Introduction to Graph Grammars

1
Introduction toGraph Grammars

• Fulvio DAntonio
• LEKS, IASI-CNR
• Rome, 14-10-03

2
Summary

• Basic concepts
• Double pushout approach
• Single pushout approach
• Tools
• References

3
Graph grammars

• Graph grammars has been invented (in early
  seventies) in order to generalize (Chomsky)
  string grammars.

• The main idea was that of extending concatenation
  of strings to a gluing of graphs

• Algebraic approaches were developed at the
  Technical University of Berlin

• The action of gluing two graphs is a
  construction, in the category of graphs and graph
  morphisms, called pushout

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Graph grammars definition

• A graph grammar is a pair
• GG (G0,P)

G0 is called the starting graph and P is a set of
production rules
L(GG) is the set of graphs that can be derived
starting with G0 and applying the rules in P
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A graph
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Definition

• A pair (V,E) of finite sets
• E ? V ? V
• E is a set of ordered pairs of vertexes.

Graphically we represent an edge (v1,v2) with an
arrow starting in v1 and ending in v2
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Another graph
This is a multigraph
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Another definition

• A pair (V,E) where V is a finite set, E is a
  finite multiset with elements in V?V

E.g. V ?v1,v2,., vn? E ?(v1,v2),.,(v1,v2),
?
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Yet another graph
a
b
This is a labelled multigraph elements without a
label are considered labelled with the null
symbol ?
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Yet another definition

• A pair (V,E) where

V is a finite set of pairs and E is a finite set
of triples.
Too complicated!
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A more elegant definition (algebraic style)

• A graph is a tuple (V,E,s,t,lV,lE)
• V and E are two finite sets (V?E?)

s,t E ? V are two mappings indicating the
source and the target of an edge
lV V ??V e lE E ??E are two mappings from from
V and E in two finite sets of labels
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Example
E
A
B
A
Notes The edges are directed Two vertexes with
the same label Multiple edges (even with the same
label!) between two vertexes
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Example 2Pacman graph (PG)
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Graph morphism informally speaking

• Given two graphs G and G we want to know if G
  contains G.

We can try to draw a correspondence between every
vertex (edge) of G and a vertex (edge) of G
This correspondence is a graph morphism (if it
respects some properties!)
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Example G is contained in G
G
G
E
A
3
3
A
2
B
2
A
B
A
1
1
This is a correct graph morphism
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Example 2
G
G
E
3
A
3
2
2
B
A
B
A
1
1
This is not a correct graph morphism
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Example 3
G
G
E
E
2,4
4
A
E
3
2
1,3
B
A
1
This is a correct non-injective graph morphism
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Graph morphism
Given G (V,E,s,t,lV,lE) and
G(V,E,s,t,lV,lE)

• a graph morphism ?
• is a pair (?1,?2), ?1V ? V, ?2 E ? E
• such as

1)labels are preserved i.e. lV(vi) lV(?1(vi))
etc.
2)incidence is preserved i.e. ?1(s(ei))
s(?2(ei))) etc.
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What is a pushout?(Very very informal)

• Gluing of two objects along a common
  substructure

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Summary

• Basic concepts
• Double pushout approach
• Single pushout approach
• Tools
• References

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Graph grammarsDouble pushout approach

• The format of a production rule is
• p L l? K ?r R
• L,K,R are graphs and l,r are two (total)
  morphisms matching K, respectively,in L and R

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Example

• movePacman

L
R
K
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Derivation

• Given a graph G,a production pL l? K ?r R and a
  graph morphism ?L ? G

1)The context graph is obtained deleting from G
all elements images of elements in L but not of
elements in K (pushout complement)
2)The final graph is obtained adding to context
graph all elements which dont have a pre-image
in K(pushout)
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Example

• movePacman

The match
The graph G
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The match
The context graph
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The match
The final graph H
G ? ?,p H G ? ?,p Gn (reflexive symmetric and
transitive closure)
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Other rules in Pacman game

• MoveGhost

Kill
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Summary

• Basic concepts
• Double pushout approach
• Single pushout approach
• Tools
• References

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Single pushout approach

• The format of a production rule is
• p L ?r R
• r is a partial graph morphism

A single derivation step is modelled by a
single-pushout diagram
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Example
1
1
r
3
3
4
2
4
2
L
R
r is a partial morphism
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Difference between the two approaches

• Double-pushout approach requires two further
  conditions for a step derivation (dangling and
  identification condition)
• Single-pushout doesnt requires such conditions
• Single pushout rules can model more situations
  than double pushout rules

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Summary

• Basic concepts
• Double pushout approach
• Single pushout approach
• Tools
• References

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Progres

• PROGRES is an integrated environment for a very
  high level programming language which has a
  formally defined semantics based on "PROgrammed
  Graph REwriting Systems"

Agg

• AGG is a rule based visual language supporting
  single pushout approach to graph transformation.
  It aims at the specification and prototypical
  implementation of applications with complex
  graph-structured data.

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Fujaba
Other tools
Grace and Graceland
Atom3
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Standards

• GXL (Graph Exchange Language)
• GTXL (Graph Transformation Exchange Language)

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References
People G.Rozenberg,A.Schurr, R.Heckel,
G.Taentzer, P.Bottoni, F.Parisi-Presicce,
A.Corradini, H.Ehrig, H.G.Kreowsky. Theory G.
Rozenberg, editor. Handbook of Graph Grammars and
Computing by Graph Transformation, Volume 1-3
Foundations. World Scientific, 1997. Corradini,
A., Montanari, U., Rossi, F., Ehrig, H., Heckel,
R., Löwe, M. Algebraic Approaches to Graph
Transformation Part I Basic Concepts and Double
Pushout Approach Corradini, A. Concurrent Graph
and Term Graph Rewriting Proc. CONCUR'96, LNCS
Tools Progres homepage http//www-i3.informatik
.rwth-aachen.de/research/projects/progres/main.htm
l Agg homepagehttp//tfs.cs.tu-berlin.de/agg/ Gra
celand homepagehttp//www.informatik.uni-bremen.d
e/theorie/GRACEland/GRACEland.html Fujaba
homepagehttp//www.fujaba.de/ Atom3http//atom3.
cs.mcgill.ca/ Standard GXL http//www.gupro.de/G
XL/