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 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

Goldblatt-Thomason Theorems for Modal Intuitionistic Logics

We prove Goldblatt-Thomason theorems for frames and models of a wide variety of modal intuitionistic logics, including ones studied by Wolter and Zakharyaschev, Goldblatt, Fischer Servi, and Plotkin and Sterling. We use the framework of dialgebraic logic to describe most of these logics and derive results in a uniform way

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

Modal Logics for Mobile Processes Revisited

We revisit the logical characterisations of various bisimilarity relations for the finite fragment of the ?-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

Dualities in modal logic

Categorical dualities are an important tool in the study of (modal) logics. They offer conceptual understanding and enable the transfer of results between the different semantics of a logic. As such, they play a central role in the proofs of completeness theorems, Sahlqvist theorems and Goldblatt-Thomason theorems. A common way to obtain dualities is by extending existing ones. For example, Jonsson-Tarski duality is an extension of Stone duality. A convenient formalism to carry out such extensions is given by the dual categorical notions of algebras and coalgebras. Intuitively, these allow one to isolate the new part of a duality from the existing part. In this thesis we will derive both existing and new dualities via this route, and we show how to use the dualities to investigate logics. However, not all (modal logical) paradigms fit the (co)algebraic perspective. In particular, modal intuitionistic logics do not enjoy a coalgebraic treatment, and there is a general lack of duality results for them. To remedy this, we use a generalisation of both algebras and coalgebras called dialgebras. Guided by the research field of coalgebraic logic, we introduce the framework of dialgebraic logic. We show how a large class of modal intuitionistic logics can be modelled as dialgebraic logics and we prove dualities for them. We use the dialgebraic framework to prove general completeness, Hennessy-Milner, representation and Goldblatt-Thomason theorems, and instantiate this to a wide variety of modal intuitionistic logics. Additionally, we use the dialgebraic perspective to investigate modal extensions of the meet-implication fragment of intuitionistic logic. We instantiate general dialgebraic results, and describe how modal meet-implication logics relate to modal intuitionistic logics

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 $\Pi_1$-persistence and show that every weak positive modal logic is $\Pi_1$-persistent. This approach leads to a new relational semantics for weak positive modal logic, for which we prove an analogue of Sahlqvist correspondence result

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

A Coalgebraic Approach to Dualities for Neighborhood Frames

We develop a uniform coalgebraic approach to J\'onsson-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 J\'{o}nsson-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

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