© Fachgebiet Softwaretechnik, Heinz Nixdorf Institut, Universität Paderborn 2.5 Graph Grammars.

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2.5 Graph Grammars

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2

Organisatorisches

Klausurtermin: Fr, 22.02.2013 Voraussichtlich 11:00 Uhr

Hinweis: Es gibt nur einen Klausurtermin! Mögliche Konflikte bis 14.12.2012 an Uwe Pohlmann

(upohl (at) upb.de) melden.

06.12.2012Graph Grammars

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3Graph Grammars 06.12.2012

Graph Grammars

Example Describing program states as graphs Describing program behavior through graph transformations

Models and graphs Graph grammars Story diagrams

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Example: Graph Grammars in every day life Most children know graph grammars Every Lego build instruction is a

graph grammar

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5Graph Grammars

Example

Example: simulation of a track-based transportation system

:Shuttle

:Track :Track :Switch

:Track

:Tracknext nextnext

:Shuttle

turnNext

Concrete syntax:

Abstract syntax:

isOn

isOn

06.12.2012

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Object Graph

The object graph represents the state of a program The behavior of a program is a change of the state

A change of the object graph

:Shuttle


:Track

:Tracknext nextnext

:Shuttle

turnNext

isOn

isOn

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Object Graph

The object graph represents the state of a program The behavior of a program is a change of the state

A change of the object graph

:Shuttle


:Track

:Tracknext nextnext

:Shuttle

turnNextisOn

isOn

isOn

Example: Moving a shuttle

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Object Graph

The program behavior can be described by rules for transforming the object graph

:Shuttle


:Track

:Tracknext nextnext

:Shuttle

turnNext

isOn

isOn

:Shuttle

:Track :Tracknext

isOnisOn

move

<<delete>>

<<create>>

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Motivation

Graph transformation rules allow us: To describe the behavior of systems in an abstract way

(e.g. RailCab)• To model and analyze them

To describe the semantics of pointer programs (OO)• To validate or verify them• To model the program behavior visually (Executable code can be

generated from them)

:Shuttle

:Track :Tracknext

isOnisOn

move

<<delete>>

<<create>>

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Graph Grammars



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Instance Model and Class Model

:Shuttle


:Track

:Tracknext nextnext

:Shuttle

turnNext

isOn

isOn

TrackShuttle isOn

1..1

Switch

next

0..1turnNext

Class Diagram:

Object graph:

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Instance Model and Class Model

Shuttle isOn

1..1

Switch

next

0..1turnNext

Track

:Shuttle


:Track

:Tracknext nextnext

:Shuttle

turnNext

isOn

isOn

Class Diagram:

Object graph: Typed Graph

Type Graph

(edges are typed, too)

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Graph Grammars



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Graph Grammars

A graph grammar consists of A set of graph grammar rules (production rules) A start graph A type graph

A graph grammar describes how to produce valid sets of graphs (sets may be infinite)

Synonyms: Graph Rewriting System or Graph Transformation System

:Track

start graph

:Shuttle

rule: move

:Track

<<create>>

isOn

:Track

isOn<<delete>>

:Track

:Track :Track

next

next

next

next:Shuttle isOn

next

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Graph Grammar Rule

A graph grammar rule consists of A left-hand side and right-hand side typed graph

move

:Shuttle

:Track:Track

:Shuttle

:Track:Track

Short hand form:

lhs: rhs:

::=

v1

v2

v3 nextnext

isOn isOn

v1

v2

v3

:Shuttle

move

:Track

<<create>>

isOn

:Track

isOn<<delete>>

identifier

next

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Graph Grammar Rule

A graph grammar rule consists of A left-hand side and right-hand side typed graph

move

:Shuttle

:Track:Track

:Shuttle

:Track:Track

lhs: rhs:

::=

v1

v2

v3 nextnext

isOn isOn

v1

v2

v3

TrackShuttle isOn

1

Switch

next

1turnNext

(edges are typed, too)

1

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Graph Grammar Rule Application

move

:Shuttle

:Track:Track

:Shuttle

:Track:Track

lhs: rhs:

