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
![Page 1: Application of TBDs](https://reader033.vdocument.in/reader033/viewer/2022051821/56816109550346895dd054d3/html5/thumbnails/1.jpg)
Application of TBDs
Technical developmentOrdered TBDsOperations on ordered TBDs (,,)Reduced ordered TBDs
Model checking == Manipulation of TBDs
![Page 2: Application of TBDs](https://reader033.vdocument.in/reader033/viewer/2022051821/56816109550346895dd054d3/html5/thumbnails/2.jpg)
p2 p3p1
Ordered TBDs
pn pn+1
![Page 3: Application of TBDs](https://reader033.vdocument.in/reader033/viewer/2022051821/56816109550346895dd054d3/html5/thumbnails/3.jpg)
Ordered TBDs
-pn+1pn+1 u
y zx
![Page 4: Application of TBDs](https://reader033.vdocument.in/reader033/viewer/2022051821/56816109550346895dd054d3/html5/thumbnails/4.jpg)
A
- B - C- B
- CA
D- D
- D
- A
- D
Example
![Page 5: Application of TBDs](https://reader033.vdocument.in/reader033/viewer/2022051821/56816109550346895dd054d3/html5/thumbnails/5.jpg)
A
- B - C- B
- CA
D- D
- D
- A
- D
Example
![Page 6: Application of TBDs](https://reader033.vdocument.in/reader033/viewer/2022051821/56816109550346895dd054d3/html5/thumbnails/6.jpg)
A
- D - C- B
- DC
D- D
- D
- D
- D
Example
D
![Page 7: Application of TBDs](https://reader033.vdocument.in/reader033/viewer/2022051821/56816109550346895dd054d3/html5/thumbnails/7.jpg)
Operations
Negation
Conjunction
Abstraction
s
s
t
sx
![Page 8: Application of TBDs](https://reader033.vdocument.in/reader033/viewer/2022051821/56816109550346895dd054d3/html5/thumbnails/8.jpg)
Negation
u
y zx
- u
y zx
![Page 9: Application of TBDs](https://reader033.vdocument.in/reader033/viewer/2022051821/56816109550346895dd054d3/html5/thumbnails/9.jpg)
Conjunction
pn+1
u
u
![Page 10: Application of TBDs](https://reader033.vdocument.in/reader033/viewer/2022051821/56816109550346895dd054d3/html5/thumbnails/10.jpg)
Conjunction
- pn+1
u
- pn+1
![Page 11: Application of TBDs](https://reader033.vdocument.in/reader033/viewer/2022051821/56816109550346895dd054d3/html5/thumbnails/11.jpg)
Conjunction
a
y zx
a
y’ z’x’
a
y zxy’ z’x’
![Page 12: Application of TBDs](https://reader033.vdocument.in/reader033/viewer/2022051821/56816109550346895dd054d3/html5/thumbnails/12.jpg)
Conjunction
- a
y zx
- a
y’ z’x’
- a
pn+1xz
x’z’
yz
y’z’
![Page 13: Application of TBDs](https://reader033.vdocument.in/reader033/viewer/2022051821/56816109550346895dd054d3/html5/thumbnails/13.jpg)
Conjunction
a
y zx
- a
y’ z’x’
- a
zxx’z’
yy’z’
![Page 14: Application of TBDs](https://reader033.vdocument.in/reader033/viewer/2022051821/56816109550346895dd054d3/html5/thumbnails/14.jpg)
Conjunction
a
y zx
b/-b
y’ z’x’
a
zx yb/-b
![Page 15: Application of TBDs](https://reader033.vdocument.in/reader033/viewer/2022051821/56816109550346895dd054d3/html5/thumbnails/15.jpg)
Conjunction
- a
y zx
b/-b
y’ z’x’
a
zx yb/-bb/-bb/-b
![Page 16: Application of TBDs](https://reader033.vdocument.in/reader033/viewer/2022051821/56816109550346895dd054d3/html5/thumbnails/16.jpg)
Abstraction
An abstraction of a TBD on a label u =Conjunction of a simplication on –u and a simplication on u
![Page 17: Application of TBDs](https://reader033.vdocument.in/reader033/viewer/2022051821/56816109550346895dd054d3/html5/thumbnails/17.jpg)
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
![Page 18: Application of TBDs](https://reader033.vdocument.in/reader033/viewer/2022051821/56816109550346895dd054d3/html5/thumbnails/18.jpg)
Simplification for a node with label u
u
y zx xz
- u
yz
u
![Page 19: Application of TBDs](https://reader033.vdocument.in/reader033/viewer/2022051821/56816109550346895dd054d3/html5/thumbnails/19.jpg)
Simplification for a node with label -u
- u
y zx xz
- u
yz
u
![Page 20: Application of TBDs](https://reader033.vdocument.in/reader033/viewer/2022051821/56816109550346895dd054d3/html5/thumbnails/20.jpg)
