Fundamentals of Algebraic Graph Transformation

Overview Tutorial on

Fundamentals of Algebraic Graph Transformation Hartmut Ehrig, Ulrike Prange : TU Berlin Karsten Ehrig : U. Leicester

July 15, 2006

1

General Introduction and Motivation

2

Category of Graphs and Gluing Construction

3

Graph Transformation Systems

4

Independence of Graph Transformations and Critical Pairs

5

Adhesive HLR Categories and Typed Attributed Graph Transformation Systems

6

Advanced Features and Conclusion

Based on

FAGT-Book, EATCS Monographs in TCS, Springer 2006

H. Ehrig, U. Prange, K. Ehrig ()

Tutorial on Graph Transformation

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What is Graph Transformation? Part I General Introduction and Motivation 1

What is Graph Transformation?

2

Aims and Paradigms of Graph Transformation

3

Graph Transformation Approaches

Rule-based modification of graphs p = (L, R) A LA

R

from Chomsky grammars on strings to graph grammars; from term rewriting to graph rewriting;

4

Application Areas of Graph Transformation Systems



from textual description to visual modeling.

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Aims and Paradigms of Graph Transformation

Graph Transformation Approaches

Computing by graph transformation is a fundamental concept for

1

Node label replacement approach

2

Hyperedge replacement approach Algebraic approach

3

visual modeling and specification,

Double Pushout (DPO) (since 1973) Single Pushout (SPO) (since 1984/90) High Level Replacement (HLR) (since 1991/2004)

model transformation, concurrency and distribution, software development.

Projects Supported by EU projects since 1990: COMPUGRAPH 1 + 2, GETGRATS, APPLIGRAPH, SEGRAVIS.

4

Logical approach

5

Theory of 2-structures

6

Programmed graph replacement approach

Literature Handbook of Graph Grammars and Computing by Graph Transformation, Vol. 1 (1997)



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Application Areas of Graph Transformation Systems

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Part II

Application areas are

Category of Graphs and Gluing Construction

model and program transformation; syntax and semantics of visual languages; visual modeling of behavior and programming; modeling, metamodeling, and model-driven architecture;

5

Graphs, Typed Graphs and Morphisms

6

Categories of Sets and Graphs

7

Mono-, Epi- and Isomorphisms

8

Pushouts as Gluing Construction

9

Composition and Decomposition of Pushouts

10

Pullbacks as Dual Construction of Pushouts

software architectures and evolution; refactoring of programs and software systems; security policies.

Literature Handbook of Graph Grammars and Computing by Graph Transformation, Vol. 2 (1999) Proc. ICGT ’02-06, LNCS H. Ehrig, U. Prange, K. Ehrig ()


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Graphs and Graph Morphisms

Example: Graphs and Graph Morphisms Example

Definition graph G = (V , E , s, t) consists of

E1

V1

fE

=

fV

E2

s2 t2

sets V of nodes and E of edges, and source and target functions s, t : E → V

Top: graph GS = (VS , ES , sS , tS ), with

s1 t1

VS = {u, v , x, y }, ES = {a, b}, GS

sS : ES → VS : a, b → u, tS : ES → VS : a, b → v

V2

Bottom: graph GT = (VT , ET , sT , tT ), with

graph morphism f : G1 → G2 , f = (fV , fE ) consists of two functions fV : V1 → V2 and fE : E1 → E2 with fV ◦ s1 = s2 ◦ fE and fV ◦ t1 = t2 ◦ fE

a b

u

v y

x

VT = {s, t}, ES = {e}, sT : ET → VT : e → s, tT : ET → VT : e → t

s

Inbetween: graph morphism f : GS → GT with

composition g ◦ f = (gV ◦ fV , gE ◦ fE ) : G1 → G3 of f : G1 → G2 and g : G2 → G3 is graph morphism

e

t

GT

fV : VS → VT : u, x → s; v , y → t fE : ES → ET : a, b → e



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Typed Graphs and Typed Graph Morphisms


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Example: Mutual Exclusion and Gluing Construction Example non−active

Definition

2

1

TG

TG :

non−active active

F2

P

F1

start crit check setTurn

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S

P

D 2

start non−active

G2


R

P

F1

R

setTurn

non−active

= type2 type1


P

m

typed graph morphism f : G1T → G2T is graph morphism f : G1 → G2 with type2 ◦ f = type1 f

R

start

typed graph G T = (G , type) over TG is graph G and graph morphism type : G → TG

G1

P

setFlag

type graph is a distinguished graph TG = (VTG , ETG , sTG , tTG )

active

P

T

start

P

2

1

P

start

T

P

G1 F1

setTurn non−active

non−active

R

active

R

1

P

T

R

start

R

T



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Notion of Category

Examples of Categories 1

Definition

objects: sets, morphisms: functions f : A → B, composition for f : A → B and g : B → C defined by (g ◦ f )(x) = g (f (x)), identity: identical mapping idA : A → A : x → x.

category C = (ObC , MorC , ◦, id) defined by class ObC of objects; set MorC (A, B) of morphisms ∀A, B ∈ ObC ;

