unit-8: algorithms for ltlsri/courses/mc2015/slides/unit8-module3.pdf · unit-8: algorithms for ltl...
TRANSCRIPT
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Unit-8: Algorithms for LTL
B. Srivathsan
Chennai Mathematical Institute
NPTEL-course
July - November 2015
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Module 3:Automaton construction
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Formula expansions
p1 U p2
{p1}
1
0
1
{p1}
1
0
1
{p1}
1
0
1
{p1}
1
0
1
{p1}
1
0
1
{p1}
1
0
1
{p1}
1
0
1
{p2}
0
1
1
{p1,p2}
1
1
1
. . .
p1
p2
p1 U p2
. . .{p1}
1
0
0
{p1}
1
0
0
{p1}
1
0
0
{p1}
1
0
0
{p1}
1
0
0
{ }
0
0
0
{ }
0
0
0
{ }
0
0
0
{ }
0
0
0
p1
p2
p1 U p2
4/7
p1 U p2
{p1}
1
0
1
{p1}
1
0
1
{p1}
1
0
1
{p1}
1
0
1
{p1}
1
0
1
{p1}
1
0
1
{p1}
1
0
1
{p2}
0
1
1
{p1,p2}
1
1
1
. . .
p1
p2
p1 U p2
. . .{p1}
1
0
0
{p1}
1
0
0
{p1}
1
0
0
{p1}
1
0
0
{p1}
1
0
0
{ }
0
0
0
{ }
0
0
0
{ }
0
0
0
{ }
0
0
0
p1
p2
p1 U p2
4/8
F (¬p1 ^ X (¬ p2 U p1))
recall that F � = true U �
true U (¬p1 ^ X (¬ p2 U p1))
{} {p2} {} {} {p1} {p1,p2} {p1,p2} . . .
p1
p2
¬p1
¬p2
¬p2 U p1
X (¬p2 U p1)
¬p1 ^ X (¬p2 U p1)
true U (¬p1 ^ X (¬p2 U p1))
0 0 0 0 1 1 1
0 1 0 0 0 1 1
1 1 1 1 0 0 0
1 0 1 1 1 0 0
1 111100
0 1 1 1 1 1 0
0 0 0 0 1 1 0
1 1 1 1 1 1 0
7/7
F (¬p1 ^ X (¬ p2 U p1))
recall that F � = true U �
true U (¬p1 ^ X (¬ p2 U p1))
{} {p2} {} {} {p1} {p1,p2} {p1,p2} . . .
p1
p2
¬p1
¬p2
¬p2 U p1
X (¬p2 U p1)
¬p1 ^ X (¬p2 U p1)
true U (¬p1 ^ X (¬p2 U p1))
0 0 0 0 1 1 1
0 1 0 0 0 1 1
1 1 1 1 0 0 0
1 0 1 1 1 0 0
1 111100
0 1 1 1 1 1 0
0 1 1 1 0 0 0
1 1 1 1 1 1 0
7/16
G F p1
recall that F � = true U � and G � = ¬ true U ¬�
¬ true U ¬(true U p1)
p1
true
true U p1
¬ true U p1
true U ¬ (true U p1)
¬ true U ¬ (true U p1)
{p1}
1
1
1
0
1
0
{p1}
1
1
1
0
1
0
{ }
0
1
0
1
1
0
{ }
0
1
0
1
1
0
{ }
0
1
0
1
1
0
{ }
0
1
0
1
1
0
{ }
0
1
0
1
1
0
{ }
0
1
0
1
1
0
{ }
0
1
0
1
1
0
6/7
G F p1
recall that F � = true U � and G � = ¬ true U ¬�
¬ true U ¬(true U p1)
p1
true
true U p1
¬ true U p1
true U ¬ (true U p1)
¬ true U ¬ (true U p1)
{p1}
1
1
1
0
0
1
{p1}
1
1
1
0
0
1
{p1}
1
1
1
0
0
1
{ }
0
1
1
0
0
1
{ }
0
1
1
0
0
1
{ }
0
1
1
0
0
1
{ }
0
1
1
0
0
1
{ }
0
1
1
0
0
1
{ }
0
1
1
0
0
1
5/7
G F p1
recall that F � = true U � and G � = ¬ true U ¬�
¬ true U ¬(true U p1)
p1
true
true U p1
¬ true U p1
true U ¬ (true U p1)
¬ true U ¬ (true U p1)
{p1}
1
1
1
0
1
0
{p1}
1
1
1
0
1
0
{ }
0
1
0
1
1
0
{ }
0
1
0
1
1
0
{ }
0
1
0
1
1
0
{ }
0
1
0
1
1
0
{ }
0
1
0
1
1
0
{ }
0
1
0
1
1
0
{ }
0
1
0
1
1
0
6/78/16
Construct an automaton with states as column vectors that can guessaccepting expansions
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Formula expansions
p1 U p2
{p1}
1
0
1
{p1}
1
0
1
{p1}
1
0
1
{p1}
1
0
1
{p1}
1
0
1
{p1}
1
0
1
{p1}
1
0
1
{p2}
0
1
1
{p1,p2}
1
1
1
. . .
p1
p2
p1 U p2
. . .{p1}
1
0
0
{p1}
1
0
0
{p1}
1
0
0
{p1}
1
0
0
{p1}
1
0
0
{ }
0
0
0
{ }
0
0
0
{ }
0
0
0
{ }
0
0
0
p1
p2
p1 U p2
4/7
p1 U p2
{p1}
1
0
1
{p1}
1
0
1
{p1}
1
0
1
{p1}
1
0
1
{p1}
1
0
1
{p1}
1
0
1
{p1}
1
0
1
{p2}
0
1
1
{p1,p2}
1
1
1
. . .
p1
p2
p1 U p2
. . .{p1}
1
0
0
{p1}
1
0
0
{p1}
1
0
0
{p1}
1
0
0
{p1}
1
0
0
{ }
0
0
0
{ }
0
0
0
{ }
0
0
0
{ }
0
0
0
p1
p2
p1 U p2
4/8
F (¬p1 ^ X (¬ p2 U p1))
recall that F � = true U �
true U (¬p1 ^ X (¬ p2 U p1))
{} {p2} {} {} {p1} {p1,p2} {p1,p2} . . .
p1
p2
¬p1
¬p2
¬p2 U p1
X (¬p2 U p1)
¬p1 ^ X (¬p2 U p1)
true U (¬p1 ^ X (¬p2 U p1))
0 0 0 0 1 1 1
0 1 0 0 0 1 1
1 1 1 1 0 0 0
1 0 1 1 1 0 0
1 111100
0 1 1 1 1 1 0
0 0 0 0 1 1 0
1 1 1 1 1 1 0
7/7
F (¬p1 ^ X (¬ p2 U p1))
recall that F � = true U �
true U (¬p1 ^ X (¬ p2 U p1))
{} {p2} {} {} {p1} {p1,p2} {p1,p2} . . .
p1
p2
¬p1
¬p2
¬p2 U p1
X (¬p2 U p1)
¬p1 ^ X (¬p2 U p1)
true U (¬p1 ^ X (¬p2 U p1))
0 0 0 0 1 1 1
0 1 0 0 0 1 1
1 1 1 1 0 0 0
1 0 1 1 1 0 0
1 111100
0 1 1 1 1 1 0
0 1 1 1 0 0 0
1 1 1 1 1 1 0
7/16
G F p1
recall that F � = true U � and G � = ¬ true U ¬�
¬ true U ¬(true U p1)
p1
true
true U p1
¬ true U p1
true U ¬ (true U p1)
¬ true U ¬ (true U p1)
{p1}
1
1
1
0
1
0
{p1}
1
1
1
0
1
0
{ }
0
1
0
1
1
0
{ }
0
1
0
1
1
0
{ }
0
1
0
1
1
0
{ }
0
1
0
1
1
0
{ }
0
1
0
1
1
0
{ }
0
1
0
1
1
0
{ }
0
1
0
1
1
0
6/7
G F p1
recall that F � = true U � and G � = ¬ true U ¬�
¬ true U ¬(true U p1)
p1
true
true U p1
¬ true U p1
true U ¬ (true U p1)
¬ true U ¬ (true U p1)
{p1}
1
1
1
0
0
1
{p1}
1
1
1
0
0
1
{p1}
1
1
1
0
0
1
{ }
0
1
1
0
0
1
{ }
0
1
1
0
0
1
{ }
0
1
1
0
0
1
{ }
0
1
1
0
0
1
{ }
0
1
1
0
0
1
{ }
0
1
1
0
0
1
5/7
G F p1
recall that F � = true U � and G � = ¬ true U ¬�
¬ true U ¬(true U p1)
p1
true
true U p1
¬ true U p1
true U ¬ (true U p1)
¬ true U ¬ (true U p1)
{p1}
1
1
1
0
1
0
{p1}
1
1
1
0
1
0
{ }
0
1
0
1
1
0
{ }
0
1
0
1
1
0
{ }
0
1
0
1
1
0
{ }
0
1
0
1
1
0
{ }
0
1
0
1
1
0
{ }
0
1
0
1
1
0
{ }
0
1
0
1
1
0
6/78/16
Construct an automaton with states as column vectors that can guessaccepting expansions
3/15
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Example 1: p1 U p2
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p1
p2
p1 U p2
0
0
0
0
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
1
0
1
1
1
Compatible states
Recall Until-compatibility
Until-compatibility
�1
�2
�1 U �2
*
1
1
1
0
1 1
0
0
0
1
0
0 0
13/16
5/15
![Page 7: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/7.jpg)
p1
p2
p1 U p2
0
0
0
0
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
1
0
1
1
1
Compatible states
Recall Until-compatibility
Until-compatibility
�1
�2
�1 U �2
*
1
1
1
0
1 1
0
0
0
1
0
0 0
13/16
5/15
![Page 8: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/8.jpg)
p1
p2
p1 U p2
0
0
0
0
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
1
0
1
1
1
Compatible states
Recall Until-compatibility
Until-compatibility
�1
�2
�1 U �2
*
1
1
1
0
1 1
0
0
0
1
0
0 0
13/16
5/15
![Page 9: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/9.jpg)
p1
p2
p1 U p2
0
0
0
0
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
1
0
1
1
1
Compatible states
Recall Until-compatibility
Until-compatibility
�1
�2
�1 U �2
*
1
1
1
0
1 1
0
0
0
1
0
0 0
13/16
5/15
![Page 10: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/10.jpg)
p1
p2
p1 U p2
0
0
0
0
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
1
0
1
1
1
Compatible states
Recall Until-compatibility
Until-compatibility
�1
�2
�1 U �2
*
1
1
1
0
1 1
0
0
0
1
0
0 0
13/16
5/15
![Page 11: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/11.jpg)
p1
p2
p1 U p2
0
0
0
0
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
1
0
1
1
1
Compatible states
Recall Until-compatibility
Until-compatibility
�1
�2
�1 U �2
*
1
1
1
0
1 1
0
0
0
1
0
0 0
13/16
5/15
![Page 12: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/12.jpg)
p1
p2
p1 U p2
0
0
0
0
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
1
0
1
1
1
Compatible states
Recall Until-compatibility
Until-compatibility
�1
�2
�1 U �2
*
1
1
1
0
1 1
0
0
0
1
0
0 0
13/16
5/16
q0
{p1}1
0
1
{p1}1
0
1
{p1}1
0
1
{p2}0
1
1
{ } 0
0
0
{p1}1
0
1
. . .
q0{p1}
1
0
1
{ }
No compatible transition
6/15
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p1
p2
p1 U p2
0
0
0
0
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
1
0
1
1
1
Compatible states
Recall Until-compatibility
Until-compatibility
�1
�2
�1 U �2
*
1
1
1
0
1 1
0
0
0
1
0
0 0
13/16
5/16
q0{p1}
1
0
1
{p1}1
0
1
{p1}1
0
1
{p2}0
1
1
{ } 0
0
0
{p1}1
0
1
. . .
q0{p1}
1
0
1
{ }
No compatible transition
6/15
![Page 14: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/14.jpg)
p1
p2
p1 U p2
0
0
0
0
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
1
0
1
1
1
Compatible states
Recall Until-compatibility
Until-compatibility
�1
�2
�1 U �2
*
1
1
1
0
1 1
0
0
0
1
0
0 0
13/16
5/16
q0{p1}
1
0
1
{p1}1
0
1
{p1}1
0
1
{p2}0
1
1
{ } 0
0
0
{p1}1
0
1
. . .
q0{p1}
1
0
1
{ }
No compatible transition
6/15
![Page 15: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/15.jpg)
p1
p2
p1 U p2
0
0
0
0
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
1
0
1
1
1
Compatible states
Recall Until-compatibility
Until-compatibility
�1
�2
�1 U �2
*
1
1
1
0
1 1
0
0
0
1
0
0 0
13/16
5/16
q0{p1}
1
0
1
{p1}1
0
1
{p1}1
0
1
{p2}0
1
1
{ } 0
0
0
{p1}1
0
1
. . .
q0{p1}
1
0
1
{ }
No compatible transition
6/15
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p1
p2
p1 U p2
0
0
0
0
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
1
0
1
1
1
Compatible states
Recall Until-compatibility
Until-compatibility
�1
�2
�1 U �2
*
1
1
1
0
1 1
0
0
0
1
0
0 0
13/16
5/16
q0{p1}
1
0
1
{p1}1
0
1
{p1}1
0
1
{p2}0
1
1
{ } 0
0
0
{p1}1
0
1
. . .
q0{p1}
1
0
1
{ }
No compatible transition
6/15
![Page 17: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/17.jpg)
p1
p2
p1 U p2
0
0
0
0
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
1
0
1
1
1
Compatible states
Recall Until-compatibility
Until-compatibility
�1
�2
�1 U �2
*
1
1
1
0
1 1
0
0
0
1
0
0 0
13/16
5/16
q0{p1}
1
0
1
{p1}1
0
1
{p1}1
0
1
{p2}0
1
1
{ } 0
0
0
{p1}1
0
1
. . .