::=

:Shuttle


:Track

:Tracknext nextnext

:Shuttle

turnNext

isOn

isOn

isOn

next next

isOn

v1

v2

v3 v1

v2

v3

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move

:Shuttle

:Track:Track

:Shuttle

:Track:Track

lhs: rhs:

::=

:Shuttle


:Track

:Tracknext nextnext

:Shuttle

turnNext

isOn

isOn

isOn

next next

isOn

1. Match lhs in host graph

v1

v2

v3 v1

v2

v3

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move

:Shuttle

:Track:Track

:Shuttle

:Track:Track

lhs: rhs:

::=

:Shuttle


:Track

:Tracknext nextnext

:Shuttle

turnNext

isOn

isOn

isOn

next next

isOn

v1

v2

v3 v1

v2

v3

2. Remove nodes and edges which are in lhs, but not in rhs

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move

:Shuttle

:Track:Track

:Shuttle

:Track:Track

lhs: rhs:

::=

:Shuttle


:Track

:Tracknext nextnext

:Shuttle

turnNext

isOn

isOn

isOn

next next

isOn

3. Create nodes and edges which are in rhs, but not in lhs

v1

v2

v3 v1

v2

v3

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Non-Determinism

Example 1: When to move which shuttle? Not determined!

move

:Shuttle

:Track:Track

:Shuttle

:Track:Track

lhs: rhs:

::=isOn

next next

isOn

v1

v2

v3 v1

v2

v3

:Shuttle

:Track :Tracknext next

isOn

:Shuttle

:Track :Tracknext

isOn

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Non-Determinism

Example 2: Go straight or turn? Not determined!

:Switch

:Track

:Track

next

:Shuttle

turnNext

isOn

:Shuttle

move

:Track

<<create>>

isOn

:Track

isOn<<delete>>

:Shuttle

turn

:Track

<<create>>

isOn

:Switch

isOn<<delete>>

next turnNext

TrackShuttle isOn

1

Switch

next

1turnNext(remember:

a switch is also a track)

1

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Negative Application Conditions Example: Do not crash with other shuttle

:Shuttle

move

:Track

<<create>>

isOn

:Track

isOn<<delete>>

next

:Shuttle

:Shuttle


isOn

:Shuttle

:Track :Tracknext

isOn

isOn

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Attribute Condition

Example 1: A broken shuttle cannot move

move

:Shuttle

:Track

<<create>>

isOn

:Track

isOn<<delete>>

next

failure=false

Shuttle isOn

1

Switch

next

1turnNext

Track

failure: boolean

1

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Attribute Condition

Example 2: May follow faster shuttles on Track

move

s1:Shuttle

:Track

<<create>>

isOn

:Track

isOn<<delete>>

next

isOn

speed ≤ s2.speed

Shuttle isOn

1

Switch

next

1turnNext

Track

broken: booleanspeed: integer

s2:Shuttle

1

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Graph Grammars: Overview

Non-deterministic rule application Negative application conditions Conditions on attribute values Inheritance in the type graph There may be other extensions depending on the problem

domain…• Ordered lists• Qualified associations• „maybe“-conditions

Now a bit more formally…

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Definition: Directed Graph

G = (V, E, s, t) V – finite set of nodes (vertices) E – finite set of edges s: E→V – source function t: E→V – target function

v2v1

v3

e3

e2

e1

Example:V = {v1, v2, v3}E = {e1, e2, e3}s(e1) = v1t(e1) = v2

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Definition: Labeled Graph

GL = (G,LV, LE)

G – Directed Graph LV: V →∑ – labeling function for vertices

LE: E →∑ – labeling function for edges

∑ – set of labels

v2

v1

e1

Example:V = {v1, v2}E = {e1}s(e1) = v1t(e1) = v2 track

shuttle

isOn

LV(v1) = “shuttle”LV(v2) = “track”LE(e1) = “isOn”

