Tree Regular Model Tree Regular Model CheckingChecking

Tree Regular Model Tree Regular Model CheckingChecking

P. Abdulla, B. Jonsson, P. Mahata P. Abdulla, B. Jonsson, P. Mahata and J. d’Orsoand J. d’Orso

Uppsala UniversityUppsala University

Presentation Overview

• Aim

• Regular model checking

• Trees, tree relations

• Transitive closures

• Results, conclusions

Aim

• With counters

• With stacks

• With channels

• parameterized

A uniform analysis framework for systems :

These systems can be characterized by finite-stateautomata.

Regular model checking

Most important operation: computing transitiveclosures.

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

Example

A simple token passing protocol:

Example

A simple token passing protocol:

Example

A simple token passing protocol:

Example

A simple token passing protocol:

Our Contribution

Regular Model Checking is nice, but…

… it’s only valid for linear or circular topologies !

Idea: extend to trees instead !

Trees

0 1

00 01 10

010

a

bc

d

a

b

d

node

label

this node is called “root”

Tree Automata

0 1

or

and

and

or

or

and

q1 q2

state

input symbol

Tree Automata

Input:

and

or

101

Run:

Tree Automata

Input:

and

or

101

Run:

q1

Transition: 0 q1

Tree Automata

Input:

and

or

101

Run:

q1q2 q2

Transition: 1 q2

Tree Automata

Input:

and

or

101

Run:

q1

q2

q2 q2

Transition: or q2q2

q1

Tree Automata

Input:

and

or

101

Run:

q1

q2

q2

q2 q2

Transition: and q2q2

q2

Tree Automata

Input:

and

or

101

Run:

q1

q2

q2

q2 q2

Accept ! q2

Tree Relations

a

b c

d

e f

Tree Relations

We represent a pair of “similar” trees by a tree onan alphabet with pairs:

a

b c

d

e f

(a,d)

(b,e) (c,f)

History Automata

x

T x

input

History Automata

x

T x

T

x x

input intermediate

History Automata

x

T x

x

x T

T

x x

input outputintermediate

History Automata

In (word) regular model checking, concept of columns

Transpose this to trees: represent a sequence of runs into single run.

x

T x

q1

q2 q3

x

x T

q4

q5 q6

T

x x

input outputintermediate

run 1 run 2

History Automata

In (word) regular model checking, concept of columns

Transpose this to trees: represent a sequence of runs into single run.

x

T x

x

x T

input output

run 1+2

q1.q4

q2.q5 q3.q6

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.

Results

We have run a prototype tool on several examples:

• Token-passing protocol (1&2 ways)

• “percolate” protocol (compute disjunctions)

• Tree arbiter (mutual exclusion)

Paper accepted at CAV’02.

Future work

• Change underlying automata representation (matrices instead of BDDs)

• Expand to new types of systems (e.g. pushdown systems)

• Create a graphical interface