Introduction to Graph Grammars Fulvio D’Antonio LEKS, IASI-CNR Rome,

Summary Basic concepts Double pushout approach Single pushout approach Tools References

Graph grammars Algebraic approaches were developed at the Technical University of Berlin The main idea was that of extending concatenation of strings to a “gluing” of graphs The action of gluing two graphs is a construction, in the category of graphs and graph morphisms, called pushout Graph grammars has been invented (in early seventies) in order to generalize (Chomsky) string grammars.

Graph grammars: definition A graph grammar is a pair: GG = (G 0,P) G 0 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 G 0 and applying the rules in P

A graph…

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 (v 1,v 2 ) with an arrow starting in v 1 and ending in v 2

Another graph … This is a multigraph

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=  v 1,v 2,…., v n  E =  (v 1,v 2 ),….,(v 1,v 2 ),… 

Yet another graph … This is a labelled multigraph: elements without a label are considered labelled with the null symbol  a b A A

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 graph is a tuple (V,E,s,t,l V,l E ): 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 l V : V  V e l E : E  E are two mappings from from V and E in two finite sets of labels

Example A A B B E Notes: The edges are directed Two vertexes with the same label Multiple edges (even with the same label!) between two vertexes

Example 2:Pacman graph (PG)

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’ A A B B E G’ G A A B This is a correct graph morphism

Example 2 A A B B E G’ G A B This is not a correct graph morphism 2

Example 3 A A B B E G’ G 1,3 2,4 This is a correct non-injective graph morphism E E

Graph morphism a graph morphism  is a pair (  1,  2 ),  1 :V  V’,  2 : E  E’ such as: 1)labels are preserved i.e. l V (v i ) = l V’ (  1 (v i )) etc. 2)incidence is preserved i.e.  1 (s(e i )) = s’(  2 (e i ))) etc. Given G =(V,E,s,t,l V,l E ) and G’=(V’,E’,s’,t’,l V’,l E’ )

What is a pushout? (Very very informal) “Gluing” of two objects along a common substructure

Summary Basic concepts Double pushout approach Single pushout approach Tools References

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

Example movePacman : L K R

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)

Example movePacman : The graph G The match L K R

The context graph The match

The final graph H The match G  ,p H G  * ,p G n (reflexive symmetric and transitive closure)

Other rules in Pacman game MoveGhost: Kill :

Summary Basic concepts Double pushout approach Single pushout approach Tools References

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

Example LR r is a partial morphism r

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

Summary Basic concepts Double pushout approach Single pushout approach Tools References

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 Atom 3 Grace and Graceland Other tools

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

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, 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: aachen.de/research/projects/progres/main.html Agg homepage: Graceland homepage: Fujaba homepage: Atom 3 : Standard: GXL: