Introduction to Graph Transformation

Introduction to Graph Transformation

Estonian Summer School on Computer and Systems Science

Lecture 1Arend Rensink, University of Twente

ESSCASS, 25 August 2008Introduction to Graph

Transformation2

Lectures & assignments

1. Introduction to Graph Transformation

A. Some puzzles and games

2. Graph-Based Operational SemanticsB. Semantics of a small language

3. Graph-Based Model Checking


Transformation3

What

• The verification question:1. Given a requirement (in some logic)2. Given a system model M (in some

language that admits a semantic of the logic)

3. Does M satisfy ? (notation: M ² )

• In this course:1. Versions of temporal logic

• Safety: invariants, no exceptions thrown• Liveness: termination, fairness

2. Graph grammars as modelling language3. Model checking as basic technique


Transformation4

Why

• Why verification?– Design/implementation-time sanity check

(precedes testing in software development)

– But: integration in design process not easy (due to absence of models)

– Hence: software verification• Why graph transformation?– Graphs are natural for the domain– Easier to grasp for non-specialist– Captures dynamic states (heap, stack)


Transformation5

Example: Circular buffer

• Data structure module as subsystem• Model the essence– Structure captured by type graph (classes)

• Buffer cells linked in cycle• Each cell may contain a value (an object)• Pointers to first and last elements• Further data invariants

– Instances captured by graphs

• Dynamics defined by operations– E.g., insertion and deletion– Captured by changes in graphs


Transformation6

Example type graph

• Node for each type• Labelled edge for each “property”– Edge label = property name

• Possible: multiplicities– These graphs are deterministic

Buffer

Cell

next

first

Objectval

last

11 1

0..1


Transformation7

Example instance graph

• Snapshot of concrete data (on the heap)– Four-cell buffer, of which 2 cells filled

• Type information & multiplicities satisfied– There are many more invariant properties!

BufferCell

Cell

Cell

Cellnext

next next

next

lastfirst

Object

Objectval

val


Transformation8

Graphs, formally

• Tuple <L,V,E>– L: set of labels– V: set of nodes (vertices)– E µ V £ L £ V: set of binary, labelled

edges– No node labels (but self-edges mimic

them)

• Example

Cell

Cell

next

Objectval

L = {Call, Object, next, val}V = {1, 2, 3}E = { (1,Cell,1), (2,Object,2), (3,Cell,3),

(1,val,2), (1,next,3) }Choice of identities is irrelevant

1

2

3


Transformation9

Graph morphisms

• Consider G = <LG,VG,EG> and H = <LH,VH,EH>

• Morphisms f: G ! H– functions fV: VG ! VH and fE: EG ! EH

– preserve structure: fE(v,a,w) = (fV(v), a, fV(w))

• Isomorphism– fV and fE are bijective– Abstraction from node identities

• Partial morphism– f does not have an image for all elements of G


Transformation10

Typing: total morphism to type graph

1

2

3

6

5 74

BufferCell

Cell

Cell

Cellnext

next next

next

lastfirst

Object

Objectval

val

Buffer

Cell

next

first

Objectval

last

1

3

2fV = { (1,3),

(2,1), (3,1), (4,1), (5,1),(6,2), (7,2)}

Type graph

Source graph

11 1

0..1

Multiplicities not regarded


Transformation11

Example invariant properties

• The Cells of a Buffer form a cycle• All Cells are connected to a Buffer• Either first is pointing to a filled Cell, or

the Buffer is empty (no Cells are filled)• Either the next of a filled Cell is filled,

or the Cell is the last• If the next Cell is filled and it is not the

first, then this Cell is also filled• No Objects are shared between Cells

(?)


Transformation12

Data structure operations

• What does a “put” operation do?– Move the last pointer to the next Cell– Attach an Object to the (now) last Cell– In this model, we create a fresh Object– Only if this next Cell did not have a val

yet!• What does a “get” operation do?– Delete the val of the first Cell– Move the first pointer to the next Cell– Only if the first Cell had a val!