q0{p1}
1
0
1
{ }
No compatible transition
6/15
![Page 18: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/18.jpg)
p1
p2
p1 U p2
0
0
0
0
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
1
0
1
1
1
Compatible states
Recall Until-compatibility
Until-compatibility
�1
�2
�1 U �2
*
1
1
1
0
1 1
0
0
0
1
0
0 0
13/16
5/16
q0{p1}
1
0
1
{p1}1
0
1
{p1}1
0
1
{p2}0
1
1
{ } 0
0
0
{p1}1
0
1
. . .
q0{p1}
1
0
1
{ }
No compatible transition
6/15
![Page 19: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/19.jpg)
p1
p2
p1 U p2
0
0
0
0
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
1
0
1
1
1
Compatible states
Recall Until-compatibility
Until-compatibility
�1
�2
�1 U �2
*
1
1
1
0
1 1
0
0
0
1
0
0 0
13/16
5/16
q0{p1}
1
0
1
{p1}1
0
1
{p1}1
0
1
{p2}0
1
1
{ } 0
0
0
{p1}1
0
1
. . .
q0{p1}
1
0
1
{ }
No compatible transition
6/15
![Page 20: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/20.jpg)
p1
p2
p1 U p2
0
0
0
0
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
1
0
1
1
1
Compatible states
Recall Until-compatibility
Until-compatibility
�1
�2
�1 U �2
*
1
1
1
0
1 1
0
0
0
1
0
0 0
13/16
5/16
q0{p1}
1
0
1
{p1}1
0
1
{p1}1
0
1
{p2}0
1
1
{ } 0
0
0
{p1}1
0
1
. . .
q0{p1}
1
0
1
{ }
No compatible transition
6/15
![Page 21: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/21.jpg)
p1
p2
p1 U p2
0
0
0
0
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
1
0
1
1
1
Compatible states
Recall Until-compatibility
Until-compatibility
�1
�2
�1 U �2
*
1
1
1
0
1 1
0
0
0
1
0
0 0
13/16
5/16
q0{p1}
1
0
1
{p1}1
0
1
{p1}1
0
1
{p2}0
1
1
{ } 0
0
0
{p1}1
0
1
. . .
q0
{p1}1
0
1
{ }
No compatible transition
6/15
![Page 22: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/22.jpg)
p1
p2
p1 U p2
0
0
0
0
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
1
0
1
1
1
Compatible states
Recall Until-compatibility
Until-compatibility
�1
�2
�1 U �2
*
1
1
1
0
1 1
0
0
0
1
0
0 0
13/16
5/16
q0{p1}
1
0
1
{p1}1
0
1
{p1}1
0
1
{p2}0
1
1
{ } 0
0
0
{p1}1
0
1
. . .
q0{p1}
1
0
1
{ }
No compatible transition
6/15
![Page 23: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/23.jpg)
p1
p2
p1 U p2
0
0
0
0
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
1
0
1
1
1
Compatible states
Recall Until-compatibility
Until-compatibility
�1
�2
�1 U �2
*
1
1
1
0
1 1
0
0
0
1
0
0 0
13/16
5/16
q0{p1}
1
0
1
{p1}1
0
1
{p1}1
0
1
{p2}0
1
1
{ } 0
0
0
{p1}1
0
1
. . .
q0{p1}
1
0
1
{ }
No compatible transition
6/15
![Page 24: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/24.jpg)
p1
p2
p1 U p2
0
0
0
0
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
1
0
1
1
1
Compatible states
Recall Until-compatibility
Until-compatibility
�1
�2
�1 U �2
*
1
1
1
0
1 1
0
0
0
1
0
0 0
13/16
5/16
q0{p1}
1
0
1
{p1}1
0
1
{p1}1
0
1
{p2}0
1
1
{ } 0
0
0
{p1}1
0
1
. . .
q0{p1}
1
0
1
{ }No compatible transition
6/15
![Page 25: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/25.jpg)
Until-compatibility
�1
�2
�1 U �2
*
1
1
1
0
1 1
0
0
0
1
0
0 0
13/16
Until-compatibility: eventuality condition
�1
�2
�1 U �2
1
0
1 1
1
0
1
1
0
1
1
0
1
1
0 . . .
Cannot happen forever that �1 U �2 = 1, �1 = 1 but �2 = 0
14/16
q0
{p1}
{p2}
{p1 ,p2}
{p1}
{p2} {p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}{ }
{p1}
1
0
1
0
1
1
1
1
1
0
0
0
1
0
0
7/15
![Page 26: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/26.jpg)
Until-compatibility
�1
�2
�1 U �2
*
1
1
1
0
1 1
0
0
0
1
0
0 0
13/16
Until-compatibility: eventuality condition
�1
�2
�1 U �2
1
0
1 1
1
0
1
1
0
1
1
0
1
1
0 . . .
