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

ICGT 2006 Hernán Melgratti@IMTLucca-

Join Calculus

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

ICGT 2006 Hernán Melgratti@IMTLucca-

Join Calculus

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

x

b

x

ax xb

ICGT 2006 Hernán Melgratti@IMTLucca-

Join Calculus

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

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

ICGT 2006 Hernán Melgratti@IMTLucca-

Join Calculus

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

x

ax def cy yx in cc

c

y

x

cy

ICGT 2006 Hernán Melgratti@IMTLucca-

Join Calculus

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

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

ICGT 2006 Hernán Melgratti@IMTLucca-

DPO Graph Grammar

The graph of types

The graph of types

The initial T-typed graph

The initial T-typed graph

The set of productions

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

ICGT 2006 Hernán Melgratti@IMTLucca-

G

DPO Rewriting Step

m

l rp : L K R

k

bD

h

dH

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Towards Dynamic Productions

ICGT 2006 Hernán Melgratti@IMTLucca-

Towards Dynamic Productions

p:

ICGT 2006 Hernán Melgratti@IMTLucca-

Towards Dynamic Productions

n

n1 n1p:

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Towards Dynamic Productions

n

n1 n1p:

Gp

ICGT 2006 Hernán Melgratti@IMTLucca-

Towards Dynamic Productions

n

n1 n1

n m

f g

p:

Gp

ICGT 2006 Hernán Melgratti@IMTLucca-

Towards Dynamic Productions

n

n1 n1 n1

n m

f

m1f1

g

p:

Gp

ICGT 2006 Hernán Melgratti@IMTLucca-

Towards Dynamic Productions

n

n1 n1 n1

n m

f

m1f1

g

p:

q:

Gp

ICGT 2006 Hernán Melgratti@IMTLucca-

Towards Dynamic Productions

n1 n1p:

Gp

n

n1

f

m1f1

q: …n m

g

ICGT 2006 Hernán Melgratti@IMTLucca-

Towards Dynamic Productions

n1 n1p:

Gp

n

n1

f

m1f1

q: …

r p

r st

n m

g

f m

g

ICGT 2006 Hernán Melgratti@IMTLucca-

Towards Dynamic Productions

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’:

ICGT 2006 Hernán Melgratti@IMTLucca-

Dynamic Graph Grammar (DGG)

The graph of types

The graph of types

The initial T-typed graph

The initial T-typed graph

The set of productions

The set of productions

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

ICGT 2006 Hernán Melgratti@IMTLucca-

G

T

Dynamic rewriting

k

b

D

m

r’

d

H

h

lL K

Tk’

K’

D’

ICGT 2006 Hernán Melgratti@IMTLucca-

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

ICGT 2006 Hernán Melgratti@IMTLucca-

Encoding a Join Process P

The graph of types

x3x2

x1

m

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

ICGT 2006 Hernán Melgratti@IMTLucca-

Encoding a Join Process P

A message xy

m

m

x y

yx

m

ICGT 2006 Hernán Melgratti@IMTLucca-

Encoding a Join Process P

A message xy

yx

m

m

x y

ICGT 2006 Hernán Melgratti@IMTLucca-

Encoding a Join Process P

A definition x1u1 |…| xkuk Pi

m

u1

x1

n1

m

uk

xk

nk

…

u1

x1

n1

uk

xk

nk

…

ICGT 2006 Hernán Melgratti@IMTLucca-

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

ICGT 2006 Hernán Melgratti@IMTLucca-

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

ICGT 2006 Hernán Melgratti@IMTLucca-

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

ICGT 2006 Hernán Melgratti@IMTLucca-

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

ICGT 2006 Hernán Melgratti@IMTLucca-

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

ICGT 2006 Hernán Melgratti@IMTLucca-

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

ICGT 2006 Hernán Melgratti@IMTLucca-

Theorem

ICGT 2006 Hernán Melgratti@IMTLucca-

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)