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Definition: Morphism

Given two graphs Gi=(Vi,Ei,si,ti), i ∈ {1, 2}

A graph morphism f: G1→G2, f = (fV , fE)

consists of two functions fV: V1 → V2

fE : E1 → E2

preserving the source and target functions: fV ◦ s1 = s2 ◦ fE and fV ◦ t1 = t2 ◦ fE

:Shuttle :Track

isOn

Shuttle TrackisOnG2:

G1:v11 v21

v12 v22

= =

e12

e11

Given two functions f

and g then f ◦ g is afunction that mapsa value x to f(g(x))

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Definition: Morphism

Given two graphs Gi=(Vi,Ei,si,ti), i ∈ {1, 2}

A graph morphism f: G1→G2, f = (fV , fE)

consists of two functions fV: V1 → V2

fE : E1 → E2

preserving the source and target functions: fV ◦ s1 = s2 ◦ fE and fV ◦ t1 = t2 ◦ fE

:Shuttle :Track

isOn

Shuttle TrackisOnG2:

G1:v11 v21

v12 v22

= =

e12

e11

Example:fV(s1(e11)) = s2(fE(e11))fV(t1(e11)) = t2(fE(e11))

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Definition: Typed Graph

A Typed Graph GTyped = (G, type) consists of a graph G and a graph morphism to a type graph type: G → GType

Shuttle isOn

1

Switch

next

1turnNext

Track

:Shuttle


:Track

:Tracknext next next

:Shuttle

turnNextisOn

isOn

Typed Graph

Type Graph

Note:A morphism is a total function (fV, fE are total), i.e. every element in the domain (typed graph) has to be related to exactly one element of theco-domain (type graph)

1

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Matching a Left-hand Side

Matching a graph inside another “host” graph Find a typed isomorphism to a subgraph of the host graph

:Shuttle

:Track:Track

lhs:

::= …

:Shuttle


isOn

isOn

nextv1

v2

v3

:Track:Track next

subgraph

isomorphism

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Definition: Subgraph

A subgraph GSub = (VSub, ESub, sSub, tSub) of G= (V, E, s, t) is a graph with VSub V

ESub E

sSub = s

tSub = t

If GSub is a subgraph of G, we may also write GSub G.

Means that the domain of the source and target function for the subgraph is reduced to the edges which are in it.

ESub

ESub

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Definition: Isomorphism

A graph morphism f: G1→G2, f = (fV , fE) with

fV, fE bijective is called a graph isomorphism.

:Shuttle

:Track:Track

lhs:

::= …

:Shuttle


isOn

isOn

nextv1

v2

v3

:Track:Track next

subgraph

isomorphism

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Definition: Typed Isomorphism Both the graph and host graph are typed over the same

type Graph GType:

Let type = (typeV, typeE) be a graph morphism and

f = (fV, fE) is an isomorphism, then f is a typed isomorphism when

• typeV(v) = typeV(fV(v))

• typeE(e) = typeE(fE(e))

:Track:Track

:Track :Tracknext

next

Track

next

1

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Definition: Graph Grammar Rule A graph grammar rule r is defined as a partial morphism r :

L→R with L GType the lhs graph of r, R GType the rhs graph of r, GType the set of all graphs typed over GType such that there is a maximum common subgraph TL of L and R, with: TL GType : TL L and TL R and

TL not empty

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Definition: Graph Grammar Rule

:Shuttle

:Track:Track

:Shuttle

:Track:Track

L(lhs graph) isOn

next next

isOn

v1

v2

v3 v1

v2

v3

R(rhs graph)

:Shuttle

:Track:Tracknextv1

v2

v3

TL(common subgraph)

partial morphismr : L→R

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Rule Application - Formally

The application of a rule r: L→R is defined by a relation appr = { (G, G‘‘)(GType x GType) | G‘‘ is defined by the three steps1. Search (on host graph G)

2. Delete (result is G‘)

3. Create (result is G‘‘)

or G = G‘‘ }

appr represents all possible applications of r

GType

An element is a possible (host) graph typed over GType

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Rule Application – Step 1

Search: Given a host graph G, find a subgraph isomorphism match: L→GSub, GSub G

:Shuttle

:Track:Track

::= …

:Shuttle


isOn

isOn

nextv1

v2

v3

:Track:Track next

Subgraph GSub

isomorphismL(lhs)

