CategoricalCategorical theory of state-based systemsCategoricalCategorical theory of state-based systemsin Sets : bisimilarityin Kleisli: trace semantics
[Hasuo,Jacobs,Sokolova LMCS´07]
in Sets : bisimilarityin Kleisli: trace semantics
[Hasuo,Jacobs,Sokolova LMCS´07]
aids compositional compositional verificationverification
aids compositional compositional verificationverification
„„bisimilarity is a bisimilarity is a congruence“congruence“
„„bisimilarity is a bisimilarity is a congruence“congruence“
Final coalgebra semantics as“process semantics”.
“Coalgebraic compositionality”
|| : CoalgF x CoalgF CoalgF o composing coalgebras/systems || : CoalgF x CoalgF CoalgF o composing coalgebras/systems
|| : Z x Z Z o composing behavior
|| : Z x Z Z o composing behavior
operation on operation on
NFAsNFAs
operation on operation on
regular regular languageslanguages
The same “algebraic theory”◦ operations (binary ||)◦ equations (e.g. associativity of ||)
is carried by◦ the category CoalgF and◦ its object Z 2 CoalgFin a nested manner!
“Microcosm principle” (Baez & Dolan)
with
o operations operations binary ||
o equationsequationse.g. assoc. of ||
XX 22 CC XX 22 CC
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
You may have seen it◦ “a monoid is in a monoidal category”
inner depends on
outer
inner depends on
outer
Answer
concurrency/concurrency/compositionalitycompositionality
mathematicsmathematics
microcosm for microcosm for concurrencyconcurrency((|| || and and ||||))
microcosm for microcosm for concurrencyconcurrency((|| || and and ||||))
parallel compositionparallel compositionvia via syncsync nat. trans. nat. trans.
parallel compositionparallel compositionvia via syncsync nat. trans. nat. trans.
2-categorical formulation2-categorical formulation
for arbitrary algebraic
theory
for arbitrary algebraic
theory
Parallel composition of coalgebrasvia
syncX,Y : FX FY F(X Y)
If the base category C has a tensor
: C x C C and F : C C comes with natural
transformationsyncX,Y : FX FY F(X Y)
then we have : CoalgF x CoalgF CoalgF
bifunctor CoalgF x CoalgF CoalgF
usually denoted by (tensor)
Theorem : CoalgF x CoalgF CoalgF : CoalgF x CoalgF CoalgF
: C x C C : C x C C
liftinglifting
????
????F
X Y
(X Y)
FX FYc d
syncX,Y
on base category
on base category
different different syncsync
different different syncsync
different different
different different
CSP-style (Hoare)
CCS-style (Milner)Assuming
C = Sets, F = Pfin( x _)C = Sets, F = Pfin( x _)
F-coalgebra = LTSF-coalgebra = LTS
x : Sets x Sets Sets x : Sets x Sets Sets
|| “composition of states/behavior” arises by coinduction
C has tensor F has syncX,Y : FX FY F(X Y)there is a final coalgebra Z FZ
C has tensor F has syncX,Y : FX FY F(X Y)there is a final coalgebra Z FZ
When is
: CoalgF x CoalgF CoalgF
associative?associative? Answer When
◦ : C x C C is associative, and◦sync is “associative”
2-categorical formulation ofthe microcosm principle
examples
omonoid in monoidal category
ofinal coalg. in CoalgF with
oreg. lang. vs. NFAs
examples
omonoid in monoidal category
ofinal coalg. in CoalgF with
oreg. lang. vs. NFAs
A Lawvere theoryLawvere theory LL is a small category s.t.
o L’s objects are natural numbers
o L has finite products
m (binary)e (nullary)m (binary)e (nullary)
assoc. of munit law assoc. of munit law
other arrows:other arrows:o projectionso composed terms
other arrows:other arrows:o projectionso composed terms
o a set with L-structure LL-set-set(product-preserving)
XX 22 CC XX 22 CC what about nested models?
o a categorycategory with L-structure LL-category-category(product-preserving)
NB. our focus is on strictstrict alg. structuresNB. our focus is on strictstrict alg. structures
Given an L-category C,
an LL-object-object X in it is a laxlax natural transformationcompatible with products.
X: carrier obj.
