Tree Regular Model Checking

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Tree Regular Model Checking

• P. Abdulla, B. Jonsson, P. Mahata and J. dOrso

Uppsala University
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Presentation Overview

• Aim
• Regular model checking
• Trees, tree relations
• Transitive closures
• Results, conclusions

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Aim
A uniform analysis framework for systems

• With counters
• With stacks
• With channels
• parameterized

These systems can be characterized by
finite-state automata.
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Regular model checking
Most important operation computing
transitive closures.

• Why ? Because it allows many analysis
• Reachability
• safety properties
• fairness properties

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Example
A simple token passing protocol
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Example
A simple token passing protocol
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Example
A simple token passing protocol
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Example
A simple token passing protocol
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Our Contribution
Regular Model Checking is nice, but
its only valid for linear or circular
topologies !
Idea extend to trees instead !
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Trees
e
this node is called root
a
0
1
a
b
01
00
10
c
label
b
d
010
node
d
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Tree Automata
input symbol
q1
q2
state
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Tree Automata
Run
Input
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Tree Automata
Run
Input
q1
q1
Transition
0
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Tree Automata
Run
Input
q1
q2
q2
q2
Transition
1
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Tree Automata
Run
Input
q2
q1
q2
q2
q1
q2
Transition
or
q2
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Tree Automata
Run
Input
q2
q2
q1
q2
q2
q2
q2
Transition
and
q2
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Tree Automata
Run
Input
q2
q2
q1
q2
q2
q2
Accept !
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Tree Relations
d
e
f
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Tree Relations
We represent a pair of similar trees by a tree
on an alphabet with pairs
d
(a,d)
e
f
(b,e)
(c,f)
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History Automata
input
x
T
x
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History Automata
input
intermediate
x
T
T
x
x
x
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History Automata
input
output
intermediate
x
x
T
T
x
x
T
x
x
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History Automata
In (word) regular model checking, concept of
columns
Transpose this to trees represent a sequence of
runs into single run.
input
output
intermediate
x
q1
x
q4
T
T
x
q2
q3
x
T
q5
q6
x
x
run 1
run 2
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History Automata
In (word) regular model checking, concept of
columns
Transpose this to trees represent a sequence of
runs into single run.
input
output
x
x
T
x
x
T
run 12
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Symbolic Automata
Property of H.A.s accept the transitive closure.
Problem infinite size !
Approach apply standard subset construction
(determinization)
Supporting data structure for sets regular
expressions.
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Results
We have run a prototype tool on several examples

• Token-passing protocol (12 ways)
• percolate protocol (compute disjunctions)
• Tree arbiter (mutual exclusion)

Paper accepted at CAV02.
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Future work

• Change underlying automata representation
  (matrices instead of BDDs)
• Expand to new types of systems (e.g. pushdown
  systems)
• Create a graphical interface