Cannot happen forever that �1 U �2 = 1, �1 = 1 but �2 = 0
14/16
q0
{p1}
{p2}
{p1 ,p2}
{p1}
{p2} {p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}{ }
{p1}
1
0
1
0
1
1
1
1
1
0
0
0
1
0
0
7/15
![Page 27: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/27.jpg)
Until-compatibility
�1
�2
�1 U �2
*
1
1
1
0
1 1
0
0
0
1
0
0 0
13/16
Until-compatibility: eventuality condition
�1
�2
�1 U �2
1
0
1 1
1
0
1
1
0
1
1
0
1
1
0 . . .
Cannot happen forever that �1 U �2 = 1, �1 = 1 but �2 = 0
14/16
q0
{p1}
{p2}
{p1 ,p2}
{p1}
{p2} {p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}{ }
{p1}
1
0
1
0
1
1
1
1
1
0
0
0
1
0
0
7/15
![Page 28: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/28.jpg)
Until-compatibility
�1
�2
�1 U �2
*
1
1
1
0
1 1
0
0
0
1
0
0 0
13/16
Until-compatibility: eventuality condition
�1
�2
�1 U �2
1
0
1 1
1
0
1
1
0
1
1
0
1
1
0 . . .
Cannot happen forever that �1 U �2 = 1, �1 = 1 but �2 = 0
14/16
q0
{p1}
{p2}
{p1 ,p2}
{p1}
{p2} {p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}{ }
{p1}
1
0
1
0
1
1
1
1
1
0
0
0
1
0
0
7/15
![Page 29: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/29.jpg)
Until-compatibility
�1
�2
�1 U �2
*
1
1
1
0
1 1
0
0
0
1
0
0 0
13/16
Until-compatibility: eventuality condition
�1
�2
�1 U �2
1
0
1 1
1
0
1
1
0
1
1
0
1
1
0 . . .
Cannot happen forever that �1 U �2 = 1, �1 = 1 but �2 = 0
14/16
q0
{p1}
{p2}
{p1 ,p2}
{p1}
{p2} {p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}{ }
{p1}
1
0
1
0
1
1
1
1
1
0
0
0
1
0
0
7/15
![Page 30: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/30.jpg)
Until-compatibility
�1
�2
�1 U �2
*
1
1
1
0
1 1
0
0
0
1
0
0 0
13/16
Until-compatibility: eventuality condition
�1
�2
�1 U �2
1
0
1 1
1
0
1
1
0
1
1
0
1
1
0 . . .
Cannot happen forever that �1 U �2 = 1, �1 = 1 but �2 = 0
14/16
q0
{p1}
{p2}
{p1 ,p2}
{p1}
{p2} {p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}{ }
{p1}
1
0
1
0
1
1
1
1
1
0
0
0
1
0
0
7/15
![Page 31: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/31.jpg)
Until-compatibility
�1
�2
�1 U �2
*
1
1
1
0
1 1
0
0
0
1
0
0 0
13/16
Until-compatibility: eventuality condition
�1
�2
�1 U �2
1
0
1 1
1
0
1
1
0
1
1
0
1
1
0 . . .
Cannot happen forever that �1 U �2 = 1, �1 = 1 but �2 = 0
14/16
q0
{p1}
{p2}
{p1 ,p2}
{p1}
{p2} {p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}{ }
{p1}
1
0
1
0
1
1
1
1
1
0
0
0
1
0
0
7/15
![Page 32: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/32.jpg)
Until-compatibility
�1
�2
�1 U �2
*
1
1
1
0
1 1
0
0
0
1
0
0 0
13/16
Until-compatibility: eventuality condition
�1
�2
�1 U �2
1
0
1 1
1
0
1
1
0
1
1
0
1
1
0 . . .
Cannot happen forever that �1 U �2 = 1, �1 = 1 but �2 = 0
14/16
q0
{p1}
{p2}
{p1 ,p2}
{p1}
{p2} {p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}{ }
{p1}
1
0
1
0
1
1
1
1
1
0
0
0
1
0
0
7/15
![Page 33: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/33.jpg)
Until-compatibility
�1
�2
�1 U �2
*
1
1
1
0
1 1
0
0
0
1
0
0 0
13/16
Until-compatibility: eventuality condition
�1
�2
�1 U �2
1
0
1 1
1
0
1
1
0
1
1
0
1
1
0 . . .
Cannot happen forever that �1 U �2 = 1, �1 = 1 but �2 = 0
14/16
q0
{p1}
{p2}
{p1 ,p2}
{p1}
{p2} {p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}{ }
{p1}
1
0
1
0
1
1
1
1
1
0
0
0
1
0
0
7/15
![Page 34: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/34.jpg)
Until-compatibility
�1
�2
�1 U �2
*
1
1
1
0
1 1
0
0
0
1
0
0 0
13/16
Until-compatibility: eventuality condition
�1
�2
�1 U �2
1
0
1 1
1
0
1
1
0
1
1
0
1
1
0 . . .