2

composition ◦ : MorC (B, C ) × MorC (A, B) → MorC (A, C ) ∀A, B, C ∈ ObC ,

3

such that: Associativity. (h ◦ g ) ◦ f = h ◦ (g ◦ f )

2

Identity. f ◦ idA = f and idB ◦ f = f

g

f

category GraphsTG objects: typed graphs, morphisms: typed graph morphisms, composition and identity: as in Graphs.

h

∀A → B → C → D 4

∀f : A → B

category Alg(Σ) objects: algebras over a given signature Σ, morphisms: homomorphisms composition and identity: componentwise on the carrier sets.

for f ∈ MorC (A, B), we write f : A → B H. Ehrig, U. Prange, K. Ehrig ()

category Graphs objects: graphs, morphisms: graph morphisms, composition and identity: componentwise on nodes and edges.

identity morphism idA ∈ MorC (A, A) ∀A ∈ ObC ; 1

category Sets

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Mono-, Epi- and Isomorphisms


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Pushouts as Gluing Construction

Definition m : B → C ∈ MorC is monomorphism if m ◦ f = m ◦ g implies f = g A

f g

B

m

C

e : A → B ∈ MorC is epimorphism if f ◦ e = g ◦ e implies f = g

A

Definition

∀f , g : A → B ∈ MorC e

B

f g

pushout (PO) over morphisms f : A → B and g : A → C defined by

C

f

B

g

=

g

C

f

D =

a pushout object D and morphisms f : C → D and g : B → D with f ◦ g = g ◦ f

∀f , g : B → C ∈ MorC

Fact In Sets, Graphs, and GraphsTG the monomorphisms (or epimorphisms or isomorphisms) are exactly those morphisms which are injective (or surjective or bijective, respectively). Tutorial on Graph Transformation

July 15, 2006

k

=

h x

X with universal property: For all objects X , morphisms h : B → X , k : C → X with k ◦ g = h ◦ f : there is a unique morphism x : D → X with x ◦ g = h and x ◦ f = k

i : A → B ∈ MorC is isomorphism if there is i −1 : B → A ∈ MorC with i ◦ i −1 = idB and i −1 ◦ i = idA


A

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The pushout construction is unique up to isomorphism.



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Pushouts in Sets

Pushouts in Graphs and GraphsTG Fact

Fact

In Graphs and GraphsTG , pushouts can be constructed componentwise for nodes and edges in Sets.

g

f

In Sets, pushout C → D ← B over f : A → B and g : A → C is

D = B ∪ C |≡ , with ≡ generated by (f (a), g (a)) ∈ ≡ f (c) = [c] ∀c ∈ C and g (b) = [b] ∀b ∈ B.

∀a ∈ A and

Example EA

Example a

5

A b

6

C

c

sA

1

d [1]

7

B 2

8


3

4

GB

sB

VA

gE

GA

EB

fE

fV

VB

gE

(1, 5), (2, 5), (2, 6), (3, 7) ∈ ≡

D [1] = [2] = [5] = [6] = {1, 2, 5, 6}, [3] = [7] = {3, 7}, [3] [8] [4] [4] = {4}, [8] = {8} July 15, 2006


gV

EC

fE

sC

sD fV

VC

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Composition and Decomposition of Pushouts

gV

ED


GC

GD

VD

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Pullbacks as Dual Construction of Pushouts

Fact PO composition: (1) and (2) POs ⇒ (1) + (2) PO PO decomposition: (1) and (1) + (2) POs ⇒ (2) PO

A

f

B

e

E

Definition

g

(1)

g

(2)

e

C

f

D

e

F

pullback (PB) over morphisms f : C → D and g : B → D defined by

X

a pullback object A and f

morphisms : A → B and g : A → C with g ◦ f = f ◦ g

Example

(1)

x k

h

= = A

f

B

g

=

g

C

f

D with universal property: For all objects X , morphisms h : X → B, k : X → C with f ◦ k = g ◦ h: there is a unique morphism x : X → A with f ◦ x = h and g ◦ x = k.

(2)

The composition and decomposition of pullbacks is dual to pushouts.



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Pullbacks in Sets, Graphs and GraphsTG

Part III Graph Transformation Systems

Fact g

f

In Sets, pullback C ← A → B over f : C → D and g : B → D: A = d∈D f −1 (d) × g −1 (d) = {(c, b) | f (c) = g (b)} ⊆ C × B with f : A → B : (c, b) → b and g : A → C : (c, b) → c.

11

Graph Productions and Transformations

12

Graph Transformation Systems, Grammars and Languages

13

Mutual Exclusion Graph Grammar

14

Applicability and Gluing Condition

15

Construction of Graph Transformations

16

Embedding of Transformations

∼

Special case: A = B ∩ C for inclusions f and g . In Graphs and GraphsTG , pullbacks can be constructed componentwise for nodes and edges in Sets.


