the microcosm principle and concurrency in coalgebras

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)