Dynamic Graph Transformation Systems

1
Dynamic Graph Transformation Systems

• Hernán Melgratti
• IMT Lucca Institute for Advance Studies
• Joint Work with
• Roberto Bruni
• Dipartimento di Informatica, Università di Pisa

2
Join Calculus

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

3
Join Calculus

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

4
Join Calculus

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

a
x
b
x
a?x? ? x?b?
5
Join Calculus

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

a
a
c
x
b
x
a?x? ? x?b?
def
in a?a? a?c?
6
Join Calculus

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

7
Join Calculus

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

a
a
b
x
c
c
y
y
x
a?x? ? def c?y? ? y?x? in c?c?
def
in a?a? a?b?
8
DPO Graph Grammar
The initial T-typed graph
The graph of types
The set of productions
Left-hand-side
Interface
Span of injective morphisms
Right-hand-side
9
DPO Rewriting Step
10
Towards Dynamic Productions
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Towards Dynamic Productions
p
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Towards Dynamic Productions
p
n1
n1
n
13
Towards Dynamic Productions
Gp
p
n1
n1
n
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Towards Dynamic Productions
Gp
p
n1
n1
g
f
n
n
m
15
Towards Dynamic Productions
Gp
p
n1
n1
n1
m1
f1
g
f
n
n
m
16
Towards Dynamic Productions
Gp
p
n1
n1
n1
m1
f1
g
f
n
n
m
q
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Towards Dynamic Productions
Gp
p
n1
n1
n1
m1
f1
g
f
n
n
m
q
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Towards Dynamic Productions
Gp
p
n1
n1
n1
m1
f1
g
f
n
n
m
q
r
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Towards Dynamic Productions
Gp
p
n1
n1
n1
m1
f1
g
g
f
f
n
n
m
m
q
q
q
r
s
t
p
r
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Dynamic Graph Grammar (DGG)
Injective Morphism
A DGG over the graph of type T ? Tp
Injective Morphism between Tp-typed Graph
21
Dynamic rewriting
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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
  xn ? n
• Any firing rule is encoded as a production

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Encoding a Join Process P

• The graph of types

m
Where fn (P ) ?? dn(P ) x1, x2, x3
x3
x1
x2
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Encoding a Join Process P

• A message x?y?

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Encoding a Join Process P

• A message x?y?

m
y
x
m
y
x
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Encoding a Join Process P

• A definition x1?u1? xk?uk? ? Pi

27
Example
x is a defined name

• P def x?u? ? def y?v? ? v?y? in y?u? x?y?
  in x?z?

z is a free name
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Example
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Theorem

• For any Join process P
• If P? P using Ji?Pi then ??Q s.t.
  and Q ? P
• If , then ??P s.t P? P using
  Ji?Pi and

30
DGG as GG

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

31
DGG as GG

• A typed graph over a refined type

Tb
Ta
f
n
m
n
n
f
m
Tb
Ta
Tb
Ta
g
f
n
m
n
n
f
m
g
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DGG as GG

• The refined version of productions

p
n1
n1
n1
m1
f1
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Theorem
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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)