exemplaric expressivity of modal logics

1

Exemplaric Expressivityof Modal Logics

Ana Sokolova University of Salzburg

joint work with

Bart Jacobs Radboud University Nijmegen

Coalgebra Day, 11-3-2008, RUN

Coalgebra Day, 11-3-2008, RUN 2

Outline Expressivity:

logical equivalence = behavioral equivalence

For three examples:

1. Transition systems2. Markov chains3. Markov processes

Boolean modal logic

Finite conjunctions probabilistic modal logic

Coalgebra Day, 11-3-2008, RUN 3

Via dual adjunctionsPredicates on

spaces

Theories on

modelsBehaviour

(coalgebras) Logics(algebras)

Dual

Coalgebra Day, 11-3-2008, RUN 4

Logical set-up

If L has an initial algebra of formulas

A natural transformation

gives interpretations

Coalgebra Day, 11-3-2008, RUN 5

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

Coalgebra Day, 11-3-2008, RUN 6

Expressivity Bijective correspondence between

and

If and the transpose of the interpretation

is componentwise mono, then expressivity.Factorisation system on

with diagonal fill-in

Coalgebra Day, 11-3-2008, RUN 7

Sets vs. Boolean algebras contravariant

powerset

Boolean algebra

s

ultrafilters

Coalgebra Day, 11-3-2008, RUN 8

Sets vs. meet semilattices

meet semilattice

s

contravariant powerset

filters

Coalgebra Day, 11-3-2008, RUN 9

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

Coalgebra Day, 11-3-2008, RUN 10

Behaviour via coalgebras Transition systems

Markov chains

Markov processes

Giry monad

Coalgebra Day, 11-3-2008, RUN 11

The Giry monad

subprobability measures

countable union of pairwise disjoint

generated by

the smallest making

measurable

Coalgebra Day, 11-3-2008, RUN 12

Logic for transition systems

Modal operator

models of boolean

logic with fin.meet

preserving modal

operators

L = GVV - forgetful

expressivity

Coalgebra Day, 11-3-2008, RUN 13

Logic for Markov chains Probabilistic modalities

models of logic with fin.conj.

andmonotone

modal operators

K = HVV - forgetful

expressivity

Coalgebra Day, 11-3-2008, RUN 14

Logic for Markov processes

General probabilistic modalities

models of logic with fin.conj.

andmonotone

modal operators

the same K

expressivity

Coalgebra Day, 11-3-2008, RUN 15

Discrete to indescrete The adjunctions are related:

discrete measure

space

forgetfulfunctor

Coalgebra Day, 11-3-2008, RUN 16

Discrete to indiscrete Markov chains as Markov processes

Coalgebra Day, 11-3-2008, RUN 17

Discrete to indiscrete

Coalgebra Day, 11-3-2008, RUN 18

Conclusions Expressivity

For three examples:

1. Transition systems2. Markov chains3. Markov processes

Boolean modal logic

Finite conjunctions probabilistic modal logic

in the setting of dual adjunctions !