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Exemplaric Expressivityof Modal Logics

Ana Sokolova University of Salzburg

joint work with

Bart Jacobs Radboud University Nijmegen

FM Group Seminar, TU/e, Eindhoven 9.2.9

FM Group Seminar, TU/e, Eindhoven 9.2.9 2

It is about... Coalgebras

Modal logics

Behaviour

functor !Generalized transition systems

In a studied setting of

dual adjunctions


Outline Expressivity:

logical equivalence = behavioral equivalence

For four examples:

1. Transition systems2. Markov chains3. Multitransition systems4. Markov processes

Boolean modal logic

Finite conjunctions „valuation“ modal logic


Via dual adjunctionsPredicates on

spaces

Theories on

modelsBehaviour

(coalgebras) Logics(algebras)

Dual


Logical set-up

If L has an initial algebra of formulas

A natural transformation

gives interpretations


Logical equivalencebehavioural equivalence

The interpretation map yields a theory map

which defines logical equivalence

behavioural equivalence is given by for some coalgebra

homomorphismsh1 and h2

Aim: expressivity


Expressivity Bijective correspondence between

and

If and the transpose of the interpretation

is componentwise abstract mono, then expressivity.

Factorisation system on

with diagonal fill-in


Sets vs. Boolean algebras contravariant

powerset

Boolean algebra

s

ultrafilters


Sets vs. meet semilattices

meet semilattice

s

contravariant powerset

filters


Measure spaces vs. meet semilattices

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


Behaviour via coalgebras Transition systems

Markov chains/Multitransition systems

Markov processes

Giry monad


What do they have in common?They are instances of the same functor

Not cancellativ

e


The Giry monad

subprobability measures

countable union of pairwise disjoint

generated by

the smallest making

measurable


Logic for transition systems

Modal operator

models of boolean

logic with fin.meet

preserving modal

operators

L = GVV - forgetful

expressivity


Logic for Markov chains Probabilistic modalities

models of logic with fin.conj.

andmonotone

modal operators

K = HVV - forgetful

expressivity


Logic for multitransition ... Graded modal logic modalities


andmonotone

modal operators

K = HVV - forgetful

expressivity


Logic for Markov processes

General probabilistic modalities


andmonotone

modal operators

the same K

expressivity


Discrete to indiscrete The adjunctions are related:

discrete measure

space

forgetfulfunctor


Discrete to indiscrete Markov chains as Markov processes


Discrete to indiscrete So we can translate chains into

processes

Behavioural equivalence

coincides

Also the logic

theories do


Or directly ...


Conclusions Expressivity

For four examples:

1. Transition systems2. Multitransition systems3. Markov chains4. Markov processes

Boolean modal logic

Finite conjunctions ``valuation´´ modal logic

in the setting of dual adjunctions !