C: L-category F: C C, lax L-functor CoalgF is an L-category
C: L-category F: C C, lax L-functor CoalgF is an L-category
lax L-functor?
lax lax natur. trans..
lax L-functor?
lax lax natur. trans..
lax naturality?lax naturality?lax naturality?lax naturality?
...
GeneralizesGeneralizes
C: L-category Z 2 C , final object
Z is an L-object
C: L-category Z 2 C , final object
Z is an L-object
GeneralizesGeneralizes
C is an L-category F : C C is a lax L-functor there is a final coalgebra Z FZ
C is an L-category F : C C is a lax L-functor there is a final coalgebra Z FZ
by coinduction
subsumessubsumes
associativeassociative
: CoalgF x CoalgF CoalgF
associativeassociative
: CoalgF x CoalgF CoalgF
associativeassociative
: C x C C
associativeassociative
: C x C C
liftinglifting
equations are built-in in L◦ as
how about „assoc“ of sync?
◦ automaticvia “coherencecoherence condition“
LL-structure -structure on CoalgF
LL-structure -structure on CoalgF
LL-structure -structure on C
LL-structure -structure on C
liftinglifting
Related to the study of bialgebraic structures [Turi-Plotkin, Bartels, Klin, …] ◦Algebraic structures on coalgebras
In the current work:◦Equations, not only operations, are also an integral
part◦Algebraic structures are nested, higher-
dimensional
“Pseudo” algebraic structures◦ monoidal category (cf. strictly monoidal category)◦ equations hold up-to-isomorphism◦ L CAT, product-preserving pseudo-functor?
Microcosm principle for full GSOS
concurrency/concurrency/compositionalitycompositionality
mathematicsmathematics
microcosm for microcosm for concurrencyconcurrency((|| || and and ||||))
microcosm for microcosm for concurrencyconcurrency((|| || and and ||||))
parallel compositionparallel compositionvia via syncsync nat. trans. nat. trans.
parallel compositionparallel compositionvia via syncsync nat. trans. nat. trans.
2-categorical formulation2-categorical formulation
for arbitrary algebraic
theory
for arbitrary algebraic
theory
Thanks for your attention!Thanks for your attention!Ichiro Hasuo (Kyoto, Japan)http://www.cs.ru.nl/~ichiro
Thanks for your attention!Thanks for your attention!Ichiro Hasuo (Kyoto, Japan)http://www.cs.ru.nl/~ichiro
Microcosm principle : same algebraic structure◦on a category C and◦on an object X 2 C
2-categorical formulation:
Concurrency in coalgebras as a CS example Thank you for your attention!
Ichiro Hasuo, Kyoto U., Japanhttp://www.cs.ru.nl/~ichiro
Thank you for your attention!Ichiro Hasuo, Kyoto U., Japanhttp://www.cs.ru.nl/~ichiro
Take F = Pfin ( £ _) in Sets.
System as coalgebra:
Behavior by coinduction:
◦ in Sets: bisimilarity◦ in certain Kleisli categories: trace equivalence
[Hasuo,Jacobs,Sokolova,CMCS’06]
the set of finitely branching edges labeled with infinite-depthtrees,
such as
xprocess
graph of x
Note:Asynchronous/interleaving compositions don’t fit in this framework◦ such as ◦ We have to use, instead of F,
the cofree comonad on F
Presentation of an algebraic theory as a category: ◦ objects: 0, 1, 2, 3, … “arities”◦ arrows: “terms (in a context)”
◦ commuting diagrams are understood as “equations”
~ unit law ~ assoc. law
◦ arises from (single-sorted) algebraic specification (, E) as the syntactic category FP-sketch
projections
operation
composed term
In a coalgebraic study of concurrency,
Nested algebraic structures ◦ on a category C and◦ on an object X 2 C arise naturally (microcosm principle)
Our contributions:◦ Syntactic formalization of microcosm principle◦ 2-categorical formalization with Lawvere theories◦ Application to coalgebras:
generic compositionality theorem
A Lawvere theory L is for ◦ operations, and ◦ equations (e.g. associativity, commutativity)
CoalgF is an L-category Parallel composition is automatically associative (for example)◦ Ultimately, this is due to the coherence condition on the lax
L-functor F
Possible application : Study of syntactic formats that ensure associativity/commutativity (future work)