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Rule Application – Step 2 (1/3)

2. Delete: Form a new graph G‘ from G by deleting all nodes and edges in GSub which are in match(L) = GSub

but not in G‘Sub = match(TL) ( GSub)

:Shuttle


isOn

:Track:Track next

GSub

:Shuttle

:Track:Track

isOn

nextv1

v2

v3

L(lhs)

:Shuttle

:Track:Tracknextv1

v2

v3

TL(common subgraph)

(Only the edge labeled “isOn” is removed)

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Delete:Remove all edges leading to deleted nodes

createAssignment:Customer

:Broker

:Shuttle :Shuttle:Shuttle

:Requestcreated

queuedRequest

respForrespFor

respFor

:Broker :Shuttle

:Request

queuedRequest

respFor

:Assignment

assigned

:Broker :Shuttle

:Request

queuedRequest

respFor

L(lhs)

:Broker :ShuttlerespFor

TL(common subgraph)

<<delete>> <<create>>assignment<<create>>

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:Broker


created

respForrespFor

respFor

:Broker :Shuttle

:Request

queuedRequest

respFor

:Assignment

assigned

:Broker :Shuttle

:Request

queuedRequest

respFor

L(lhs)


TL(common subgraph)


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:Broker


respForrespFor

respFor

:Broker :Shuttle

:Request

queuedRequest

respFor

:Assignment

assigned

:Broker :Shuttle

:Request

queuedRequest

respFor

L(lhs)


TL(common subgraph)


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Create:Add nodes and edges to G‘, forming an extended graph G‘‘, such that there is an isomorphism matchR: R → G‘‘Sub from R to a new subgraph G‘‘Sub G‘‘ such that(G‘Sub =) match(TL) matchR(R) (= G‘‘Sub)

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Create: G‘Sub = match(TL) matchR(R) = G‘‘Sub

:Shuttle

:Track :Tracknext next :Track:Track next

G‘‘Sub

:Shuttle

:Track:Tracknextv1

v2

v3

TL(common subgraph)

:Shuttle

:Track:Tracknextv1

v2

v3

isOn

isOn

R(lhs)

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Create:G‘Sub = match(TL) matchR(R) = G‘‘Sub

:Customer

:Broker


respForrespFor

respFor


TL(common subgraph)


:Assignment

assigned

:Assignment

R(lhs)

G‘‘Sub

G‘Sub

assignment

assigned

assignment

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Rule Application - Formally

The application of a rule r: L→R is defined by a relation appr = { (G, G‘‘)(GType x GType) | G‘‘ is defined by the three steps1. Search (on host graph G)

2. Delete (result is G‘)

3. Create (result is G‘‘)

or G = G‘‘ }

A concrete rule application is repre-sented by a tuple (G, G‘‘) appr where G is the host graph before applying the rule and G‘‘ is the host graph after rule application.

GType

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What about Inheritance?

GType = (GL, I)

GL – Directed labeled Graph

I V x V – inheritance relation (cycle free)

v2v1

v3

e3

e2

e1

Example:V = {v1, v2, v3}E = {e1, e2, e3}s(e1) = v1t(e1) = v2I = {(v2, v3)}

But now the formalization of the matching becomes more complicated…

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Summary

Graph transformation rules allow us: To describe the behavior of systems (e.g. RailCab) in an

abstract, formal way• To model and analyze them

To describe the semantics of OO-programs• To validate or verify them• To model the program behavior visually (Executable code

can be generated from them)

:Shuttle

:Track :Tracknext

isOnisOn

move

<<delete>>

<<create>>

© Fachgebiet Softwaretechnik, Heinz Nixdorf Institut, Universität Paderborn 2.5 Graph Grammars.

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Transcript of © Fachgebiet Softwaretechnik, Heinz Nixdorf Institut, Universität Paderborn 2.5 Graph Grammars.