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Graph Productions and Transformations


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Recap: Example Mutual Exclusion Example

Definition

non−active

l

P

setFlag

(typed) graph production r

p = (L ← K → R) consists of (typed) graphs L, K , and R,

L

l

K

r

m

(1)

k

(2)

R n

R

P

F1

R

m

2

1

TG :

non−active active

F2

P

F1

start crit check setTurn

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S

P

D 2

start non−active

∗

(typed) graph transformation, G0 ⇒ Gn , is sequence G0 ⇒ G1 ⇒ . . . ⇒ Gn of direct (typed) graph transformations

July 15, 2006

R

setTurn

non−active


P

start

injective (typed) graph morphisms l g f G D H and r p,m direct (typed) graph transformation G =⇒ H given by POs (1) and (2), production p and match m : L → G


active

P

T

start

P

2

1

P

start

T

P

G1 F1


non−active

R

active

R

1

P

T

R

start

R

T



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Example: Mutual Exclusion 2

Example: Mutual Exclusion 3

Example

Example

non−active

setFlag :

P

active

R

P

R

P

F1

R

non−active

⇓enter

non−active non−active

start

setTurn

non−active

non−active

S:

1

non−active

P

R

P T

start

P setTurn1:

P T

R

P

P

T

R

P

R

P

T

T

R

P

P

T R

P

F1

non−active

T

G1 :

R

1

R

P

R

T

P

R T

⇓setFlag

1

non−active

G2 :

1

R

P

F1

P

P

R T

setTurn

2

crit

⇓exit

R

P T

non−active

active

active

F1

start

active

F2

⇓setTurn1

F2

P

G4 :

2

setTurn

2

crit

G5 :

2

1

P

F1

R

T

P

2

start

setTurn

check

start

crit


P

R

non−active

F2

P

R

active

active

F1

start

P crit

P

⇓setFlag

T R

active

exit:

R

check

P check

T P

setTurn

enter :

1

active

F2

start

check

T

G3 :

start

P

setTurn

setTurn2:

2

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Graph Transformation Systems, Grammars and Languages


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Example: Language of Mutual Exclusion Grammar S: Example

Definition (typed) graph transformation system GTS = (TG , P) consists of (type graph TG and) set of (typed) graph productions P

setFlag exit

(typed) graph grammar GG = (GTS, S) consists of GTS and (typed) start graph S

process 1 active setTurn flag =1 turn

process 2 non−act. start flag =0

setTurn2

(typed) graph language L of GG is defined by

process 1 active check flag =1 turn

∗

L = {G | ∃ (typed) graph transformation S ⇒ G }.


setFlag

Example typed graph transformation system ME = (TG , P) typed graph grammar MutualExclusion = (TG , P, S)


setFlag

setFlag



process 2 non−act. start flag =0 setFlag turn process 1 non−act. setFlag start flag =0 process 1 process 2 active non−act. setTurn start flag =1 flag =0 turn

setFlag

process 1 non−act. start flag =0 turn

process 2 active setTurn flag =1

setFlag

setFlag


process 1 active setTurn flag =1 exit



exit

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exit

setTurn2


process 2 active check flag =1 turn enter

setFlag

setTurn2

enter

process 1 active crit flag =2 turn

setFlag

setTurn1


setTurn2

P = {setFlag , setTurn1, setTurn2, enter , exit}.



setTurn1

enter


process 1 non−act. start flag =0 turn



setFlag

enter





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Applicability and Gluing Condition

Example and Counterexample: Gluing Condition

Problem: When is production p applicable via match m?

Counterexample

Example

Definition

setFlag non−active

production p is applicable to G via match m if context graph D exists such that (1) is PO

L

K

l

R

r

P

active

R

P

R

start

(1)

m

k

G gluing points GP = lV (VK ) ∪ lE (EK ) = l (K )

D

f

F1

R

2

1

S

P

P

∅

m

D 2

start non−active

identification points IP =

∅

P

m

non−active

Gluing condition: IP ∪ DP ⊆ GP

P setTurn

T

P

2

R

1

P

T

start

non−active

G1

P

F1

2

R

1

P

T

e % e % e % e % e % e D% %e e % e % e % e %

S

P

start non−active


non−active

start

active

R

1

P

T

R

start

start

{v ∈ VL | ∃w ∈ VL , w = v : mV (v ) = mV (w )} ∪ {e ∈ EL | ∃f ∈ EL , f = e : mE (e) = mE (f )}

dangling points DP =

GP = l (K ): both nodes in L,

GP = l (K ) = ∅,

{v ∈ VL | ∃e ∈ EG \mE (EL ) : sG (e) = mV (v ) or tG (e) = mV (v )}

IP = ∅,

IP = ∅,

DP = node labelled R.

DP = node labeled P.

Fact



gluing condition is not satisfied, because DP GP

gluing condition is satisfied with IP ∪ DP ⊆ GP

Context graph D with the PO (1) exists iff the gluing condition is satisfied. If D exists, it is unique up to isomorphism. July 15, 2006

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July 15, 2006


Construction of Direct Graph Transformations

Embedding of Graph Transformations

Fact

Problem: When can graph transformation t : G0 ⇒ Gn be extended to ∗ t : G0 ⇒ Gn for k0 : G0 → G0 ?

∗

L

l

K

r

R

m

(1)

k

(2)

n

p,m

If gluing condition is satisfied, G ⇒ H can be constructed in two steps:

2

Theorem ∗

G D H Delete those nodes and edges in G that are reached by the match m, but keep the image of K , i.e. D = (G \m(L)) ∪ m(l (K )), such that G = L +K D in PO (1). f

1

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g

Given transformation t : G0 ⇒ Gn and consistent morphism k0 : G0 → G0 w.r.t. t, then there is an extension diagram (1) over t and k0 . ∗ t G0 Gn

Add those nodes and edges that are newly created in R, i.e.

H = D ∪ (R\r (K )), such that H = R +K D in PO (2). This construction is unique up to isomorphism.



k0

(1)

G0

t

kn

∗ Gn

Problem: What is consistency of k0 with respect to t?

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Boundary, Context and Consistency

Example: Embedding of Graph Transformation

Definition Example

boundary B is smallest subgraph of G0 containing IP and DP of k0 : G0 → G0

bn

non−active

context C is smallest subgraph of G0 such that G0 = G0 +B C

1

P

1

t

B

consistency: boundary B is preserved by G0 ⇒ Gn leading to bn : B → Gn

S

T

non−active

R

P

2

start

non−active

active

∗

P

G5

start

1

P

F1

R

C bn

B In case of consistency we also have Gn = Gn +B C

b0

(2) C


G0

t

k0

(1)

G0

t

∗ Gn

1

non−active

P

1

T

R

G0

P

start

k5

T

R

T

R

non−active

P

start

2

setTurn

k0

Fact

T

2

active

∗

P

G5

start

1

F1

R

T

R

T

non−active

P

2

setTurn

P

start

kn

∗ Gn


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Parallel and Sequential Independence

Part IV Independence of Graph Transformations and Critical Pairs

Problem: When are direct transformations commutable?

Definition p1 ,m1

p2 ,m2

17

Parallel and Sequential Independence

18

Local Church-Rosser Theorem

19

Parallelism Theorem

G =⇒ H1 and G =⇒ H2 are parallel independent if m1 (L1 ) ∩ m2 (L2 ) ⊆ m1 (l1 (K1 )) ∩ m2 (l2 (K2 )). p2 ,m2 p1 ,m1 G =⇒ H1 =⇒ H2 are sequentially independent for comatch n1 : R1 → H1 if n1 (R1 ) ∩ m2 (L2 ) ⊆ n1 (r1 (K1 )) ∩ m2 (l2 (K2 )).

20

Global Determinism and Confluence

Fact

21

Critical Pairs

p1 ,m1

p2 ,m2

R

r1

1

22

Local Confluence Theorem - Critical Pair Lemma

23

Functional Behavior of Graph Transformation Systems

K

l1

1 1 G =⇒ H1 and G =⇒ H2 are n j k parallel independent iff there exf ist morphisms i : L1 → D2 and H1 g D1 j : L2 → D1 such that f2 ◦ i = m1 and f1 ◦ j = m2 . 1

1

1

L1

L2

m1 m2

G

l2

K2

r2

n2

k2

i f2

D2

R2

g2

H2

Similar for sequential independence. H. Ehrig, U. Prange, K. Ehrig ()


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Examples: Parallel and Sequential Independence

Local Church-Rosser Theorem

Example

Theorem

Example setFlag non−active

K1 ...

P

R

P

setFlag non−active

L1

R

P

start

1

P

2

R

...

R

P

...

T P

R

crit

1

P

P

P

2

P

P

2 g1

non−active

R

P

1

T

R

1

P

T

R

P

P

P

p2 ,m

R

...


p1 ,m

p1 ,m1

p1 ,m

p2 ,m2

G

G

D2

active

F2

T

2

F2

1

P

p1 ,m1

p2 ,m2

p2 ,m2

p1 ,m1

H1

crit

f2

R

P

T

R

sequentially independent: n2 (R1 ) ∩ m3 (L2 ) = {node labelled R} = n2 (r1 (K1 )) ∩ m3 (l2 (K2 )) July 15, 2006

H2 G

start


p2 ,m

G =⇒ H2 =⇒1 G are sequentially independent.

2

parallel independent: m(L1 ) ∩ m (L2 ) = {node labelled R} = m(l1 (K1 )) ∩ m (l2 (K2 ))

p2 ,m2

m3

crit non−active 1

start

R

p1 ,m1

If G =⇒ H1 and G =⇒ H2 are parallel independent, there are G and H1 =⇒2 G and H2 =⇒1 G such that G =⇒ H1 =⇒2 G and

start

active

start

f2

T

2

start

start

R F2

D1

active

D2

K2

non−active

n2

non−active

S

start non−active

f1

T

P

R1

m

m

non−active

R

non−active

P

K2

start

D1 2

L2

1

setFlag L2

enter T K1

p1 ,m1

p2 ,m2

If G =⇒ H1 =⇒ G are sequentially independent, there are H2 and p2 ,m2

p1 ,m

p1 ,m1

p2 ,m2

G =⇒ H2 =⇒1 G such that G =⇒ H1 and G =⇒ H2 are parallel independent.

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Example: Local Church-Rosser Theorem


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Overview of Parallelism and Concurrency

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SKIP

Example non−active

P

1

T

start

R

1

P

start

P

R

P T

Solution: 1. Parallel composition p1 + p2 (independency) 2. E-concurrent production p1 ∗E p2 (dependency)

Theorem

setFlag ,m

active

active

F1

2

start

S

setFlag ,m

non−active

Problem: How to compose productions p1 and p2 ?

non−active

G

G1

2

1

P

setTurn

setTurn

P

R T

2

p1

start

p2 p1 +p2

H1

setFlag ,m

setFlag ,m

p2 active 1

R

P

setTurn


X F1

T Tutorial on Graph Transformation

P

H H2

p1

p1

G

p2 p 1 ∗E p 2

G

G

active

F1

Concurrency

Parallelism G

non−active

F1

2

setTurn

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Parallel Productions and Transformations

Parallelism Theorem

Definition l

r

l

r

1 1 2 2 Given productions p1 = (L1 ← K1 → R1 ) and p2 = (L2 ← K2 → R2 ), the parallel production p1 + p2 is defined by the disjoint union

l1 ∪l2

r1 ∪r2

Theorem Given a transformation system GTS, we have: 1

p1 + p2 = (L1 ∪ L2 ←− K1 ∪ K2 −→ R1 ∪ R2 ) The application of a parallel (typed) graph production is called a parallel direct (typed) graph transformation.

2

Example

3 non−active

P enter + setFlag :

start

R

P

T

P check

active

R

setTurn

T R

P

R

F1


P

R

T

P crit


F1

R

Synthesis. If G ⇒ H1 ⇒ G via productions (p1 , p2 ) are sequentially independent, then there is a parallel transformation G ⇒ G via the parallel production p1 + p2 . Analysis. For G ⇒ G via p1 + p2 there are sequentially independent G ⇒ H1 ⇒ G via (p1 , p2 ) and G ⇒ H2 ⇒ G via (p2 , p1 ). G Bijective correspondence. p1 p2 The synthesis and analysis p1 +p2 constructions are inverse to H1 H2 each other up to p2 p1 isomorphism. G

F2

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Concurrency Theorem

Example: Concurrency Theorem

Problem: How to compare productions and transformations in general?

Example non−active

Theorem Given productions p1 and p2 with E -concurrent production p1 ∗E p2 , then: 1

2

3

Synthesis. Given an E -related transformation sequence G ⇒ H ⇒ G via (p1 , p2 ), then there is a synthesis construction leading to a direct transformation G ⇒ G via p1 ∗E p2 . Analysis. Given a direct transformation G ⇒ G via p1 ∗E p2 , then there is an analysis construction leading to an E -related transformation sequence G ⇒ H ⇒ G via (p1 , p2 ). Bijective correspondence. The H synthesis and analysis p1 p2 constructions are inverse to each p 1 ∗E p 2 G G other up to isomorphism.

1

P

T

R

start


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P

2

setTurn

G1 setTurn1,m1

non−active

G2

setFlag ∗E setTurn1,m

S

2

P T

setFlag ,m

non−active

start

1

P

active

F1 R

P T

start

2

check

setFlag ∗E setTurn1: non−active

P E :

non−active

T

active

P

P R

F1

setTurn


R

start

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active

F1

P

1

non−active

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non−active

T

non−active

P

P R

non−active

T

P R

P

start


T

active

P

R F1

check

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Global Determinism and Confluence

Termination and Confluence

Problem: How to achieve global determinism of a GTS in spite of local nondeterminism?

Definition

Definition

∗

A GTS is (locally) confluent if, for all (direct) transformations G ⇒ H1 ∗ ∗ ∗ and G ⇒ H2 , there is X and transformations H1 ⇒ X and H2 ⇒ X .

A GTS is terminating if there is no infinite sequence of graph ∗ tn transformations (tn : G ⇒ Gn )n∈N with tn+1 = G ⇒ Gn ⇒ Gn+1 .

G

G ∗

∗ H2

H1

Lemma Every terminating and locally confluent GTS is also confluent.

H2

H1

∗ X ∗

Problem: How to obtain local confluence in spite of parallel dependent graph transformations?

∗ X ∗

Lemma Every confluent GTS is globally deterministic. H. Ehrig, U. Prange, K. Ehrig ()

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Critical Pairs

Local Confluence Theorem - Critical Pair Lemma

Definition

Theorem

p1 ,o1

p2 ,o2

Pair P1 ⇐= K =⇒ P2 of direct transformations is called a critical pair if it is parallel dependent and minimal, i.e. the matches o1 : L1 → K and o2 : L2 → K are jointly surjective.

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A (typed) graph transformation system GTS is locally confluent if all its critical pairs are strictly confluent.

Example Critical Pair 1

Critical Pair 2 non−active

non−active

active

P

F1 R

P

non−active

R

T

P

R

P

setTurn

non−active

active

P T

start

setTurn

non−active

P

R

non−active

P

F1 R

P

R

T

P

start

setTurn

start

P T

R

setTurn

P1

active

P

non−active

F1

o1

o2

K1

P

P setTurn

R

P1

active

P

F1

P setTurn


R

P

P K2 R

setFlag start

T

o2

non−active

R

setTurn1 start

T

P2

P

setFlag setTurn

non−active

P check

setTurn

T R

P

T R

setTurn


non−active

P

P2 R

setTurn1 start

P setTurn

start

T R

start

T

P

R

setFlag

non−active setTurn1

setTurn

F1

P

P2

P

P1

active

P

F1

K2 R

T

P

R

non−active

setFlag

setTurn1 setTurn

R

P

P2 R

check setTurn

o1

P1

active

P P

check

non−active

strict confluent critical pairs

P

non−active

P

R

K1

P

start

T R

non−active ∗

setTurn

start

R

P

T R

P

start

T R

non−active ∗

id check

P

setTurn

P start

T

P

T

P2

setTurn

R

P

check

P2 R

P

T R

id check

T R

check

check

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Functional Behavior of Graph Transformation Systems

Part V Adhesive HLR Categories and Typed AGT Systems

Theorem If a GTS is terminating and locally confluent, it has functional behavior: 1

2

∗

For each G , there is a terminating transformation G ⇒ H in GTS, and H is unique up to isomorphism. ∗

Motivation for a Categorical Framework

25

Van Kampen Squares

26

Adhesive and Adhesive HLR Categories

27

Adhesive HLR Systems

28

Typed Attributed Graphs

29

Typed Attributed Graph Transformation Systems

∗

Each pair G ⇒ H1 and G ⇒ H2 can be extended to terminating ∗ ∗ ∗ ∗ transformations G ⇒ H1 ⇒ H and G ⇒ H2 ⇒ H with the same H.

Theorem Every layered typed graph grammar GG = (TG , P, G0 ) with injective matches terminates if it satisfies suitable layer conditions for deleting and nondeleting productions.


24


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Motivation for a Categorical Framework



Overview of Adhesive HLR Categories

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SKIP

Axioms: Adhesive HLR categories (C, M) satisfy: Problem: Up to now we have graph transformation theory for Graphs, GraphsTG only But the theory is required also for hypergraph transformation systems, Petri net transformation systems, transformation systems of algebraic specifications and typed attributed graph transformation systems.

2

pushouts (POs) and pullbacks (PBs) along M-morphisms

3

compatibility of POs and PBs by van Kampen squares

Theorem

Replace Graphs by suitable category. Suitable is Adhesive HLR category based on van Kampen property.


M ⊆ Monos closed under composition and decomposition

Examples: Sets, Graphs, GraphsTG , PTNets, AHLNets, (AGraphsATG , M)

Solution:


1

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1

construction of adhesive HLR categories by product, slice, coslice, comma and functor categories

2

properties of adhesive HLR categories are sufficient for all constructions and results of Parts II-IV



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Van Kampen Squares

Adhesive Categories

Idea: compatibility of pushouts and pullbacks

Definition

Definition

Pushout (1) is a van Kampen square if, for any commutative cube with (1) in the bottom and where the back faces are pullbacks, we have:

category C is an adhesive category if:

The top face is pushout iff the front faces are pullbacks.

Example {0, 1}

{∗}

A

m

f

(1)

C

n

B

A

f

g C

m a

n

D

D

c f

C

B

g

A

b

g

B


{0, 1, 2}

{∗, 2}

D Tutorial on Graph Transformation

C has pushouts along monomorphisms (i.e. pushouts where at least one of the given morphisms is a monomorphism)

2

C has pullbacks

3

Pushouts along monomorphisms are VK squares

Examples: Sets, Graphs, GraphsTG , Hypergraphs, ElemNets Counterexamples: PTNets, AGraphsATG

m

d

n

(2) {∗}

{∗, 2, 3} {0, 1}

{0, 1, 2, 3}

1

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Idea for generalization: Replace monomorphisms by subclass M and pullbacks by those along monomorphisms H. Ehrig, U. Prange, K. Ehrig ()


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Construction and Properties of AHLR categories

(Weak) Adhesive HLR Categories

Theorem If (C, M1 ) and (D, M2 ) are (weak) adhesive HLR categories, then the following categories are (weak) adhesive HLR categories:

Definition category C with a morphism class M is called a (weak) adhesive HLR category if: 1

2

3

M is a class of monomorphisms closed under isomorphisms, composition (f : A → B ∈ M, g : B → C ∈ M ⇒ g ◦ f ∈ M), and decomposition (g ◦ f ∈ M, g ∈ M ⇒ f ∈ M)

1

the product category (C × D, M1 × M2 ), the slice category (C\X , M1 ∩ C\X ) and the coslice category (X \C, M1 ∩ X \C),

2

the functor category ([X, C], M − functor transformations),

3

the comma category (ComCat(F , G ; I), (M1 × M2 ) ∩ MorComCat ), where F : C → X preserves pushouts along M1 -morphisms and G : D → X preserves pullbacks (along M2 -morphisms).

C has pushouts and pullbacks along M-morphisms, and M-morphisms are closed under pushouts and pullbacks Pushouts in C along M-morphisms are (weak) VK squares

In (weak) adhesive categories, the following properties hold:

Examples: (PTNets, Mmono ), (Spec, Mstrict ), (AHLNets, Mmono ), (AGraphsATG , M) (see below)



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1

Pushouts along M-morphisms are pullbacks

2

M pushout–pullback decomposition lemma

3

Cube pushout–pullback lemma

4

Uniqueness of pushout complements



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Adhesive HLR Systems

Motivation for Typed AGT Systems

Definition adhesive HLR system AHS = (C, M, P) consists of (weak) adhesive HLR category (C, M) and set of productions P with morphisms in M adhesive HLR grammar AHG = (AHS, S) is an adhesive HLR system together with a distinguished start object S

Typed attributed graph transformation (AGT) is integration of typed graph transformation with data types as attributes for modeling of visual languages (VL) with alphabet of a VL given by attributed type graph, syntax given by generating grammar, operational semantics given by simulating grammar and animation.

language L of an adhesive HLR grammar is defined by ∗

L = {G | ∃ transformation S ⇒ G }.

model transformations between VLs by typed AGT systems.

The categorical framework allows to prove the results from Parts III and IVj for all the instantiations (like Graphs, HyperGraphs, PTNets etc.).


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Typed Attributed Graphs

Example: Typed Attributed Graphs

Definition

Example EG

E-graph G is defined by G = (VG , VD , EG , sourceEA ENA , EEA , (sourcej , targetj )j∈{G ,NA,EA} ). EEA

sourceG targetG

ENA targetNA

∪

param

Method

mname

par2

2

p2 Parameter par3

c

Nat

order

nat

Notations for typed attributed graphs: pname

type

m:Method

:param

par1 :Parameter

mname=add noOfPars=3

order =1

pname=p1 kind=in

:param

par2 :Parameter

order =3

:type

pname=p2 kind=in

:param

Ds = VD .

order =2

string

AGraphsATG = category of typed attributed graphs (G , D) and typed attributed graph morphisms (fG , fD ) consistent w.r.t. SD over an attributed type graph ATG H. Ehrig, U. Prange, K. Ehrig ()

3

kind

noOfPars

Let DSIG = (SD , OPD ) be a data signature with SD ⊆ SD . Attributed graph AG = (G , D) is E-graph G and DSIG -algebra D with s∈SD

in

parameterDirectionKind

return

VD

p1

1

sourceNA

targetEA

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par1

m

add

Type graph ATG :

VG

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cname

Class

DSIG = CHAR + STRING + NAT + sorts : parameterDirectionKind opns : in, out, inout, return :→ parameterDirectionKind SD = {string , nat, parameterDirectionKind}

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:type

par3 :Parameter kind=return


c:Class

:type

cname=Nat

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Typed Attributed Graph Transformation

Theory for Typed AGT Systems Idea: Instantiation of theory for adhesive HLR systems

Definition

Theorem l

r

typed attributed graph production p = (L ← K → R) consists of typed attributed graphs L, K , R with common DSIG -termalgebra TDSIG (X ) with variables X , and l , r ∈ M, with M = {f = (fG , fD ) | fg injective, fD isomorphism of DSIG -algebras} direct typed attributed graph transforp,m mation G =⇒ H from G to H via the match m is given by the following double pushout diagram in AGraphsATG

L

l

K

r

R

m

(1)

k

(2)

n

G

f

D

g

H

(AGraphsATG , M) is an adhesive HLR category.

Corollary In AGraphsATG , the following results hold: Construction of typed attributed graph transformations, Embedding Theorem, Local Church-Rosser Theorem, Parallelism Theorem, Concurrency Theorem, Local Confluence Theorem.



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Overview of Constraints and Application Conditions

Part VI

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SKIP

Advanced Features and Conclusion Idea of graph constraints: Restriction of graphs / graph languages by existence or nonexistence of suitable graph patterns, called constraints c

30

Graph Constraints

31

Application Conditions

32

Typed AGT Systems with Inheritance

33

Modeling of Visual Languages

Idea of application conditions: p,m Restriction of direct graphs transformations G ⇒ H by suitable conditions for the match m (e.g. gluing condition)

34

Model Transformations

Theorem

35

Implementation of Typed AGT Systems in AGG

Construction of application condition acc for match m from graph constraint c such that H |= c if m |= acc.

36

Conclusion



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Graph Constraints

Application Conditions

Problem: How to define graph constraints?

Problem: How to define application conditions?