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
![Page 21: Application of TBDs](https://reader033.vdocument.in/reader033/viewer/2022051821/56816109550346895dd054d3/html5/thumbnails/21.jpg)
Existential Abstraction on u
zu
![Page 22: Application of TBDs](https://reader033.vdocument.in/reader033/viewer/2022051821/56816109550346895dd054d3/html5/thumbnails/22.jpg)
Properties
s1
s2
t2
s1
s1 s2 t1 t2
s2
s1
t1
s2uu
![Page 23: Application of TBDs](https://reader033.vdocument.in/reader033/viewer/2022051821/56816109550346895dd054d3/html5/thumbnails/23.jpg)
Observation: comp(s)
spn
::
p1
pn+1
![Page 24: Application of TBDs](https://reader033.vdocument.in/reader033/viewer/2022051821/56816109550346895dd054d3/html5/thumbnails/24.jpg)
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
![Page 25: Application of TBDs](https://reader033.vdocument.in/reader033/viewer/2022051821/56816109550346895dd054d3/html5/thumbnails/25.jpg)
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
![Page 26: Application of TBDs](https://reader033.vdocument.in/reader033/viewer/2022051821/56816109550346895dd054d3/html5/thumbnails/26.jpg)
u
T’ - zT
T zT
- z
T
Reduction Rules for u
- z T- z - z
z Tz T
![Page 27: Application of TBDs](https://reader033.vdocument.in/reader033/viewer/2022051821/56816109550346895dd054d3/html5/thumbnails/27.jpg)
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
![Page 28: Application of TBDs](https://reader033.vdocument.in/reader033/viewer/2022051821/56816109550346895dd054d3/html5/thumbnails/28.jpg)
- u
T’ - zT
T zT
z
- T
Reduction Rules for -u
- z T- z z
z Tz - T
![Page 29: Application of TBDs](https://reader033.vdocument.in/reader033/viewer/2022051821/56816109550346895dd054d3/html5/thumbnails/29.jpg)
- 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
![Page 30: Application of TBDs](https://reader033.vdocument.in/reader033/viewer/2022051821/56816109550346895dd054d3/html5/thumbnails/30.jpg)
u
~y - z~x
~y z~x
- u u
- z
~x
- z
~y
Explanation on Some Rules (Semantics)
![Page 31: Application of TBDs](https://reader033.vdocument.in/reader033/viewer/2022051821/56816109550346895dd054d3/html5/thumbnails/31.jpg)
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
![Page 32: Application of TBDs](https://reader033.vdocument.in/reader033/viewer/2022051821/56816109550346895dd054d3/html5/thumbnails/32.jpg)
u
~y - z~x
- u u
- z - z
Explanation on Some Rules
- y z- x
y zx
- x
x
- y
y
![Page 33: Application of TBDs](https://reader033.vdocument.in/reader033/viewer/2022051821/56816109550346895dd054d3/html5/thumbnails/33.jpg)
u
~y - z~x - z
Explanation on Some Rules
- x z- x
x zx
- x
x
-u/u
![Page 34: Application of TBDs](https://reader033.vdocument.in/reader033/viewer/2022051821/56816109550346895dd054d3/html5/thumbnails/34.jpg)
u
T’ - zT
T zT
- z
T
Explanation on Some Rules
![Page 35: Application of TBDs](https://reader033.vdocument.in/reader033/viewer/2022051821/56816109550346895dd054d3/html5/thumbnails/35.jpg)
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
![Page 36: Application of TBDs](https://reader033.vdocument.in/reader033/viewer/2022051821/56816109550346895dd054d3/html5/thumbnails/36.jpg)
u
~y - z~x
- u u
- z - z
Explanation on Some Rules
- y z- x - x - y
y zx x y
![Page 37: Application of TBDs](https://reader033.vdocument.in/reader033/viewer/2022051821/56816109550346895dd054d3/html5/thumbnails/37.jpg)
u
~y - z~x
-u/u
- z
Explanation on Some Rules
- x z- x - x
x zx x
![Page 38: Application of TBDs](https://reader033.vdocument.in/reader033/viewer/2022051821/56816109550346895dd054d3/html5/thumbnails/38.jpg)
u
T’ - zT
T zT
- z
T
Explanation on Some Rules
- z T- z - z
z Tz T
![Page 39: Application of TBDs](https://reader033.vdocument.in/reader033/viewer/2022051821/56816109550346895dd054d3/html5/thumbnails/39.jpg)
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