Cannot happen forever that �1 U �2 = 1, �1 = 1 but �2 = 0
14/16
q0
{p1}
{p2}
{p1 ,p2}
{p1}
{p2} {p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}{ }
{p1}
1
0
1
0
1
1
1
1
1
0
0
0
1
0
0
7/15
![Page 35: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/35.jpg)
Until-compatibility
�1
�2
�1 U �2
*
1
1
1
0
1 1
0
0
0
1
0
0 0
13/16
Until-compatibility: eventuality condition
�1
�2
�1 U �2
1
0
1 1
1
0
1
1
0
1
1
0
1
1
0 . . .
Cannot happen forever that �1 U �2 = 1, �1 = 1 but �2 = 0
14/16
q0
{p1}
{p2}
{p1 ,p2}
{p1}
{p2} {p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}{ }
{p1}
1
0
1
0
1
1
1
1
1
0
0
0
1
0
0
7/15
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Until-compatibility
�1
�2
�1 U �2
*
1
1
1
0
1 1
0
0
0
1
0
0 0
13/16
Until-compatibility: eventuality condition
�1
�2
�1 U �2
1
0
1 1
1
0
1
1
0
1
1
0
1
1
0 . . .
Cannot happen forever that �1 U �2 = 1, �1 = 1 but �2 = 0
14/16
q0
{p1}
{p2}
{p1 ,p2}
{p1}
{p2} {p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}{ }
{p1}
1
0
1
0
1
1
1
1
1
0
0
0
1
0
0
7/15
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Until-compatibility
�1
�2
�1 U �2
*
1
1
1
0
1 1
0
0
0
1
0
0 0
13/16
Until-compatibility: eventuality condition
�1
�2
�1 U �2
1
0
1 1
1
0
1
1
0
1
1
0
1
1
0 . . .
Cannot happen forever that �1 U �2 = 1, �1 = 1 but �2 = 0
14/16
q0
{p1}
{p2}
{p1 ,p2}
{p1}
{p2} {p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}
{ }
{p1}
1
0
1
0
1
1
1
1
1
0
0
0
1
0
0
7/15
![Page 38: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/38.jpg)
Until-compatibility
�1
�2
�1 U �2
*
1
1
1
0
1 1
0
0
0
1
0
0 0
13/16
Until-compatibility: eventuality condition
�1
�2
�1 U �2
1
0
1 1
1
0
1
1
0
1
1
0
1
1
0 . . .
Cannot happen forever that �1 U �2 = 1, �1 = 1 but �2 = 0
14/16
q0
{p1}
{p2}
{p1 ,p2}
{p1}
{p2} {p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}{ }
{p1}
1
0
1
0
1
1
1
1
1
0
0
0
1
0
0
7/15
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Until-compatibility
�1
�2
�1 U �2
*
1
1
1
0
1 1
0
0
0
1
0
0 0
13/16
Until-compatibility: eventuality condition
�1
�2
�1 U �2
1
0
1 1
1
0
1
1
0
1
1
0
1
1
0 . . .
Cannot happen forever that �1 U �2 = 1, �1 = 1 but �2 = 0
14/16
q0
{p1}
{p2}
{p1 ,p2}
{p1}
{p2} {p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}{ }
{p1}
1
0
1
0
1
1
1
1
1
0
0
0
1
0
0
7/15
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Until-compatibility
�1
�2
�1 U �2
*
1
1
1
0
1 1
0
0
0
1
0
0 0
13/16
Until-compatibility: eventuality condition
�1
�2
�1 U �2
1
0
1 1
1
0
1
1
0
1
1
0
1
1
0 . . .
Cannot happen forever that �1 U �2 = 1, �1 = 1 but �2 = 0
14/16
q0
{p1}
{p2}
{p1 ,p2}
{p1}
{p2} {p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}{ }
{p1}
1
0
1
0
1
1
1
1
1
0
0
0
1
0
0
7/15
![Page 41: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/41.jpg)
Until-compatibility
�1
�2
�1 U �2
*
1
1
1
0
1 1
0
0
0
1
0
0 0
13/16
Until-compatibility: eventuality condition
�1
�2
�1 U �2
1
0
1 1
1
0
1
1
0
1
1
0
1
1
0 . . .
Cannot happen forever that �1 U �2 = 1, �1 = 1 but �2 = 0
14/16
q0
{p1}
{p2}
{p1 ,p2}
{p1}
{p2} {p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}
{ }
{p1}
{p1}
{p2}
{p1 ,p2}{ }
{p1}
1
0
1
0
1
1
1
1
1
0
0
0
1
0
0
7/15
![Page 42: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/42.jpg)
Example 2: (X p1) U p2
8/15
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*
*
*
*
p1
p2
X p1
(X p1) U p2
q0{ }
0
0
1
1
{ p1}1
0
1
1
{ p1,p2}1
1
1
0
{ p2}0
1
1
1
. . .
