1. Ana Sokolova University of Salzburg joint work with Bart Jacobs Radboud University Nijmegen Coalgebra Day, 11-3-2008, RUN 1 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A A A A A A

2. Coalgebra Day, 11-3-2008, RUN  Expressivity: logical equivalence = behavioral equivalence  For three examples: 1.Transition systems 2.Markov chains 3.Markov processes Boolean modal logic Finite conjunctions probabilistic modal logic 2

3. Coalgebra Day, 11-3-2008, RUN 3 Predicates on spaces Theories on models Behaviour (coalgebras) Behaviour (coalgebras) Logics (algebras) Logics (algebras) Dual

4. Coalgebra Day, 11-3-2008, RUN 4  If L has an initial algebra of formulas  A natural transformation gives interpretations

5. Coalgebra Day, 11-3-2008, RUN 5  The interpretation map yields a theory map  which defines logical equivalence  behavioural equivalence is given by for some coalgebra homomorphisms h 1 and h 2 Aim: expressivity

6.  Bijective correspondence between and Coalgebra Day, 11-3-2008, RUN 6 If and the transpose of the interpretation is componentwise mono, then expressivity. Factorisation system on with diagonal fill-in Factorisation system on with diagonal fill-in

7. Coalgebra Day, 11-3-2008, RUN 7 contravariant powerset Boolean algebras ultrafilters

8. Coalgebra Day, 11-3-2008, RUN 8 meet semilattices contravariant powerset filters

9. Coalgebra Day, 11-3-2008, RUN 9 measure spaces ¾ -algebra: “measurable” subsets closed under empty, complement, countable union maps a measure space to its ¾ -algebra filters on A with ¾ -algebra generated by

10. Coalgebra Day, 11-3-2008, RUN 10  Transition systems  Markov chains  Markov processes Giry monad

11. Coalgebra Day, 11-3-2008, RUN 11 subprobability measures countable union of pairwise disjoint generated by generated by the smallest making measurable

12. Coalgebra Day, 11-3-2008, RUN  Modal operator 12 models of boolean logic with fin.meet preserving modal operators L = GV V - forgetful expressivity

13. Coalgebra Day, 11-3-2008, RUN  Probabilistic modalities 13 models of logic with fin.conj. and monotone modal operators K = HV V - forgetful expressivity

14. Coalgebra Day, 11-3-2008, RUN  General probabilistic modalities 14 models of logic with fin.conj. and monotone modal operators the same K expressivity

15.  The adjunctions are related: Coalgebra Day, 11-3-2008, RUN 15 discrete measure space discrete measure space forgetful functor

16.  Markov chains as Markov processes Coalgebra Day, 11-3-2008, RUN 16

17. Coalgebra Day, 11-3-2008, RUN 17

18. Coalgebra Day, 11-3-2008, RUN 18  Expressivity  For three examples: 1.Transition systems 2.Markov chains 3.Markov processes Boolean modal logic Finite conjunctions probabilistic modal logic in the setting of dual adjunctions !