dynamic graph transformation systems
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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
ICGT 2006 Hernán Melgratti@IMTLucca-
Towards Dynamic Productions
ICGT 2006 Hernán Melgratti@IMTLucca-
Towards Dynamic Productions
p:
ICGT 2006 Hernán Melgratti@IMTLucca-
Towards Dynamic Productions
n
n1 n1p:
ICGT 2006 Hernán Melgratti@IMTLucca-
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)
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