Categorical Categorical theory of state-based systems in Sets : bisimilarity in Kleisli: trace semantics [Hasuo,Jacobs,Sokolova LMCS´07] in Sets : bisimilarity in Kleisli: trace semantics [Hasuo,Jacobs,Sokolova LMCS´07]

is everywhere computer networks multi-core processors modular, component-based design of complex systems is hard to get right so easy to get into deadlocks exponentially growing complexity cf. Edward Lee. Making Concurrency Mainstream. Invited talk at CONCUR 2006.

compositional verification aids compositional verification „bisimilarity is a congruence“

 Final coalgebra semantics as “process semantics”.  “Coalgebraic compositionality” On coalgebras || : Coalg F x Coalg F  Coalg F On states || : Z x Z  Z || : Coalg F x Coalg F  Coalg F o composing coalgebras/systems || : Coalg F x Coalg F  Coalg F o composing coalgebras/systems || : Z x Z  Z o composing behavior || : Z x Z  Z o composing behavior operation on NFAs operation on regular languages

 The same “algebraic theory” ◦ operations (binary ||) ◦ equations (e.g. associativity of ||)  is carried by ◦ the category Coalg F and ◦ its object Z 2 Coalg F in a nested manner!  “Microcosm principle” (Baez & Dolan) with operations o operations binary || o equations e.g. assoc. of || operations o operations binary || o equations e.g. assoc. of || X 2 CX 2 CX 2 CX 2 C X 2 CX 2 CX 2 CX 2 C

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 Dolan Higher-Dimensional Algebra III: n-Categories and the Algebra of Opetopes Adv. Math. 1998

 You may have seen it ◦ “a monoid is in a monoidal category” inner depends on outer

Answer

microcosm for concurrency (|| and ||)(|| and ||) microcosm for concurrency (|| and ||)(|| and ||) parallel composition via sync nat. trans. parallel composition via sync nat. trans. 2-categorical formulation for arbitrary algebraic theory

Parallel composition of coalgebras via sync X, 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 transformation sync X,Y : FX ­ FY  F(X ­ Y)  then we have ­ : Coalg F x Coalg F  Coalg F bifunctor Coalg F x Coalg F  Coalg F usually denoted by ­ (tensor) bifunctor Coalg F x Coalg F  Coalg F usually denoted by ­ (tensor) Theorem ­ : Coalg F x Coalg F  Coalg F ­ : C x C  C lifting

?? F X ­ YX ­ Y (X ­ Y)(X ­ Y) FX ­ FY c ­ dc ­ d sync X, Y ­ on base category different sync different different ­

 CSP-style (Hoare)  CCS-style (Milner) Assuming C = Sets, F = P fin (  x _) F-coalgebra = LTS x : Sets x Sets  Sets

 || “composition of states/behavior” arises by coinduction

 C has tensor ­  F has sync X,Y : FX ­ FY  F(X ­ Y)  there is a final coalgebra Z  FZ  C has tensor ­  F has sync X,Y : FX ­ FY  F(X ­ Y)  there is a final coalgebra Z  FZ o by finality o yields o by finality o yields

 When is ­ : Coalg F x Coalg F  Coalg Fassociative?  Answer When ◦ ­ : C x C  C is associative, and ◦ sync is “associative”

2-categorical formulation of the microcosm principle

examples o monoid in monoidal category o final coalg. in Coalg F with ­ o reg. lang. vs. NFAs examples o monoid in monoidal category o final coalg. in Coalg F with ­ o reg. lang. vs. NFAs

Lawvere theory L A Lawvere theory L is a small category s.t. o L’s objects are natural numbers o L has finite products Lawvere theory L A Lawvere theory L is a small category s.t. o L’s objects are natural numbers o L has finite products

operations as arrows equations as commuting diagrams m (binary) e (nullary) m (binary) e (nullary) assoc. of m unit law assoc. of m unit law other arrows: o projections o composed terms other arrows: o projections o composed terms

L -set o a set with L -structure  L -set (product-preserving) X 2 CX 2 CX 2 CX 2 C X 2 CX 2 CX 2 CX 2 C what about nested models?

category L -category o a category with L -structure  L -category (product-preserving) strict NB. our focus is on strict alg. structures

Given an L -category C, L -object an L -object X in it lax is a lax natural transformation compatible with products. Given an L -category C, L -object an L -object X in it lax is a lax natural transformation compatible with products. DefinitionComponents X : carrier obj.

 C : L -category  F : C  C, lax L -functor  Coalg F is an L -category  C : L -category  F : C  C, lax L -functor  Coalg F is an L -category lax L-functor? lax lax natur.. trans. lax L-functor? lax lax natur.. trans. lax naturality?... Generalizes  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 Generalizes

 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 o by finality o yields o by finality o yields by coinduction subsumes

associative ­ : Coalg F x Coalg F  Coalg Fassociative associative ­ : C x C  Cassociative lifting

 equations are built-in in L ◦ as  how about „assoc“ of sync? ◦ automatic coherence via “coherence condition“ L-structure on Coalg F L-structure on Coalg F L-structure on C L-structure on C lifting

 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

microcosm for concurrency (|| and ||)(|| and ||) microcosm for concurrency (|| and ||)(|| and ||) parallel composition via sync nat. trans. parallel composition via sync nat. trans. 2-categorical formulation for arbitrary algebraic theory Thanks for your attention! Ichiro Hasuo (Kyoto, Japan) Thanks for your attention! Ichiro Hasuo (Kyoto, Japan)

 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., Japan Thank you for your attention! Ichiro Hasuo, Kyoto U., Japan

Take F = P fin (  £ _) 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-depth trees, such as x process 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)  Coalg F 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)

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