Definition

Definition

atomic (typed) graph constraint is of form PC (a), where a : P → C is (typed) graph morphism

P

C

a

p = q (typed) graph constraint is Boolean formula over atomic (typed) graph constraints G G |= PC (a), if, for every injective morphism p : P → G , there exists injective morphism q : C → G such that q ◦ a = p. This can be extended to boolean formulas.

Example R

a

P

T

G satisfies this constraint if, for each resource node R, there is a turn variable that connects it to a process.

R C

P



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atomic application condition over L is given by P(x, ∨i ∈I xi )

L

m |= P(x, ∨i ∈I xi ), if, for all injective m = p : X → G with p ◦ x = m, there exist i ∈ I and injective qi : Ci → G with G qi ◦ xi = p. This can be extended to boolean formulas. negative application condition NAC (x) is a L morphism x : L → X . A morphism m : L → G satisfies NAC (x) if there does not exist an injective p : X → G with p ◦ x = m.


Construction of Application Conditions

Definition

Theorem

application condition A(p) = (AL , AR ) for production p consists of left application condition AL over L and right application condition AR over R p,m

G ⇒ H with comatch n : R → H satisfies A(p) if m |= AL and n |= AR .

=

p

qi

x m

Ci

xi

X

= p| G

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Productions with Application Conditions

X

x

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For each graph constraint c for H, there is an equivalent right application condition racc for the comatch n. For each right application condition racc for comatch n, there is an equivalent left application condition lacc for match m. r l L K R

Example m

(1)

k

(2)

n

G

f

D

g

H

addResource with NAC (x): P x

active


P

P


P

T

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H |= c, if m |= lacc.



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Typed AGT Systems with Inheritance

Modeling of Visual Languages

{abstract} has

Screen

visible

Figure

Example

Bool

addTransition() [X={}] LHS

height width

RHS 1:StateMachine

id

NAC=RHS

Problem: How to handle inheritance?

x y

Circle

Generation of Statecharts by typed attributed graph grammar

radius width height

Nat

has id: Nat

Rectangle Screen

endx

width: Nat height: Nat

endy

x: Nat y: Nat visible: Bool

String

has id: Nat

:sm2conf :sm2step :from :to 2:Conf 3:Conf 4:Step

:conf2state

Conf

5:State Step

trigger

Line

Transition

trans2act

end cond2state

endx: Nat endy: Nat x: Nat y: Nat visible: Bool

Boolean

RHS 1:StateMachine

Cond

SC type graph

1:StateMachine

:sm2conf

:sm2conf :sm2step :to :from 2:Conf 3:Conf 4:Step :conf2state 5:State

:sm2conf

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Model Transformation from Statecharts to Petri Nets ATGS

incS

ATG

incT

ATGT

typeS

typeGS

typeGi

typeGT

typeT

GS

...

pi

Gi

pj

SC type graph

StateMachine smname: String

:conf2state State

PreArc presrc

from Conf

trigger

Step to

conf2state

Event ename: String

step2trans begin

trans2act

:step2trans

:conf2state 5:State

Transition :end

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Implementation of Typed AGT Systems in AGG Screenshot of SC 2PN Model Transformation

GT

pl

pretgt

Trans

Place plname: String token: Boolean

postsrc

act2event

stname: String end isInit: Boolean Transition


:to 3:Conf

4:Step

PN type graph

sm2step

sm2conf

State

...

pk


:sm2conf :sm2step

2:Conf :from

:begin

July 15, 2006


Transition :end

[X={}]

LHS

Action

NAC=RHS trans2cond

State

act2event

addTransitionLeftState()

begin

isInit

:conf2state

:step2trans

5:State

Event

:to 3:Conf

4:Step

:begin step2trans

conf2state

:sm2conf :sm2step

2:Conf :from

to String

State


Transition :end

:sm2conf

:conf2state

ename

from

width: Nat height: Nat x: Nat y: Nat visible: Bool

State

1:StateMachine

:sm2conf

sm2conf

stname

radius: Nat x: Nat y: Nat visible: Bool

:conf2state

:step2trans

RHS 1:StateMachine

sm2step

Rectangle

:to 3:Conf

4:Step

:begin

LHS

StateMachine

Circle

:sm2conf :sm2step

2:Conf :from

State

NAC=RHS

smname

has id: Nat

:sm2conf

addTransitionRightState() [X={}]

only in abstract closure

Line has id: Nat

:sm2conf :sm2step :from :to 2:Conf 3:Conf 4:Step

:conf2state

{abstract}

Figure

1:StateMachine

:sm2conf

posttgt

PostArc

Action

cond2state trans2cond Cond r1

r2

r3

r4

r5

r6

RefEvent

RefState

SC 2PN type graph ATG H. Ehrig, U. Prange, K. Ehrig ()

RefStep


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Conclusion

Fundamental aspects of algebraic graph transformation: 1 Algebraic concept of graph transformation based on gluing construction and double pushouts. 2

Basic results concerning independence, parallelism, critical pairs.

3

Categorical framework of AHLR systems for unified constructions and proofs.

4

Typed attributed graph transformation for modeling of software systems, model transformations of visual languages.



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