Ana Sokolova University of Salzburg joint work with Bart Jacobs Radboud University Nijmegen Coalgebra Day, , 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

Coalgebra Day, , 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

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

Coalgebra Day, , RUN 4  If L has an initial algebra of formulas  A natural transformation gives interpretations

Coalgebra Day, , 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

 Bijective correspondence between and Coalgebra Day, , 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

Coalgebra Day, , RUN 7 contravariant powerset Boolean algebras ultrafilters

Coalgebra Day, , RUN 8 meet semilattices contravariant powerset filters

Coalgebra Day, , 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

Coalgebra Day, , RUN 10  Transition systems  Markov chains  Markov processes Giry monad

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

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

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

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

 The adjunctions are related: Coalgebra Day, , RUN 15 discrete measure space discrete measure space forgetful functor

 Markov chains as Markov processes Coalgebra Day, , RUN 16

Coalgebra Day, , RUN 17

Coalgebra Day, , 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 !