called a meet-simulation. Meet-simulations distinguish themselves from simulations because they relate pairs of states from one model to single states from another. En route to this result we use a more traditional notion of simulations and prove a Hennessy-Milner style theorem for it, using an analogue of modal saturation called meet-compactness.

math.LO↗
arXiv2022

Coalgebraic Geometric Logic: Basic Theory

Using the theory of coalgebra, we introduce a uniform framework for adding modalities to the language of propositional geometric logic. Models for this logic are based on coalgebras for an endofunctor on some full subcategory of the category of topological spaces and continuous functions. We investigate derivation systems, soundness and completeness for such geometric modal logics, and we specify a method of lifting an endofunctor on Set, accompanied by a collection of predicate liftings, to an endofunctor on the category of topological spaces, again accompanied by a collection of (open) predicate liftings. Furthermore, we compare the notions of modal equivalence, behavioural equivalence and bisimulation on the resulting class of models, and we provide a final object for the corresponding category.

math.LO↗
arXiv2022

A Coalgebraic Approach to Dualities for Neighborhood Frames

We develop a uniform coalgebraic approach to Jónsson-Tarski and Thomason type dualities for various classes of neighborhood frames and neighborhood algebras. In the first part of the paper we construct an endofunctor on the category of complete and atomic Boolean algebras that is dual to the double powerset functor on $\mathsf{Set}$. This allows us to show that Thomason duality for neighborhood frames can be viewed as an algebra-coalgebra duality. We generalize this approach to any class of algebras for an endofunctor presented by one-step axioms in the language of infinitary modal logic. As a consequence, we obtain a uniform approach to dualities for various classes of neighborhood frames, including monotone neighborhood frames, pretopological spaces, and topological spaces. In the second part of the paper we develop a coalgebraic approach to Jónsson-Tarski duality for neighborhood algebras and descriptive neighborhood frames. We introduce an analogue of the Vietoris endofunctor on the category of Stone spaces and show that descriptive neighborhood frames are isomorphic to coalgebras for this endofunctor. This allows us to obtain a coalgebraic proof of the duality between descriptive neighborhood frames and neighborhood algebras. Using one-step axioms in the language of finitary modal logic, we restrict this duality to other classes of neighborhood algebras studied in the literature, including monotone modal algebras and contingency algebras. We conclude the paper by connecting the two types of dualities via canonical extensions, and discuss when these extensions are functorial.

cs.LO↗
arXiv2022

Modal meet-implication logic

We extend the meet-implication fragment of propositional intuitionistic logic with a meet-preserving modality. We give semantics based on semilattices and a duality result with a suitable notion of descriptive frame. As a consequence we obtain completeness and identify a common (modal) fragment of a large class of modal intuitionistic logics. We recognise this logic as a dialgebraic logic, and as a consequence obtain expressivity-somewhere-else. Within the dialgebraic framework, we then investigate the extension of the meet-implication fragment of propositional intuitionistic logic with a monotone modality and prove completeness and expressivity-somewhere-else for it.

math.LO↗
arXiv2021

Hennessy-Milner Properties via Topological Compactness

We give Hennessy-Milner classes for intuitionistic, dual-intuitionistic and bi-intuitionistic logic interpreted in intuitionistic Kripke models, and generalise these results to modal (dual- and bi-)intuitionistic logics. Our main technical tools are image-compact and pre-image-compact relations that provide a semantical description of modal saturation properties.

math.LO↗