• What does an “extend” operation do?– Insert a new Cell after the current last


Transformation13

Graph transformations

• The operations are graph changes– These can be captured as changes

• What has to be removed• What has to be added

– In addition: where in the host graph• E.g., for “get”: (simultaneously)– Identify b and c connected by a first edge– Remove the val-edge and Object at c – Remove the first edge pointing to c– Add a first edge from b to the next w.r.t.

c • This constitutes a transformation rule


Transformation14

Example production rule

Single-graph representation of <put>

Buffer Cell

Cell

Object

next

val

val

last

last

Object

blue = eraser:to be matched and

deleted

green = creator:

to be added

black = reader:to be matched and

preserved

red = embargo:forbidden


Transformation15

Equivalent rule syntax

Buffer Cell

Cell

nextlast

Object

val

Right Hand Side(RHS)

Left Hand Side(LHS)

Buffer Cell

Cell

nextlast

Negative Application Condition (NAC)

Buffer Cell

Cell

nextlast

Object

val

Buffer Cell

Cell

Object

next

val

vallast

last

Object

Single-graph:

Multi-graph:Partial graph morphisms


Transformation16

Production rules

• Partial morphism p: L R– L \ dom(p): elements to be deleted– R \ cod(p): elements to be added– Non-injectivity (f(x)=f(y) for x y):

merging

• Rule application to host graph G: – Find total matching m: L G

• Multiple matchings are possible– Subtract m-image of L from G– Add R disjointly to result

• Partial morphisms h: G H, m’: R H


Transformation17

Example rule application

Buffer

Cell Cell

Cell

first | last

next

next

next

Objectval

Buffer

Cell Cell

Cell

last

firstnext

next

next

Objectval

Object

val

Buffer

Cell

Cell

Object

next

val

val

last

last Object

matching

tran

sition


Transformation18

Transitions: Partial morphisms

1

2

3

6

5 7

4

BufferCell

Cell

Cell

Cell

next

next next

next

last

first

Object

Objectval

val

1

2

3

6

5

4

BufferCell

Cell

Cell

Cell

next

next next

nextlastfirst

Objectval

1 cell filled

2 cells filled

<put>

<get>

fV = { (1,1),(2,2), (3,3), (4,4),

(5,5),(6,6)}

fE partial on edge (1,last,2)

fV = { (1,1),(2,5), (3,2), (4,3), (5,4),(7,6)}

fV partial on node 6 fE partial on several edges


Transformation19

Negative application conditions

Buffer Cell

Cell

nextlast

Object

val

Buffer Cell

Cell next

last

BufferCell

Cell

Cell

Cellnext

next next

next

lastfirst

Object

Objectval

val

Objectval

Object

val

• Here the rule should not be applicable!

How to rule out this matching?

LHS RHS

Buffer Cell

Cellnext

last

Object

val

NAC

m

n

There exists a total morphism

g: NAC G such that

m = g n(g factors m through n)

Gg


Transformation20

Kinds of NACs

• Merge embargoes: force injectivity– Non-injective morphism n: LHS NAC– n(v1) = n(v2) rules out m(v1) = m(v2); hence

v1 and v2 must be matched injectively

• Edge embargoes: forbid edges– Edge not in image of n: LHS NAC– (v1,lab,v2) LHS and (n(v1),lab,n(v2)) NAC

imply (m(v1),lab,m(v2)) G

• General NACs: forbid larger subgraphs– Multiple NACs per rule


Transformation21

forbidden

Graph Productions

Production rule

source graph

matching

Graph transition

src(t) tgt(t)morph(t) target

graph

pushout

NACNACNACs

(SPO = Single Pushout Approach)

LHS RHSrule morphism

(partial)


Transformation22

Example production rule (again)Single-graph representation of <put>

Buffer Cell

Cell

Object

next

val

val

last

last

Object

blue = eraser:LHS, not RHS;

to be matched and deleted

green = creator:

RHS, not LHS;to be added

black = reader:LHS and RHS;

to be matched and preserved

red = embargo:

NAC, not LHS;forbidden

blue = eraser:to be matched and

deleted

green = creator:

to be added

black = reader:to be matched and

preserved

red = embargo:forbidden


Transformation23

Graphs as states

• Every graph represents a snapshot– State of the system

• Every rule application changes the graph– Transition of the system

• Together form a state/transition system– Captures the system behaviour– Basis for verification

• Exercise: state space of a 4-cell buffer – No “extend” operations

• Will “put-get” return to the same graph?


