application of tbds
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
Application of TBDs
Technical developmentOrdered TBDsOperations on ordered TBDs (,,)Reduced ordered TBDs
Model checking == Manipulation of TBDs
p2 p3p1
Ordered TBDs
pn pn+1
Ordered TBDs
-pn+1pn+1 u
y zx
A
- B - C- B
- CA
D- D
- D
- A
- D
Example
A
- B - C- B
- CA
D- D
- D
- A
- D
Example
A
- D - C- B
- DC
D- D
- D
- D
- D
Example
D
Operations
Negation
Conjunction
Abstraction
s
s
t
sx
Negation
u
y zx
- u
y zx
Conjunction
pn+1
u
u
Conjunction
- pn+1
u
- pn+1
Conjunction
a
y zx
a
y’ z’x’
a
y zxy’ z’x’
Conjunction
- a
y zx
- a
y’ z’x’
- a
pn+1xz
x’z’
yz
y’z’
Conjunction
a
y zx
- a
y’ z’x’
- a
zxx’z’
yy’z’
Conjunction
a
y zx
b/-b
y’ z’x’
a
zx yb/-b
Conjunction
- a
y zx
b/-b
y’ z’x’
a
zx yb/-bb/-bb/-b
Abstraction
An abstraction of a TBD on a label u =Conjunction of a simplication on –u and a simplication on u
A
- B - C- B
- CA
D- D
- D
- A
- D
Simplification on a Label u/-u
Select all non-terminal nodes labeled with singed/unsigned uReplace the selected nodes with a simpler one according to given rules
Simplification for a node with label u
u
y zx xz
- u
yz
u
Simplification for a node with label -u
- u
y zx xz
- u
yz
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
zu
Existential Abstraction on u
zu
Properties
s1
s2
t2
s1
s1 s2 t1 t2
s2
s1
t1
s2uu
Observation: comp(s)
spn
::
p1
pn+1
Quantified Boolean Formulas
Consider formulas with variables p1, p2, …, pn
pi
pi
pn+1 pn+1- pn+1
s st s
x
φ φΨ x. φ
φ is valid comp( ) holds s
u
y - pn+1x
x
Reduced Ordered TBDs
- pn+1 pn+1
- pn+1 y pn+1
x x y
x y y
y x y
x pn+1
y pn+1
Non-terminal
y pn+1
x>0
Not allowed
u
T’ - zT
T zT
- z
T
Reduction Rules for u
- z T- z - z
z Tz T
u
T z- z
Reduction Rules for uu
z T- z
- z zT z T- z
T’ TT T’ Tz
T’ T’T z T’T
- u
T’ - zT
T zT
z
- T
Reduction Rules for -u
- z T- z z
z Tz - T
- u
T z- z
Reduction Rules for -u- u
z T- z
- z zT z T- z
T’ TT T’ Tz
T’ T’T z T’T
- u
T zz
u
- T z- zz zT - z z- T
u
~y - z~x
~y z~x
- u u
- z
~x
- z
~y
Explanation on Some Rules (Semantics)
u
~y - z~x
- u u
- z - z
Explanation on Some Rules (1)
- y z- x
- y zx
y z- xy zx
- xx
- xx
- y
- yy
y
u
~y - z~x
- u u
- z - z
Explanation on Some Rules
- y z- x
y zx
- x
x
- y
y
u
~y - z~x - z
Explanation on Some Rules
- x z- x
x zx
- x
x
-u/u
u
T’ - zT
T zT
- z
T
Explanation on Some Rules
u
~y - z~x
- u u
- z - z
Explanation on Some Rules (2)
- y z- x
- y zx
y z- xy zx
- xx
- xx
- y
- yy
y
u
~y - z~x
- u u
- z - z
Explanation on Some Rules
- y z- x - x - y
y zx x y
u
~y - z~x
-u/u
- z
Explanation on Some Rules
- x z- x - x
x zx x
u
T’ - zT
T zT
- z
T
Explanation on Some Rules
- z T- z - z
z Tz T
Boolean Diagram Model Checking
m variables for representing states2m variables for representing transitions
Let n=2m
Construct a TBD for the formula representing the initial statesConstruct a TBD for the formula representing the transition relation
The rest follows from the CTL model checking techniques