the microcosm principle and concurrency in coalgebra
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The Microcosm Principle and Concurrency in Coalgebra. Ichiro Hasuo Kyoto University, Japan PRESTO Promotion Program, Japan Bart Jacobs Radboud University Nijmegen, NL Technical University Eindhoven, NL. Ana Sokolova University of Salzburg, Austria. - PowerPoint PPT PresentationTRANSCRIPT
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The Microcosm Principle and Concurrency in Coalgebra
Ana Sokolova University of Salzburg, Austria
Ichiro Hasuo Kyoto University, Japan PRESTO Promotion Program, Japan
Bart Jacobs Radboud University Nijmegen, NL Technical University Eindhoven, NL
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Categorical theory of state-based systems
A short review of coalgebra/coinduction
c a t e g o r i c a l l y
system coalgebra
behavior-preserving
map
morphism of coalgebras
behavior coinduction (via final coalgebra)
in Sets : bisimilarityin Kleisli: trace semantics
[Hasuo,Jacobs,Sokolova LMCS´07]
Theory of coalgebra
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Coalgebra example – LTS
C = Sets, F = Pfin( x _)
F-coalgebra = LTS
coalgebra c: X FX
states X = { 1,2,3,4} labels = {a,b}
transitions c(1) = {(a,2), (b,3)}, c(2) = , …
1
2 3
4
a
a
a
b
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is everywhere
computer networks multi-core processors
modular, component-based design of complex systems
is hard to get right
exponentially growing complexity need for a compositional verification
Concurrency C || Drunning C and D in parallel
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Behavior of C || Dis determined by
behavior of C and behavior of D
Compositionalityaids compositional
verification
Conventional presentation
behav iora l equiva lenceo bisimilarityo trace equivalenceo ...
„bisimilarity is a congruence“
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• Final coalgebra semantics asprocess semantics.
• Coalgebraic compositionality
Compositionality in coalgebra
|| : CoalgF x CoalgF CoalgF composing coalgebras/systems
|| : Z x Z Z composing behavior
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the microcosm principlewith
o operations binary ||
o equationse.g. assoc. of ||
algebraic theoryX 2 C outer interpretation
inner interpretation
Nested algebraic structures:
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We name this principle the microcosm principle, after the theory, common in pre-modern correlative cosmologies, that every feature of the microcosm (e.g. the human soul) corresponds to some feature of the macrocosm.
John Baez & James DolanHigher-Dimensional Algebra III:
n-Categories and the Algebra of OpetopesAdv. Math. 1998
Microcosm in macrocosm
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The microcosm principle: you may have seen it
monoid in a monoidal category
inner depends on outer
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Formalizing the microcosm principle
What do we mean by “microcosm principle”?
mathematical definition of such nested models?
algebraic theory as Lawvere theory
outer modelas prod.-pres. functor
inner modelas lax natural trans.
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Outline
microcosm for concurrency(|| and ||)
2-categorical formulation
generic compositionality
theorem
for arbitrary algebraic
theory
parallel composition
via sync nat. trans.
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Parallel composition of coalgebrasvia sync
Part 1
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Parallel composition of coalgebras
Theorem
: CoalgF x CoalgF CoalgF
: C x C C
liftingF with
syncsyncX,Y : FX FY F(X Y)
Aim
bifunctor CoalgF x CoalgF CoalgF
usually denoted by (tensor)
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Parallel composition via sync
on the base category
different sync
different
syncX,Y : FX FY F(X Y)
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CSP-style (Hoare)
CCS-style (Milner)Assuming
Examples of sync : FX FY F(X Y)
C = Sets, F = Pfin( x _)F-coalgebra = LTS
x : Sets x Sets Sets
: CoalgF x CoalgF CoalgF
liftingF with
sync
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|| “composition of states/behavior” arises by coinduction
Inner compositionAim
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for by and || by
Compositionality theoremTheorem
: CoalgF x CoalgF CoalgF
: C x C Clifting
F with sync
Assumptions: ,sync, final exists
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Equational properties
associative: CoalgF x CoalgF CoalgF
associative
: C x C C
lifting
F with “associative“ sync
commutativity?
arbitrary algebraic theory?
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2-categorical formulation ofthe microcosm principle
Part 2
for arbitrary algebraic theory
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Lawvere theory L
A Lawvere theory L is a small categoryo with objects natural numbers o that has finite products
Definition
a category representing an algebraic theory
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Lawvere theorya l ge b ra i c t h e o r y a s c a te g o r y L
operations as arrows
equations as commuting diagrams
m (binary)e (nullary)
assoc. of munit law
other arrows:o projectionso composed terms
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Models for a Lawvere theory L
a set with L-structure, L-setStandard: set-theoretic model
(product-preserving)
binary op. on X
X 2 C what about nested models?
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Outer model: L-category
o a category with L-structure, L-categoryouter model
(product-preserving)
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Given an L-category C, an L-object X in it is a lax natural transformationcompatible with products.
Inner model: L-objectDefinition
components lax naturality
X: carrier obj.
inner alg. str. by
mediating 2-cells
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Resultslax L-functor = F with sync
CoalgF is an L-category
L-category C
liftinglax L-
functor F
The final object of an L-category is an L-object
lax L-functor?
lax natur. trans.
lax naturality?
Theorem
TheoremEquations are built in!
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In a situation
Compositionality theorem
The final object of an L-category is an
L-object
Assumptions: C is an L-category, F is lax L-functor, final exists
by coinduction
The behaviour functor beh is a strict L-functor
CoalgF is an L-category
L-category C
liftinglax L-
functor F
Theorem
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Related and future work: bialgebras
Bialgebraic structures
[Turi-Plotkin, Bartels, Klin, …] algebraic structures on coalgebras
In the current work
Equations, not only operations ,are an integral partThe algebraic structures are nested, higher dimensional
Missing
Full GSOS expressivity
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Conclusion
o operations o equations
algebraic theoryX 2 C
inner interpretation
outer interpretation
Microcosm principle
2-categorical formulation
Concurrency in coalgebra as motivation and CS
example