1. Introduction to Graph Transformation
Kaminski, Seidl et al. Muscholl
Estonian Summer School on Computer and Systems Science Lecture 1 Arend Rensink, University of Twente

2. Lectures & assignments
Introduction to Graph Transformation
Some puzzles and games
Graph-Based Operational Semantics
Semantics of a small language
Graph-Based Model Checking
ESSCASS, 25 August 2008
Introduction to Graph Transformation

3. Introduction to Graph Transformation
What
The verification question:
Given a requirement  (in some logic)
Given a system model M (in some language that admits a semantic of the logic)
Does M satisfy ? (notation: M ² )
In this course:
Versions of temporal logic
Safety: invariants, no exceptions thrown
Liveness: termination, fairness
Graph grammars as modelling language
Model checking as basic technique
ESSCASS, 25 August 2008
Introduction to Graph Transformation

4. Introduction to Graph Transformation
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)
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Introduction to Graph Transformation

5. 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
ESSCASS, 25 August 2008
Introduction to Graph Transformation

6. Introduction to Graph Transformation
Example type graph
next
val
Cell
Object 1
0..1
first
last 1 1
Buffer
Node for each type
Labelled edge for each “property”
Edge label = property name
Possible: multiplicities
These graphs are deterministic
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Introduction to Graph Transformation

7. Example instance graph
Buffer
Cell
next
last
first
Object
val
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!
ESSCASS, 25 August 2008
Introduction to Graph Transformation

8. Introduction to Graph Transformation
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
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
Cell
next
Object
val 1 2 3
ESSCASS, 25 August 2008
Introduction to Graph Transformation

9. Introduction to Graph Transformation
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
ESSCASS, 25 August 2008
Introduction to Graph Transformation

10. Typing: total morphism to type graph1 2 3 6 5 7 4
Buffer
Cell
next
last
first
Object
val
Source graph
Type graph
Buffer
Cell
next
first
Object
val
last 1 3 2
fV = { (1,3),
(2,1), (3,1), (4,1), (5,1),
(6,2), (7,2)} 1
0..1 1 1
Multiplicities not regarded
ESSCASS, 25 August 2008
Introduction to Graph Transformation

11. 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 (?)
ESSCASS, 25 August 2008
Introduction to Graph Transformation

12. 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
ESSCASS, 25 August 2008
Introduction to Graph Transformation

13. 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
ESSCASS, 25 August 2008
Introduction to Graph Transformation

14. Example production rule
Single-graph representation of <put>
black = reader: to be matched and preserved
red = embargo:
forbidden
Cell
blue = eraser:
to be matched and deleted
last
next
Object
val
Buffer
Cell
last
val
Object
green = creator:
to be added
ESSCASS, 25 August 2008
Introduction to Graph Transformation

15. Equivalent rule syntax
Single-graph:
Buffer
Cell
Object
next
val
last
Multi-graph:
Partial graph morphisms
Negative Application Condition (NAC)
Left Hand Side (LHS)
Right Hand Side (RHS)
Cell
Object
Cell
Cell
Object
next
next
next
last
val
last
val
last
Buffer
Cell
Buffer
Cell
Buffer
Cell
ESSCASS, 25 August 2008
Introduction to Graph Transformation

16. Introduction to Graph Transformation
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
ESSCASS, 25 August 2008
Introduction to Graph Transformation

17. Example rule application
Buffer
Cell
Object
next
val
last
matching
Buffer
Cell
first | last
next
Object
val
Buffer
Cell
last
first
next
Object
val
transition
ESSCASS, 25 August 2008
Introduction to Graph Transformation

18. Transitions: Partial morphisms
1 cell filled 1 2 3 6 5 4
Buffer
Cell
next
last
first
Object
val
fV = { (1,1),
(2,2), (3,3), (4,4), (5,5),
(6,6)}
fE partial on edge (1,last,2)
<put> 1 2 3 6 5 7 4
Buffer
Cell
next
last
first
Object
val
<get>
fV = { (1,1),
(2,5), (3,2), (4,3), (5,4),
(7,6)}
fV partial on node 6
fE partial on several edges
2 cells filled
ESSCASS, 25 August 2008
Introduction to Graph Transformation

19. Negative application conditions
Here the rule should not be applicable!
Buffer
Cell
next
last
Object
val
NAC
LHS
RHS n
Buffer
Cell
next
last
Object
val
Cell
next
last
Buffer
Cell g m
How to rule out this matching? G
There exists a total morphism g: NAC  G such that m = g  n
(g factors m through n)
Buffer
Cell
next
last
first
Object
val
ESSCASS, 25 August 2008
Introduction to Graph Transformation

20. Introduction to Graph Transformation
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
ESSCASS, 25 August 2008
Introduction to Graph Transformation

