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

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

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

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

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“

• Final coalgebra semantics asprocess semantics.

• Coalgebraic compositionality

Compositionality in coalgebra

|| : CoalgF x CoalgF CoalgF composing coalgebras/systems

|| : Z x Z Z composing behavior

the microcosm principlewith

o operations binary ||

o equationse.g. assoc. of ||

algebraic theoryX 2 C outer interpretation

inner interpretation

Nested algebraic structures:

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

The microcosm principle: you may have seen it

monoid in a monoidal category

inner depends on outer

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.

Outline

microcosm for concurrency(|| and ||)

2-categorical formulation

generic compositionality

theorem

for arbitrary algebraic

theory

parallel composition

via sync nat. trans.

Parallel composition of coalgebrasvia sync

Part 1

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)

Parallel composition via sync

­­on the base category

different sync

different ­

syncX,Y : FX ­ FY F(X ­ Y)

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

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

Inner compositionAim

for ­­by and || by

Compositionality theoremTheorem

­­­: CoalgF x CoalgF CoalgF

­­­: C x C Clifting

F with sync

Assumptions: ­ ,­sync, final exists

Equational properties

associative­­­: CoalgF x CoalgF CoalgF

associative

­­­: C x C C

lifting

F with “associative“ sync

commutativity?

arbitrary algebraic theory?

2-categorical formulation ofthe microcosm principle

Part 2

for arbitrary algebraic theory

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

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

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?

Outer model: L-category

o a category with L-structure, L-categoryouter model

(product-preserving)

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

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!

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

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

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