101 research outputs found
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
Topology, randomness and noise in process calculus
Formal models of communicating and concurrent systems are one of the most important topics in formal methods, and process calculus is one of the most successful formal models of communicating and concurrent systems. In the previous works, the author systematically studied topology in process calculus, probabilistic process calculus and pi-calculus with noisy channels in order to describe approximate behaviors of communicating and concurrent systems as well as randomness and noise in them. This article is a brief survey of these works. © Higher Education Press 2007
Coalgebras of topological types
In This work, we focus on developing the basic theory of coalgebras over the category Top (the category of topological spaces with continuous maps). We argue that, besides Set, the category Top is an interesting base category for coalgebras. We study some endofunctors on Top, in particular, Vietoris functor and P-Vietoris Functor (where P is a set of propositional letters) that due to
Hofmann et. al. [42] can be considered as the topological versions of the powerset functor and P-Kripke functor, respectively. We define the notion of compact Kripke structures and we prove that Kripke homomorphisms preserve compactness. Our definition of "compact Kripke structure" coincides with the notion of "modally saturated structures" introduced in Fine [27]. We prove that the class of compact Kripke structures has Hennessy-Milner property. As a consequence we show that in this class of Kripke structures, bihavioral equivalence, modal equivalence and Kripke bisimilarity all coincide.Furthermore, we generalize the notion of descriptive structures defined in Venema et. al. [11] by introducing a notion Vietoris models. We identify Vietoris models as coalgebras for the P-Vietoris functor on the category Top. One can see that each compact Kripke model can be modified to a Vietoris model. This yields an adjunction between the category of Vietoris structures (VS) and the category of compact Kripke structurs (CKS). Moreover, we will prove that the category of Vietoris models has a terminal object. We study the concept of a Vietoris bisimulation between Vietoris models, and we will prove that the closure of a Kripke bisimulation between underlying Kripke models of two Vietoris models is a Vietoris bisimulation. In the end, it will be shown that in the class of Vietoris models, Vietoris bisimilarity, bihavioral equivalence, modal equivalence, all coincide
Abstractions of Stochastic Hybrid Systems
In this paper we define a stochastic bisimulation concept for a very general class of stochastic hybrid systems, which subsumes most classes of stochastic hybrid systems. The definition of this bisimulation builds on the concept of zigzag morphism defined for strong Markov processes.
The main result is that this stochastic bisimulation is indeed an equivalence relation. The secondary result is that this bisimulation relation for the stochastic hybrid system models used in this paper implies the same
kind of bisimulation for their continuous parts and respectively for their jumping structures
Coalgebraic Geometric Logic
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 T on some full subcategory of the category Top of topological spaces and continuous functions. 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. Furthermore, 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
Logical Relations for Monadic Types
Logical relations and their generalizations are a fundamental tool in proving
properties of lambda-calculi, e.g., yielding sound principles for observational
equivalence. We propose a natural notion of logical relations able to deal with
the monadic types of Moggi's computational lambda-calculus. The treatment is
categorical, and is based on notions of subsconing, mono factorization systems,
and monad morphisms. Our approach has a number of interesting applications,
including cases for lambda-calculi with non-determinism (where being in logical
relation means being bisimilar), dynamic name creation, and probabilistic
systems.Comment: 83 page
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