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Application of TBDs. Technical development Ordered TBDs Operations on ordered TBDs ( ,,) Reduced ordered TBDs. M odel checking == Manipulation of TBDs. Ordered TBDs. p 1. p 2. p 3. p n. p n+1. Ordered TBDs. p n+1. - p n+1. u. x. y. z. Example. A. - B. - B. - C.

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Application of tbds
Application of TBDs

Technical development

Ordered TBDs

Operations on ordered TBDs (,,)

Reduced ordered TBDs

Model checking == Manipulation of TBDs


Ordered tbds
Ordered TBDs

p1

p2

p3

pn

pn+1


Ordered tbds1
Ordered TBDs

pn+1

-pn+1

u

x

y

z


Example
Example

A

- B

- B

- C

A

- C

- D

- D

- D

D

- A


Example1
Example

A

- B

- B

- C

A

- C

- D

- D

- D

D

- A


Example2
Example

A

- B

- D

- C

D

C

- D

- D

- D

- D

D

- D


Operations
Operations

Negation

Conjunction

Abstraction

s

s

t

  • x

s


Negation
Negation

u

- u

x

y

z

x

y

z


Conjunction
Conjunction

pn+1

u

u


Conjunction1
Conjunction

- pn+1

- pn+1

u


Conjunction2
Conjunction

a

a

x

y

z

x

y

z

x’

y’

z’

a

x’

y’

z’


Conjunction3
Conjunction

- a

- a

x

y

z

pn+1

x

y

z

z

x’

y’

- a

z’

z’

x’

y’

z’


Conjunction4
Conjunction

a

- a

x

y

z

x

y

z

x’

y’

z’

z’

- a

x’

y’

z’


Conjunction5
Conjunction

a

a

x

y

z

x

y

z

b/-b

b/-b

x’

y’

z’


Conjunction6
Conjunction

- a

a

x

y

z

x

y

z

b/-b

b/-b

b/-b

b/-b

x’

y’

z’


Abstraction
Abstraction

An abstraction of a TBD on a label u

=

Conjunction of

a simplication on –u and a simplication on u


Simplification on a label u u
Simplification on a Label u/-u

A

- B

- B

- C

A

- C

- D

- D

- D

D

- A

Select all non-terminal nodes labeled with singed/unsignedu

Replace the selected nodes with a simpler one according to given rules




Abstraction on u
Abstraction on u

Given a TBD.

(1) Make a simplification on –u and a simplification on u

(2) Make a conjunction of the two simplifications

u

z



Properties
Properties

s1

s2

t1

t2

s1

s2

s1

s2

t1

t2

  • u

  • u

s1

s2


Observation comp s
Observation: comp(s)

p1

:

:

pn

s

pn+1


Quantified boolean formulas
Quantified Boolean Formulas

Consider formulas with variables p1,p2,…,pn

pi

φ

φΨ

x. φ

pi

s

s

  • x

t

s

- pn+1

pn+1

pn+1

φ is valid comp( ) holds

s


Reduced ordered tbds
Reduced Ordered TBDs

u

x

y

- pn+1

Not allowed

x

- pn+1

pn+1

x

pn+1

- pn+1

y

pn+1

y

pn+1

x

x

y

Non-terminal

x

y

y

y

pn+1

y

x

y

x>0


Reduction rules for u
Reduction Rules for u

u

T

T’

- z

- z

- z

- z

T

- z

z

z

T

T

T

T

z

T


Reduction rules for u1
Reduction Rules for u

u

u

- z

  • T

z

- z

z

T

T

  • - z

z

- z

z

T

T

  • T’

T’

T

z

T’

T

  • T’

T

z

T’

T


Reduction rules for u2
Reduction Rules for -u

- u

T

T’

- z

z

- z

- z

T

z

z

z

T

- T

T

T

z

- T


Reduction rules for u3
Reduction Rules for -u

- u

- u

- z

  • T

z

- z

z

T

T

  • - z

z

- z

z

T

T

  • T’

T’

T

z

T’

T

  • T’

T

z

T’

T

- u

u

z

  • T

z

- z

- T

z

T

  • z

z

- T

- z

z


Explanation on some rules semantics
Explanation on Some Rules (Semantics)

u

- u

u

~x

~y

- z

- z

- z

~x

~y

z

~x

~y


Explanation on some rules 1
Explanation on Some Rules (1)

u

- u

u

~x

~y

- z

- z

- z

- x

- y

z

- x

- y

x

- y

z

x

- y

- x

y

z

- x

y

x

y

z

x

y


Explanation on some rules
Explanation on Some Rules

u

- u

u

~x

~y

- z

- z

- z

- x

- y

z

- x

- y

x

y

z

x

y


Explanation on some rules1
Explanation on Some Rules

u

-u/u

~x

~y

- z

- z

- x

- x

z

- x

x

x

z

x


Explanation on some rules2
Explanation on Some Rules

u

T

T’

- z

- z

T

T

z

T


Explanation on some rules 2
Explanation on Some Rules (2)

u

- u

u

~x

~y

- z

- z

- z

- x

- y

z

- x

- y

x

- y

z

x

- y

- x

y

z

- x

y

x

y

z

x

y


Explanation on some rules3
Explanation on Some Rules

u

- u

u

~x

~y

- z

- z

- z

- x

- y

z

- x

- y

x

y

z

x

y


Explanation on some rules4
Explanation on Some Rules

u

-u/u

~x

~y

- z

- z

- x

- x

z

- x

x

x

z

x


Explanation on some rules5
Explanation on Some Rules

u

T

T’

- z

- z

- z

- z

T

- z

z

z

T

T

T

T

z

T


Boolean diagram model checking
Boolean Diagram Model Checking

m variables for representing states

2m variables for representing transitions

Let n=2m

Construct a TBD for the formula representing the initial states

Construct a TBD for the formula representing the transition relation

The rest follows from the CTL model checking techniques


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