21. Introduction to Graph Transformation
Graph Productions
Production rule
NAC
NACs
LHS
RHS
rule morphism
(partial)
source graph
matching
target graph
pushout
forbidden
Graph transition
src(t)
tgt(t)
morph(t)
(SPO = Single Pushout Approach)
ESSCASS, 25 August 2008
Introduction to Graph Transformation

22. Example production rule (again)
Single-graph representation of <put>
black = reader: LHS and RHS; to be matched and preserved
black = reader: to be matched and preserved
red = embargo:
NAC, not LHS; forbidden
red = embargo:
forbidden
blue = eraser:
LHS, not RHS; to be matched and deleted
Cell
blue = eraser:
to be matched and deleted
last
next
Object
val
Buffer
Cell
last
val
green = creator:
RHS, not LHS; to be added
Object
green = creator:
to be added
ESSCASS, 25 August 2008
Introduction to Graph Transformation

23. Introduction to Graph Transformation
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?
ESSCASS, 25 August 2008
Introduction to Graph Transformation

24. Graph transition system
Buffer
Cell
next
first, last
Object
val
<put>
<get>
<put>
<get>
Buffer
Cell
next
first
last
Buffer
Cell
next
last
first
Object
val
The states grow
The size of states is unbounded
The number of states is potentially infinite
Morphisms are not inverse
This is SPO; a similar situation holds after DPO
Buffer
Cell
next
last
first
Object
val
<put>
<get>
Buffer
Cell
next
last
first
Object
val
<put>
<get>
ESSCASS, 25 August 2008
Introduction to Graph Transformation

25. Introduction to Graph Transformation
GROOVE Demo
ESSCASS, 25 August 2008
Introduction to Graph Transformation

26. Example rule application
Buffer
Cell
next
last 3 2 1
Buffer
Cell
next
last
Object
val 4 5 6 7 R p
(1,4), (2,5), (3,6) m m’
(1,1), (2,2), (3,3)
(4,1), (5,2), (6,3), (7,7) G 1 2 3 6 5 7 4
Buffer
Cell
next
last
first
Object
val H 1 2 3 6 5 4
Buffer
Cell
next
last
first
Object
val h
(1,1), (2,2), (3,3), (4,4), (5,5) (6,6) 
General requirement: h  m = m’  p
ESSCASS, 25 August 2008
Introduction to Graph Transformation

27. Rule application: variation 1
Buffer
Cell
next
last 3 2 1
Buffer
Cell
next
last
Object
val 4 5 6 7 R p
(1,4), (2,5), (3,6) m m’
(1,1), (2,2), (3,3)
(4,2), (5,5), (6,1), (7,9) G 2 5 1 4 3 9 8
Buffer
Cell
next
last
first
Object
val H’ 1 2 3 6 5 4
Buffer
Cell
next
last
first
Object
val h
(1,2), (2,5), (3,1), (4,8), (5,3) (6,4) 
Also correct: H  H’
ESSCASS, 25 August 2008
Introduction to Graph Transformation

28. Rule application: variation 2
Buffer
Cell
next
last 3 2 1
Buffer
Cell
next
last
Object
val 4 5 6 7 R p
(1,4), (2,5), (3,6) m m’
(1,1), (2,2), (3,3)
(4,1), (5,2), (6,3) G 1 2 3 6 5 4
Buffer
Cell
next
last
first
Object
val 2 H
Cell
next
next h
first 3
Cell 1
Buffer
Cell
last 5 4
next
next
Cell
Object 6
val
(1,1), (2,2), (3,3), (4,4), (5,5) 
What’s wrong? h  m = m’  p fulfilled!
ESSCASS, 25 August 2008
Introduction to Graph Transformation

29. 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
matching (m)
rule (p)
diagram commutes (hm = m’p) 
that’s the one!   
wrong alternative 
ESSCASS, 25 August 2008
Introduction to Graph Transformation

30. Introduction to Graph Transformation
Example pushouts B A b 1 2 1 C A c 2 B A b 1 2 1 C A c 3 A a 1 2 1 A 2
(1,4),(2,5)
(1,4),(2,5)
(1,4),(2,5) B A b 4 5 6 a C A c 4 5 B 6 b a B A b 4 5 6 a C A c 4 7 B 6 b
A B b 4 5 B A b 4 5 a A x 1 2 3 y A 1 2 3
x,y A a 1 2 1 C A c 2 A a 1 2 1 C A c 3
(1,4),(2,4)
(1,4),(2,4) B A b 4 5 a B C b 4 5 c B A b 4 5 a B C 6 5
(1,4),(2,5),(3,5) A 4 5
x,y A 4 5
x,y
Lessons: - Pushouts always exist - Leave no dangling edges - Deletion always wins
x-edge not mapped!
ESSCASS, 25 August 2008
Introduction to Graph Transformation

31. 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
ESSCASS, 25 August 2008
Introduction to Graph Transformation

32. 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
ESSCASS, 25 August 2008
Introduction to Graph Transformation

33. Introduction to Graph Transformation
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
ESSCASS, 25 August 2008
Introduction to Graph Transformation