An Introduction to Graph Rewriting

1
An Introduction to Graph Rewriting

• Thomas Huining Fenghttp//www.eecs.berkeley.edu/
  tfeng/CHESS, UC Berkeley
• May 1, 2007
• Inspired by Tutorial Introduction to Graph
  Transformation A Software Engineering
  Perspective. Luciano Baresi, Reiko Heckel. ICGT
  2002

2
PacMan a Motivating Example
PacMan
Field
Field
Field
Ghost
Marble
Field
Field
Field
Field
Field
Field
Type
1
1..4
1
0..1
1
0..1
Field
Ghost
PacMan
1
0..1
Marble
3
Our 1st Rule pmove
PacMan
PacMan
LHS (Left Hand Side)
RHS (Right Hand Side)
b Field
a Field
a Field
b Field
Redex finding is a sub-graph isomorphism problem).
Multiple redexes?
Redex
PacMan
Field
Field
Field
Ghost
Marble
Field
Field
Field
Field
Host Graph
Field
Field
4
Our 2nd Rule gmove
Ghost
Ghost
b Field
a Field
a Field
b Field
PacMan
Field
Field
Field
Ghost
Marble
Field
Field
Field
Field
Field
Field
5
Combining pmove and gmove

• pmove and gmove do not interfere with each
  other.
• I.e., applying any one does not affect
  applicability of the other.

6
catch Rule
PacMan
Field
Field
Field
Ghost
Marble
Field
Field
Field
Field
Field
Field
7
Important Decision Which Rule to Choose?

• catch is actually a sub-case of gmove, because
• LHS(gmove) ? LHS(catch)
• Assign priority (almost always the same for
  sub-cases)
• P(catch) gt P(gmove)

8
Attribute Binding collect Rule
PacMan 3 marble
PacMan 4 marble
Field
Field
Field
Ghost
Marble
Field
Field
Field
Field
Field
Field
9
Attribute Binding collect Rule

• Here, m on the LHS serves as a bound variable on
  the RHS, it refers to the value bound to.

PacMan 3 marble
Field
Field
Field
Ghost
Marble
Field
Field
Field
Field
Field
Field
10
Completing the PacMan Game
Priority
3
interfering
3
sub-case
1
Ghost
Ghost
PacMan
b Field
a Field
a Field
b Field
sub-case
2
11
The Graph Rewriting Problem

• Common components in the problem
• Host graph
• A set of rewriting rules
• LHS and RHS
• Attribute binding and transfer
• Application condition
• Embedding information

Host Graph Redex
PacMan m marble
PacMan m1 marble
Marble
b Field
a Field
b Field
a Field
m lt BAG_SIZE
12
The Graph Rewriting Problem

• Common components in the problem
• Host graph
• A set of rewriting rules
• LHS and RHS
• Attribute binding and transfer
• Application condition
• Embedding information
• Rule application
• In the host graph, find all redexes according to
  the rules.
• Match LHS
• Check application condition
• Choose a redex and a rule to apply.
• Apply the rule.
• Generate a substitution (unique) isomorphic to
  RHS
• Compute new attributes
• Make substitution (bridge w.r.t embedding
  information)

13
More on Application Decision

• We have seen priorities (as a partial order of
  rules).
• Applications may require other approaches.
• More flexible rule choosing
• Combine with control structure
• Event-driven
• Combine with state machine

14
More on Application Decision

• We have seen priorities (as a total order of
  rules).
• Applications may require other approaches.
• Maze walking (goal-directed searching)

1
1..4
1
0..1
1
0..1
Field
Entrance
Kid
1
0..1
Exit
15
More on Application Decision

• We have seen priorities (as a total order of
  rules).
• Applications may require other approaches.
• Shortest path (optimizing an objective function)

16
Conclusion

• Key idea is simple.
• User-friendly.
• Expressive.
• Transformation problems
• Analysis problems
• Search problems
• Optimization problems
• Complexity due to sub-graph isomorphism on the
  LHS.
• Rules need to be carefully designed and chosen at
  run time.
• Depending on the problem and the goal.
• Problem formulation and algorithm are application
  dependent.