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Exemplaric Expressivityof Modal Logics

Ana Sokolova University of Salzburg

joint work with

Bart Jacobs Radboud University Nijmegen

Dagstuhl Seminar on Coalgebraic Logics, 7.12.9

Dagstuhl Seminar on Coalgebraic Logics, 7.12.9 2

It is about... Coalgebras

Modal logics

Behaviour

functor !

In a studied setting of

dual adjunctions

Dagstuhl Seminar on Coalgebraic Logics, 7.12.9 3

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

Dagstuhl Seminar on Coalgebraic Logics, 7.12.9 4

Via dual adjunctionsPredicates on

spaces

Theories on

modelsBehaviour

(coalgebras) Logics(algebras)

Dual

Dagstuhl Seminar on Coalgebraic Logics, 7.12.9 5

Logical set-up

If L has an initial algebra of formulas

A natural transformation

gives interpretations

Dagstuhl Seminar on Coalgebraic Logics, 7.12.9 6

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

Dagstuhl Seminar on Coalgebraic Logics, 7.12.9 7

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

Dagstuhl Seminar on Coalgebraic Logics, 7.12.9 8

Sets vs. Boolean algebras contravariant

powerset

Boolean algebra

s

ultrafilters

Dagstuhl Seminar on Coalgebraic Logics, 7.12.9 9

Sets vs. meet semilattices

meet semilattice

s

contravariant powerset

filters

Dagstuhl Seminar on Coalgebraic Logics, 7.12.9 10

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

Dagstuhl Seminar on Coalgebraic Logics, 7.12.9 11

Behaviour via coalgebras Transition systems

Markov chains/Multitransition systems

Markov processes

Giry monad

Dagstuhl Seminar on Coalgebraic Logics, 7.12.9 12

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

Not cancellativ

e

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The Giry monad

subprobability measures

countable union of pairwise disjoint

generated by

the smallest making

measurable

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Logic for transition systems

Modal operator

models of boolean

logic with fin.meet

preserving modal

operators

L = GVV - forgetful

expressivity

Dagstuhl Seminar on Coalgebraic Logics, 7.12.9 15

Logic for Markov chains Probabilistic modalities

models of logic with fin.conj.

andmonotone

modal operators

K = HVV - forgetful

expressivity

Dagstuhl Seminar on Coalgebraic Logics, 7.12.9 16

Logic for multitransition ... Graded modal logic modalities

models of logic with fin.conj.

andmonotone

modal operators

K = HVV - forgetful

expressivity

Dagstuhl Seminar on Coalgebraic Logics, 7.12.9 17

Logic for Markov processes

General probabilistic modalities

models of logic with fin.conj.

andmonotone

modal operators

the same K

expressivity

Dagstuhl Seminar on Coalgebraic Logics, 7.12.9 18

Discrete to indiscrete The adjunctions are related:

discrete measure

space

forgetfulfunctor

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Discrete to indiscrete Markov chains as Markov processes

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Discrete to indiscrete So we can translate chains into

processes

Behavioural equivalence

coincides

Also the logic

theories do

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Or directly ...

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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 !