dynamic graph transformation systems

Dynamic Graph Transformation Systems

Hernán MelgrattiIMT Lucca Institute for Advance Studies

Joint Work with

Roberto BruniDipartimento di Informatica, Università di Pisa

ICGT 2006 Hernán Melgratti@IMTLucca-

Join Calculus

Join processes can be seen as dynamic and reconfigurable, coloured nets


Join Calculus

Join processes can be seen as dynamic and reconfigurable, coloured netsa

x

b

x

ax xb


Join Calculus


x

b

x

ax xb

ac

defin aa | ac

a

x

b

x

ax xb

b

def

in ab | cb

bc

a

x

b

x

ax xb

a

defin aa | cb

bc


Join Calculus


x

ax def cy yx in cc

c

y

x

cy


Join Calculus


x

ax def cy yx in cc

ab

def

c

y

x

cy

in aa | ab

a

x

ax def cy yx in cc

b

def

c

y

x

cy

in ab | cc

c yc

y

cy ay


DPO Graph Grammar

The graph of types

The graph of types

The initial T-typed graph


The set of productions


T-typed Graphs

T-typed Graphs

Left-hand-sideLeft-hand-side

InterfaceInterface

Right-hand-side

Right-hand-side

Span of injective

morphisms

Span of injective

morphisms


G

DPO Rewriting Step

m

l rp : L K R

k

bD

h

dH


Towards Dynamic Productions



p:



n

n1 n1p:



n

n1 n1p:

Gp



n

n1 n1

n m

f g

p:

Gp



n

n1 n1 n1

n m

f

m1f1

g

p:

Gp



n

n1 n1 n1

n m

f

m1f1

g

p:

q:

Gp



n1 n1p:

Gp

n

n1

f

m1f1

q: …n m

g



n1 n1p:

Gp

n

n1

f

m1f1

q: …

r p

r st

n m

g

f m

g



n1 n1p:

Gp

n

n1

f

m1f1

q: …

r p

r st

f n m

g

m

g

q: …

p

r st

s’t’

f’

m’

g’

q’: …

q:q:

q’:


Dynamic Graph Grammar (DGG)

The graph of types

The graph of types





T-typed Graphs

T-typed Graphs

A DGG over the graph of type T

Tp

A DGG over the graph of type T

Tp

Injective MorphismInjective

MorphismInjective Morphism

between Tp-typed Graph Injective Morphism

between Tp-typed Graph


G

T

Dynamic rewriting

k

b

D

m

r’

d

H

h

lL K

Tk’

K’

D’


Encoding the Join Calculus

A channel (or place) x is encoded as a node n

The actual name of the channel is given by an arc x:n n

Any firing rule is encoded as a production


Encoding a Join Process P

The graph of types

x3x2

x1

m

Where fn (P ) dn(P ) = { x1, x2, x3 }



A message xy

m

m

x y

yx

m



A message xy

yx

m

m

x y



A definition x1u1 |…| xkuk Pi

m

u1

x1

n1

m

uk

xk

nk

…

u1

x1

n1

uk

xk

nk

…


Example

P = def xu def yv vy in yu | xy in xz

xz

m

z is a free name

x is a defined name

m

nu

x nx

x z

m

xz

mx nx

nu

yu

y

x nx

m

m

nu

u

m

m

nv

y ny y ny

nv

m

m

nv

y ny


Example

m

nu

x nx x nx

nu

y

x nx

m

m

nu

u

m

m

nv

y ny y ny

nv

m

m

nv

y ny

x

m

z

x

z

u

ym

m

x

z

u

y

m

m

u’

y’m


Theorem

For any Join process P•If P P’ using JiPi then Q s.t.

and Q P’

•If , then P’ s.t P P’ using JiPi and


DGG as GG

We start by defining a graph of types for representing the tree of types created dynamically

n f m

gTa

Tb

A chain of types

nn f m g

Ta Tb

Refined Type


DGG as GG

A typed graph over a refined type

n f m

gTa

Tb

nn f m g

Ta Tb

n f m

nn f m

Ta Tb


DGG as GG

The refined version of productionsn1 n1p: n1 m1f1

nn f m

Ta Tb

nn

Ta

nn

Ta

nn f m g

Ta Tb


Theorem


Final Remarks DGG offers a convenient level of

abstraction for describing reflexive systems

DGG can be simulated by ordinary GG

Future works:•To study independent derivations,

parallelism, process semantics, unfolding semantics and event structure semantics

•To show that concurrency is preserved by our encoding

•To consider other approaches (like SPO)

dynamic graph transformation systems

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