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Introduction to Graph Grammars. Fulvio D’Antonio LEKS, IASI-CNR Rome, 14-10-03. Summary. Basic concepts Double pushout approach Single pushout approach Tools References. Graph grammars.

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introduction to graph grammars

Introduction toGraph Grammars

Fulvio D’Antonio

LEKS, IASI-CNR

Rome, 14-10-03

summary
Summary
  • Basic concepts
  • Double pushout approach
  • Single pushout approach
  • Tools
  • References
graph grammars
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

graph grammars definition
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

definition
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

another graph
Another graph …

This is a multigraph

another definition
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),…

yet another graph

A

A

Yet another graph …

a

b

This is a labelled multigraph: elements without a label are considered labelled with the null symbol 

yet another definition
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!

a more elegant definition algebraic style
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

example

B

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

graph morphism informally speaking
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!)

example g is contained in g

B

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

example 2

B

Example 2

G’

G

E

3

A

3

2

2

B

A

B

A

1

1

This is not a correct graph morphism

example 3

B

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

graph morphism
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), 1:V  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.

what is a pushout very very informal
What is a pushout?(Very very informal)
  • “Gluing” of two objects along a common substructure
summary20
Summary
  • Basic concepts
  • Double pushout approach
  • Single pushout approach
  • Tools
  • References
graph grammars double pushout approach
Graph grammars:Double 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
example22
Example
  • movePacman :

L

R

K

derivation
Derivation
  • Given: a graph G,a production p:L 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 don’t have a pre-image in K(pushout)

example24

L

R

K

Example
  • movePacman :

The match

The graph G

slide25

The match

The context graph

slide26

The match

The final graph H

G  ,p H

G * ,p Gn (reflexive symmetric and transitive closure)

summary28
Summary
  • Basic concepts
  • Double pushout approach
  • Single pushout approach
  • Tools
  • References
single pushout approach
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

example30
Example

1

1

r

3

3

4

2

4

2

L

R

r is a partial morphism

difference between the two approaches
Difference between the two approaches
  • Double-pushout approach requires two further conditions for a step derivation (dangling and identification condition)
  • Single-pushout doesn’t requires such conditions
  • Single pushout rules can model more situations than double pushout rules
summary32
Summary
  • Basic concepts
  • Double pushout approach
  • Single pushout approach
  • Tools
  • References
progres
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.

fujaba
Fujaba

Other tools

Grace and Graceland

Atom3

standards
Standards
  • GXL (Graph Exchange Language)
  • GTXL (Graph Transformation Exchange Language)
references
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.html

Agg homepage:http://tfs.cs.tu-berlin.de/agg/

Graceland homepage:http://www.informatik.uni-bremen.de/theorie/GRACEland/GRACEland.html

Fujaba homepage:http://www.fujaba.de/

Atom3:http://atom3.cs.mcgill.ca/

Standard:

GXL: http://www.gupro.de/GXL/