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

Generalized Vietoris Bisimulations

We introduce and study bisimulations for coalgebras on Stone spaces [14]. Our notion of bisimulation is sound and complete for behavioural equivalence, and generalizes Vietoris bisimulations [4]. The main result of our paper is that bisimulation for a $\mathbf{Stone}$ coalgebra is the topological closure of bisimulation for the underlying $\mathbf{Set}$ coalgebra

Model-theoretic characterization of intuitionistic propositional formulas

Notions of k-asimulation and asimulation are introduced as asymmetric counterparts to k-bisimulation and bisimulation, respectively. It is proved that a first-order formula is equivalent to a standard translation of an intuitionistic propositional formula iff it is invariant with respect to k-asimulations for some k, and then that a first-order formula is equivalent to a standard translation of an intuitionistic propositional formula iff it is invariant with respect to asimulations. Finally, it is proved that a first-order formula is intuitionistically equivalent to a standard translation of an intuitionistic propositional formula iff it is invariant with respect to asimulations between intuitionistic models.Comment: 16 pages, 0 figures. arXiv admin note: substantial text overlap with arXiv:1202.119

Monadic Second-Order Logic and Bisimulation Invariance for Coalgebras

Generalizing standard monadic second-order logic for Kripke models, we introduce monadic second-order logic interpreted over coalgebras for an arbitrary set functor. Similar to well-known results for monadic second-order logic over trees, we provide a translation of this logic into a class of automata, relative to the class of coalgebras that admit a tree-like supporting Kripke frame. We then consider invariance under behavioral equivalence of formulas; more in particular, we investigate whether the coalgebraic mu-calculus is the bisimulation-invariant fragment of monadic second-order logic. Building on recent results by the third author we show that in order to provide such a coalgebraic generalization of the Janin-Walukiewicz Theorem, it suffices to find what we call an adequate uniform construction for the functor. As applications of this result we obtain a partly new proof of the Janin-Walukiewicz Theorem, and bisimulation invariance results for the bag functor (graded modal logic) and all exponential polynomial functors. Finally, we consider in some detail the monotone neighborhood functor, which provides coalgebraic semantics for monotone modal logic. It turns out that there is no adequate uniform construction for this functor, whence the automata-theoretic approach towards bisimulation invariance does not apply directly. This problem can be overcome if we consider global bisimulations between neighborhood models: one of our main technical results provides a characterization of the monotone modal mu-calculus extended with the global modalities, as the fragment of monadic second-order logic for the monotone neighborhood functor that is invariant for global bisimulations

The Second Order Traffic Fine: Temporal Reasoning in European Transport Regulations

We argue that European transport regulations can be formalized within the Sigma^1_1 fragment of monadic second order logic, and possibly weaker fragments including linear temporal logic. We consider several articles in the regulation to verify these claims

Bisimulation for Weakly Expressive Coalgebraic Modal Logics

Research on the expressiveness of coalgebraic modal logics with respect to semantic equivalence notions has so far focused mainly on finding logics that are able to distinguish states that are not behaviourally equivalent (such logics are said to be expressive). In other words, the notion of behavioural equivalence is taken as the starting point, and the expressiveness of the logic is evaluated against it. However, for some applications, modal logics that are not expressive are of independent interest. Such an example is given by contingency logic. We can now turn the question of expressiveness around and ask, given a modal logic, what is a suitable notion of semantic equivalence? In this paper, we propose a notion of Lambda-bisimulation which is parametric in a collection Lambda of predicate liftings. We study the basic properties of Lambda-bisimilarity, and prove as our main result a Hennessy-Milner style theorem, which shows that (for finitary functors) Lambda-bisimilarity exactly matches the expressiveness of the coalgebraic modal logic arising from Lambda

Coalgebraic Fuzzy geometric logic

The paper aims to develop a framework for coalgebraic fuzzy geometric logic by adding modalities to the language of fuzzy geometric logic. Using the methods of coalgebra, the modal operators are introduced in the language of fuzzy geometric logic. To define the modal operators, we introduce a notion of fuzzy-open predicate lifting. Based on coalgebras for an endofunctor $T$ on the category $\textbf{Fuzzy-Top}$ of fuzzy topological spaces and fuzzy continuous maps, we build models for the coalgebraic fuzzy geometric logic. Bisimulations for the defined models are discussed in this work