9/15
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*
*
*
*
p1
p2
X p1
(X p1) U p2
q0
{ }0
0
1
1
{ p1}1
0
1
1
{ p1,p2}1
1
1
0
{ p2}0
1
1
1
. . .
9/15
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*
*
*
*
p1
p2
X p1
(X p1) U p2
q0{ }
0
0
1
1
{ p1}1
0
1
1
{ p1,p2}1
1
1
0
{ p2}0
1
1
1
. . .
9/15
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*
*
*
*
p1
p2
X p1
(X p1) U p2
q0{ }
0
0
1
1
{ p1}1
0
1
1
{ p1,p2}1
1
1
0
{ p2}0
1
1
1
. . .
9/15
![Page 47: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/47.jpg)
*
*
*
*
p1
p2
X p1
(X p1) U p2
q0{ }
0
0
1
1
{ p1}1
0
1
1
{ p1,p2}1
1
1
0
{ p2}0
1
1
1
. . .
9/15
![Page 48: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/48.jpg)
*
*
*
*
p1
p2
X p1
(X p1) U p2
q0{ }
0
0
1
1
{ p1}1
0
1
1
{ p1,p2}1
1
1
0
{ p2}0
1
1
1
. . .
9/15
![Page 49: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/49.jpg)
*
*
*
*
p1
p2
X p1
(X p1) U p2
q0{ }
0
0
1
1
{ p1}
1
0
1
1
{ p1,p2}1
1
1
0
{ p2}0
1
1
1
. . .
9/15
![Page 50: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/50.jpg)
*
*
*
*
p1
p2
X p1
(X p1) U p2
q0{ }
0
0
1
1
{ p1}1
0
1
1
{ p1,p2}1
1
1
0
{ p2}0
1
1
1
. . .
9/15
![Page 51: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/51.jpg)
*
*
*
*
p1
p2
X p1
(X p1) U p2
q0{ }
0
0
1
1
{ p1}1
0
1
1
{ p1,p2}
1
1
1
0
{ p2}0
1
1
1
. . .
9/15
![Page 52: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/52.jpg)
*
*
*
*
p1
p2
X p1
(X p1) U p2
q0{ }
0
0
1
1
{ p1}1
0
1
1
{ p1,p2}1
1
1
0
{ p2}0
1
1
1
. . .
9/15
![Page 53: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/53.jpg)
*
*
*
*
p1
p2
X p1
(X p1) U p2
q0{ }
0
0
1
1
{ p1}1
0
1
1
{ p1,p2}1
1
1
0
{ p2}0
1
1
1
. . .
9/15
![Page 54: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/54.jpg)
*
*
*
*
p1
p2
X p1
(X p1) U p2
q0{ }
0
0
1
1
{ p1}1
0
1
1
{ p1,p2}1
1
1
0
{ p2}
0
1
1
1
. . .
9/15
![Page 55: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/55.jpg)
*
*
*
*
p1
p2
X p1
(X p1) U p2
q0{ }
0
0
1
1
{ p1}1
0
1
1
{ p1,p2}1
1
1
0
{ p2}0
1
1
1
. . .
9/15
![Page 56: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/56.jpg)
*
*
*
*
p1
p2
X p1
(X p1) U p2
q0{ }
0
0
1
1
{ p1}1
0
1
1
{ p1,p2}1
1
1
0
{ p2}0
1
1
1
. . .
9/15
![Page 57: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/57.jpg)
Coming next: Construction for an arbitrary LTL formula φ
10/15
![Page 58: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/58.jpg)
Step 1: List down subformulae of φ
*
*
*
*
p1
p2
X p1
(X p1) U p2
*
*
*
p1
p2
p1 U p2
11/15
![Page 59: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/59.jpg)
Step 1: List down subformulae of φ
*
*
*
*
p1
p2
X p1
(X p1) U p2
*
*
*
p1
p2
p1 U p2
11/15
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Step 2: Check AND-NOT and Until compatibility
0
1
0
0
p1
p2
X p1
(X p1) U p2
0
0
1
p1
p2
p1 U p2
Incompatible states!
Remove incompatible states and add a new state {q0}
12/15
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Step 2: Check AND-NOT and Until compatibility
0
1
0
0
p1
p2
X p1
(X p1) U p2
0
0
1
p1
p2
p1 U p2
Incompatible states!
Remove incompatible states and add a new state {q0}
12/15
![Page 62: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/62.jpg)
Step 2: Check AND-NOT and Until compatibility
0
1
0
0
p1
p2
X p1
(X p1) U p2
0
0
1
p1
p2
p1 U p2
Incompatible states!
