1. Ana Sokolova University of Salzburg joint work with Bart Jacobs Radboud University Nijmegen Dagstuhl Seminar on Coalgebraic Logics, 7.12.9 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.  Coalgebras  Modal logics Dagstuhl Seminar on Coalgebraic Logics, 7.12.9 2 Behaviour functor ! In a studied setting of dual adjunctions

3. Dagstuhl Seminar on Coalgebraic Logics, 7.12.9  Expressivity: logical equivalence = behavioral equivalence  For four examples: 1.Transition systems 2.Markov chains 3.Multitransition systems 4.Markov processes Boolean modal logic Finite conjunctions „valuation“ modal logic 3

4. Dagstuhl Seminar on Coalgebraic Logics, 7.12.9 4 Predicates on spaces Theories on models Behaviour (coalgebras) Behaviour (coalgebras) Logics (algebras) Logics (algebras) Dual

5. Dagstuhl Seminar on Coalgebraic Logics, 7.12.9 5  If L has an initial algebra of formulas  A natural transformation gives interpretations

6. Dagstuhl Seminar on Coalgebraic Logics, 7.12.9 6  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

7.  Bijective correspondence between and Dagstuhl Seminar on Coalgebraic Logics, 7.12.9 7 If and the transpose of the interpretation is componentwise abstract mono, then expressivity. Factorisation system on with diagonal fill-in Factorisation system on with diagonal fill-in

8. Dagstuhl Seminar on Coalgebraic Logics, 7.12.9 8 contravariant powerset Boolean algebras ultrafilters

9. Dagstuhl Seminar on Coalgebraic Logics, 7.12.9 9 meet semilattices contravariant powerset filters

10. Dagstuhl Seminar on Coalgebraic Logics, 7.12.9 10 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

11. Dagstuhl Seminar on Coalgebraic Logics, 7.12.9 11  Transition systems  Markov chains/Multitransition systems  Markov processes Giry monad

12. Dagstuhl Seminar on Coalgebraic Logics, 7.12.9 12 They are instances of the same functor Not cancellative

13. Dagstuhl Seminar on Coalgebraic Logics, 7.12.9 13 subprobability measures countable union of pairwise disjoint generated by generated by the smallest making measurable

14. Dagstuhl Seminar on Coalgebraic Logics, 7.12.9  Modal operator 14 models of boolean logic with fin.meet preserving modal operators L = GV V - forgetful expressivity

15. Dagstuhl Seminar on Coalgebraic Logics, 7.12.9  Probabilistic modalities 15 models of logic with fin.conj. and monotone modal operators K = HV V - forgetful expressivity

16. Dagstuhl Seminar on Coalgebraic Logics, 7.12.9  Graded modal logic modalities 16 models of logic with fin.conj. and monotone modal operators K = HV V - forgetful expressivity

17. Dagstuhl Seminar on Coalgebraic Logics, 7.12.9  General probabilistic modalities 17 models of logic with fin.conj. and monotone modal operators the same K expressivity

18.  The adjunctions are related: Dagstuhl Seminar on Coalgebraic Logics, 7.12.9 18 discrete measure space discrete measure space forgetful functor

19.  Markov chains as Markov processes Dagstuhl Seminar on Coalgebraic Logics, 7.12.9 19

20.  So we can translate chains into processes Dagstuhl Seminar on Coalgebraic Logics, 7.12.9 20 Behavioural equivalence coincides Also the logic theories do

21. Dagstuhl Seminar on Coalgebraic Logics, 7.12.9 21

22. Dagstuhl Seminar on Coalgebraic Logics, 7.12.9 22  Expressivity  For four examples: 1.Transition systems 2.Multitransition systems 3.Markov chains 4.Markov processes Boolean modal logic Finite conjunctions ``valuation´´ modal logic in the setting of dual adjunctions !