dynamic graph transformation systems
DESCRIPTION
Dynamic Graph Transformation Systems. Hernán Melgratti IMT Lucca Institute for Advance Studies Joint Work with Roberto Bruni Dipartimento di Informatica, Università di Pisa. Join Calculus. Join processes can be seen as dynamic and reconfigurable, coloured nets. Join Calculus. - PowerPoint PPT PresentationTRANSCRIPT
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)