Coalgebraic logic over general measurable spaces – a survey

Abstract

Link to this article: http://journals.cambridge.org/abstract_S0960129510000526How to cite this article: ERNST-ERICH DOBERKAT and CHRISTOPH SCHUBERT (2011). Coalgebraic logic over general measurable spaces -a survey.In this survey we discuss the generalisation of stochastic Kripke models for general modal logics through predicate liftings for functors over general measurable spaces. We derive results on expressivity and show that selection arguments allow us to incorporate the discussion of bisimilarity, provided the underlying spaces are assumed to be standard Borel.

A stochastic interpretation of propositional dynamic logic: expressivity

“…Third, an alternative to the proof presented would have been through a coalgebraic approach. This would work by defining for each modal operator [n] q a suitable predicate lifting, and by investigating the corresponding logic over coalgebras the functor of which sends the measurable space S to the measurable space 6 (S) with dynamics s i-)-(K7i(s)) ne ci(i]), see [9,Section 4.4] or [12]. Balancing the-considerableeffort of doing so against modifying a construction which works with conventional methods resulted in the proof presented above.…”

“…The technique of generating an interpretation from the model for a fragment seems to be interesting in itself; another application can be found in [12]. The question arises whether it can be applied to dynamic coalgebraic logics, i.e., to dynamic logics in which the modal operators are given through predicate liftings.…”

A Stochastic Interpretation of Propositional Dynamic Logic: Expressivity

“…Thus B is a contravariant functor. This has been discussed extensively in [10,7]. Given a morphism f :…”

“…Thus B is a contravariant functor. This has been discussed extensively in [10,7]. Given a morphism f : S → T in category M, we obtain a morphism S (f ) :…”

“…The models are bisimilar iff we can find a mediating model for them, and they are HM equivalent iff we can find for each state in K 1 another state in K 2 which satisfies exactly the same formulas, and vice versa. The reader is referred to [14,4,6,10] for an extensive discussion stressing different angles. Kripke models have been defined over the category of measurable spaces, the discussion of bisimilarity, however, requires some differentiation with respect to the base category for the state space.…”