Transformation24

Graph transition system

BufferCell

Cell

Cell

Cellnext

next next

nextfirst, last

Objectval

<put>

<get>

BufferCell

Cell

Cell

Cellnext

next next

next

last

first

Object

Objectval

val

Objectval

<put>

<get>

BufferCell

Cell

Cell

Cellnext

next next

next

lastfirst

Object

Objectval

val

Objectval

Object

val

<put>

<get>

<put>

<get>BufferCell

Cell

Cell

Cellnext

next next

nextfirst

last

BufferCell

Cell

Cell

Cellnext

next next

next

lastfirst

Object

Objectval

val


Transformation25

GROOVE Demo


Transformation26

Example rule application

1

2

36

54

BufferCell

Cell

Cell

Cell

next

next next

nextlastfirst

Objectval

Buffer Cell

Cell

nextlast

3

2

1 Buffer Cell

Cell

nextlast

Object

val

4

5

6

7L R

G

1

2

36

5 74

BufferCell

Cell

Cell

Cell

next

next next

next

last

first

Object

Objectval

val

H

p

m m’

h

(1,4), (2,5), (3,6)

(1,1), (2,2), (3,3)

(1,1), (2,2), (3,3), (4,4), (5,5) (6,6)

(4,1), (5,2), (6,3), (7,7)

• General requirement: h m = m’ p


Transformation27

Rule application: variation 1

1

2

36

54

BufferCell

Cell

Cell

Cell

next

next next

nextlastfirst

Objectval

Buffer Cell

Cell

nextlast

3

2

1 Buffer Cell

Cell

nextlast

Object

val

4

5

6

7L R

G

2

5

14

3 98

BufferCell

Cell

Cell

Cell

next

next next

next

last

first

Object

Objectval

val

H’

p

m m’

h

(1,4), (2,5), (3,6)

(1,1), (2,2), (3,3)

(1,2), (2,5), (3,1), (4,8), (5,3) (6,4)

(4,2), (5,5), (6,1), (7,9)

• Also correct: H H’


Transformation28

Rule application: variation 2

1

2

36

54

BufferCell

Cell

Cell

Cell

next

next next

nextlastfirst

Objectval

Buffer Cell

Cell

nextlast

3

2

1 Buffer Cell

Cell

nextlast

Object

val

4

5

6

7L R

G

1

2

3

54

BufferCell Cell

Cellnext

nextnext

next

last

first

H

p

m m’

h

(1,4), (2,5), (3,6)

(1,1), (2,2), (3,3)

(1,1), (2,2), (3,3), (4,4), (5,5)

(4,1), (5,2), (6,3)

• What’s wrong? h m = m’ p fulfilled!

ObjectCellval 6


Transformation29

Criteria for the target graph

• Should be the “minimal complete” one– Complete: elements kept when possible– Minimal: no spurious elements

• Universal property: Pushout

– Minimal because it’s “smaller” than the others– Complete because the subdiagrams commute

that’s the one!

rule (p)

matching (m)

diagramcommutes

(hm = m’p) wrong alternative


Transformation30

Example pushouts

BAb

1 2 1 CAc

2

BAb

4 5

B 6b

aCA

c4 5

B 6b

a

BAb

1 2 1 CAc

3

BAb

4 5

B 6b

aCA

c4 7

B 6b

AAa

1 2 1 CAc

2

BAb

4 5a

BCb

4 5c

(1,4),(2,4)

(1,4),(2,5)

AAa

1 2 1 CAc

3

BAb

4 5a

BC6 5

(1,4),(2,4)

(1,4),(2,5)

Aa

1 2 1 A 2

AB

b

4 5

(1,4),(2,5)

BAb

4 5a

A

x1

2 3

y

A1

2 3

x,y

A4

5

x,y

A4

5

x,y

(1,4),(2,5),(3,5)

x-edge not mapped!

Lessons: - Pushouts always exist- Leave no dangling edges- Deletion always wins


Transformation31

Principles of SPO transformation• What happens upon node deletion– All incident edges are deleted

• What happens upon node merging– Incident edges are copied to merged

nodes

• What happens for non-injective matches– Node deletion wins over preservation– Edge creation wins over deletion


Transformation32

Assignment A: Hands-On

• Download GROOVE– sf.net/projects/groove

• Model the following games/puzzles:– Wolf-Goat-Cabbage (WGC)– Solitaire– Pacman– Ludo (see course notes)

(Answers are available as downloads)• Specify properties of WGC– Type graphs– Invariants


Transformation33

Seen today

• Graphs and morphisms– Graphs: Tuples of nodes, labels, edges– Morphisms: node and edge mappings– State invariants as graph properties

• Graph transformation rules– Single-graph and multi-graph representation– Rule applications as pushouts– Negative application conditions

• GROOVE tool– State space generation – Graphs as states– Graph productions as transitions

Introduction to Graph Transformation

Documents

Transcript of Introduction to Graph Transformation