Ana Sokolova University of Salzburg joint work with Bart Jacobs Radboud University Nijmegen Coalgebra Day 1132008 RUN 1 TexPoint fonts used in EMF Read the TexPoint manual before you delete this box ID: 742682
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Exemplaric Expressivityof Modal Logics
Ana Sokolova University of Salzburgjoint work withBart Jacobs Radboud University Nijmegen
Coalgebra Day, 11-3-2008, RUN
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TexPoint fonts used in EMF.
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OutlineCoalgebra Day, 11-3-2008, RUN
Expressivity: logical equivalence = behavioral equivalence For three examples:Transition systemsMarkov chains
Markov processes
Boolean modal logic
Finite conjunctions probabilistic modal logic
2Slide3
Via dual adjunctions
Coalgebra Day, 11-3-2008, RUN3
Predicates on spaces
Theories on models
Behaviour
(
coalgebras
)
Logics
(algebras)
Dual Slide4
Logical set-upCoalgebra Day, 11-3-2008, RUN
4
If L has an initial algebra of formulas
A natural transformation
gives interpretationsSlide5
Logical equivalencebehavioural equivalence
Coalgebra Day, 11-3-2008, RUN5 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: expressivitySlide6
Expressivity
Bijective correspondence between andCoalgebra Day, 11-3-2008, RUN
6
If
and the transpose of the interpretation is
componentwise
mono, then expressivity
.
Factorisation system on
with diagonal fill-in
Slide7
Sets vs. Boolean algebras
Coalgebra Day, 11-3-2008, RUN7
contravariant powerset
Boolean algebras
ultrafiltersSlide8
Sets vs. meet
semilatticesCoalgebra Day, 11-3-2008, RUN8
meet semilattices
contravariant powerset
filtersSlide9
Measure spaces vs. meet semilattices
Coalgebra Day, 11-3-2008, RUN9
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 bySlide10
Behaviour via coalgebrasCoalgebra Day, 11-3-2008, RUN
10
Transition systems
Markov chains
Markov processes
Giry monadSlide11
The Giry monadCoalgebra Day, 11-3-2008, RUN
11
subprobability measures
countable union of pairwise disjoint
generated by
the smallest making
measurableSlide12
Coalgebra Day, 11-3-2008, RUNLogic for transition systems
Modal operator12
models of boolean logic with fin.meet preserving modal operators
L = GV
V -
forgetful
expressivitySlide13
Coalgebra Day, 11-3-2008, RUNLogic for Markov chains
Probabilistic modalities13
models of logic with fin.conj. and
monotone modal operators
K = HV
V -
forgetful
expressivitySlide14
Coalgebra Day, 11-3-2008, RUNLogic for Markov processes
General probabilistic modalities14
models of logic with fin.conj. and
monotone modal operators
the same
K
expressivitySlide15
Discrete to indescrete
The adjunctions are related:Coalgebra Day, 11-3-2008, RUN15
discrete
measure
space
forgetful
functorSlide16
Discrete to indiscreteMarkov chains as Markov processes
Coalgebra Day, 11-3-2008, RUN16Slide17
Discrete to indiscreteCoalgebra Day, 11-3-2008, RUN
17
Slide18
ConclusionsCoalgebra Day, 11-3-2008, RUN
18 Expressivity For three examples:Transition systemsMarkov chainsMarkov processes
Boolean modal logic
Finite conjunctions probabilistic modal logic
in the setting of dual adjunctions !