Graph Transformation and Software Technology Chapter 1
Proc (A4) ; if [product available] then Proc(A5) else Proc(A8) ⦠receive order ; ... using two simple rules: delete isolated nodes remove successive.
Graph Transformation and Software Technology Chapter 1: Motivation Reiko Heckel Software Technology Group
Outline Why it is fun Where it comes from What it is good for
Concept of the course
Why it is fun: Programming By Example StageCast (www.stagecast.com): a visual programming environment for kids (from 8 years on), based on behavioral rules associated to graphical objects visual pattern matching simple internal control structures (priority, sequence, nondeterminism, ...) external keyboard control
Î
Rule-based behavior modeling is a natural and intuitive paradigm!
Example: A simple PacMan game; concrete (StageCast) vs. abstract (graph-based) presentation.
States of the PacMan Game: Graph-Based Presentation :Ghost :Field
:Field
:Field
:Field
:Field
:PacMan marbles=3 :Marble
:Field
instance graph (represents a single state; abstracts from spatial layout)
typing
Ghost *
cardinalities (specify additional constraints on well-typed instance graphs)
1
Field 1 * Marble
1 *
PacMan marbles:int
type graph (specifies legal instance graphs Æ state space)
Rules of the PacMan Game: Graph-Based Presentation, PacMan pm:PacMan marbles=m
:Marble
f1:Field
f2:Field
pm:PacMan f1:Field
pm:PacMan marbles=m+1
collect f1:Field
pm:PacMan
movePM f2:Field
f2:Field
f1:Field
f2:Field
PacMan’s rules: collect has priority over movePM
Rules of the PacMan Game: Graph-Based Presentation, Ghost g:Ghost
:PacMan
f1:Field
f2:Field
g:Ghost f1:Field
g:Ghost
kill f1:Field
g:Ghost
moveGhost f2:Field
f2:Field
f1:Field
f2:Field
Ghost’s rules: kill has priority over moveGhost
Graph Transformation :Field
:Ghost
:Field
:Field
:Field
:Field
:PacMan marbles=3 marbles=4 :Marble
:Field
typing
Ghost *
1
Field 1 * Marble
1 *
PacMan marbles:int
collect ; kill
Where it all comes from … Chomsky Grammars
Term Rewriting
Petri Nets
Graph Transformation and Graph Grammars
Chomsky Grammars: Transforming Strings Production A Æ aAb as graph grammar production (represents path expression anbn) 2:
A
3:
1:
a
2:
A
3:
b
4:
Graph grammars: formal language theory for graphical structures Language
hierarchies (e.g. what are context free graph grammars, and how useful can they be) Decidability and complexity results Parsing algorithms
Petri Nets: Transforming (Multi-) Sets A PT net transition as graph transformation rule A
2 C
B
a1:A a2:A
b1:B
b2:B
c:C
Graph transformation: modeling and analysis of concurrent systems Relation
with Petri nets (use of analysis tools, transfer of concepts) Independence, causality, and conflicts of rules Semantics of concurrency
Term Rewriting: Transforming Trees or DAGs Term rewrite rules (directed equations):
x*0Æ0 x * s(y) Æ (x * y) + x
Transformation of trees or directed acyclic graphs (DAGs)? or
Graph rewriting: efficient implementation of equational
deduction (in functional programming, code optimization in compiler and DBMS)
Relation with term rewriting Termination and confluence
.. and what it is good for Chomsky Grammars
Term Rewriting
Petri Nets
Graph Transformation and Graph Grammars
Diagram Languages
Graph-based Computation
Model and Program Transformation
9
Behaviour Modelling and Visual Programming
Graph-based Language Definition Example: activity diagrams compiled to imperative program, in two phases 1. 2.
if [product available] then Proc(A5) else Proc(A8) … receive order ; check availability ; if [product available] then calculate prize; send receipt else notify client
check availability
4
2
[product available]
[product not available]
calculate prize notify client
1
3
6
8 send receipt
7
5
.. and what it is good for Chomsky Grammars
Term Rewriting
Petri Nets
Graph Transformation and Graph Grammars
9 Diagram
Languages
Graph-based Computation
Model and Program Transformation
9 Behaviour
Modelling and Visual Programming
Model (and Program) Transformation
Explain statechart in terms of equivalent finite automaton
new auction offer something auction running
make bid
make bid no bid
some bid timeout
Flattening Rule: Incoming Transition
s1
a
s2 b s3
s1
a
s2 b s3
Transition from state s1 to the boundary of a complex state represents a transition from s1 to the initial state of the complex state
Flattening Rule: Outgoing Transitions
s1
s2 a
b s3
a
s2 b
s1 a
s3
Transition from boundary of complex state to state s1 represents corresponding transitions from all substates to s1
.. and what it is good for Chomsky Grammars
Term Rewriting
Petri Nets
Graph Transformation and Graph Grammars
9 Diagram Languages
9 Graph-based Computation
Model and Program Transformation
9 Behaviour Modelling and Visual Programming
Bridges of Königsberg (Euler 1736)
walk across each of the bridges exactly once, returning to the starting place
Euler Graph
contains only nodes with even numbers of links
can be reduced to a loop
using two simple rules:
remove successive pair of links
delete isolated nodes
Chinese Postman (Kwan 1962)
travel across each link to deliver mail, but minimize the amount of additional walking
Eulerization
add links between pairs of nodes with an odd number of links each x
x+1
if x and y are odd, add link in between y
y+1
Graph is Eulerian as soon as rule is no longer applicable
Application in Software Engineering: Model-Based Testing
Test a web site by following each link at least once, but minimize additional navigation
.. and what it is good for Chomsky Grammars
Tree Transformations
Petri Nets
Graph Transformation and Graph Grammars
9
9
Graph-based Language Definition
Graph-based Computation
9 Model and Program Transformation
9 Behaviour Modelling and Visual Programming
The Message
Graphs are everywhere: they provide representations for many real-world and computer science problems Program
states Diagrams City maps …
Graph transformation rules are the most natural way to model the manipulation of graphs
Concept of the Course Application-oriented mix Foundations Theory Applications Tools
For example Ægraph
grammars Æcontext-freeness Ædiagram languages ÆDiaGen
Next lectures: Graph
Transformation in a Nutshell: Introduction to basic oncepts Mathematical Background: How to build a formalism
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