Remove incompatible states and add a new state {q0}
12/15
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Step 3: Add transitions satisfying
Word, X and Until compatibility
0
0
1
1
p1
p2
X p1
(X p1) U p2
{p1,p2}1
1
*
1
q0{p1}
1
0
1
1
From q0 add compatible transitions to states where last entry is 1
13/15
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Step 3: Add transitions satisfying
Word, X and Until compatibility
0
0
1
1
p1
p2
X p1
(X p1) U p2
{p1,p2}1
1
*
1
q0{p1}
1
0
1
1
From q0 add compatible transitions to states where last entry is 1
13/15
![Page 65: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/65.jpg)
Step 3: Add transitions satisfying
Word, X and Until compatibility
0
0
1
1
p1
p2
X p1
(X p1) U p2
{p1,p2}1
1
*
1
q0{p1}
1
0
1
1
From q0 add compatible transitions to states where last entry is 1
13/15
![Page 66: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/66.jpg)
Step 4: Accepting states should ensure Until-eventuality condition
For every Until subformula φ1 U φ2, define
Fφ1 U φ2: set of states where φ1 U φ2 = 0 or φ2 = 1
Fp1 U p2:
0
0
0
1
0
0
0
1
1
1
1
1
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![Page 67: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/67.jpg)
Step 4: Accepting states should ensure Until-eventuality condition
For every Until subformula φ1 U φ2, define
Fφ1 U φ2: set of states where φ1 U φ2 = 0 or φ2 = 1
Fp1 U p2:
0
0
0
1
0
0
0
1
1
1
1
1
14/15
![Page 68: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/68.jpg)
Step 4: Accepting states should ensure Until-eventuality condition
For every Until subformula φ1 U φ2, define
Fφ1 U φ2: set of states where φ1 U φ2 = 0 or φ2 = 1
Fp1 U p2:
0
0
0
1
0
0
0
1
1
1
1
1
14/15
![Page 69: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/69.jpg)
Final automatonAφ
É Compatible states + q0
É Compatible transitions
É If no Until subformula, then every state is accepting
É Otherwise, accepting set for each Until subformula: {F1,F2, . . . ,Fk}
É This will be a Generalized NBA (GNBA)
Every GNBA can be converted to an equivalent NBA (Unit 6 - Module 4)
In general, this algorithm gives NBA which is exponential in size offormula
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![Page 70: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/70.jpg)
Final automatonAφÉ Compatible states + q0
É Compatible transitions
É If no Until subformula, then every state is accepting
É Otherwise, accepting set for each Until subformula: {F1,F2, . . . ,Fk}
É This will be a Generalized NBA (GNBA)
Every GNBA can be converted to an equivalent NBA (Unit 6 - Module 4)
In general, this algorithm gives NBA which is exponential in size offormula
15/15
![Page 71: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/71.jpg)
Final automatonAφÉ Compatible states + q0
É Compatible transitions
É If no Until subformula, then every state is accepting
É Otherwise, accepting set for each Until subformula: {F1,F2, . . . ,Fk}
É This will be a Generalized NBA (GNBA)
Every GNBA can be converted to an equivalent NBA (Unit 6 - Module 4)
In general, this algorithm gives NBA which is exponential in size offormula
15/15
![Page 72: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/72.jpg)
Final automatonAφÉ Compatible states + q0
É Compatible transitions
É If no Until subformula, then every state is accepting
É Otherwise, accepting set for each Until subformula: {F1,F2, . . . ,Fk}
É This will be a Generalized NBA (GNBA)
Every GNBA can be converted to an equivalent NBA (Unit 6 - Module 4)
In general, this algorithm gives NBA which is exponential in size offormula
15/15
![Page 73: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/73.jpg)
Final automatonAφÉ Compatible states + q0
É Compatible transitions
É If no Until subformula, then every state is accepting
É Otherwise, accepting set for each Until subformula: {F1,F2, . . . ,Fk}
É This will be a Generalized NBA (GNBA)
Every GNBA can be converted to an equivalent NBA (Unit 6 - Module 4)
In general, this algorithm gives NBA which is exponential in size offormula
15/15
![Page 74: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/74.jpg)
Final automatonAφÉ Compatible states + q0
É Compatible transitions
É If no Until subformula, then every state is accepting
É Otherwise, accepting set for each Until subformula: {F1,F2, . . . ,Fk}
É This will be a Generalized NBA (GNBA)
Every GNBA can be converted to an equivalent NBA (Unit 6 - Module 4)
In general, this algorithm gives NBA which is exponential in size offormula
15/15
![Page 75: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/75.jpg)
Final automatonAφÉ Compatible states + q0
É Compatible transitions
É If no Until subformula, then every state is accepting
É Otherwise, accepting set for each Until subformula: {F1,F2, . . . ,Fk}
É This will be a Generalized NBA (GNBA)
Every GNBA can be converted to an equivalent NBA (Unit 6 - Module 4)
In general, this algorithm gives NBA which is exponential in size offormula
15/15
![Page 76: Unit-8: Algorithms for LTLsri/Courses/MC2015/Slides/Unit8-Module3.pdf · Unit-8: Algorithms for LTL B. Srivathsan Chennai Mathematical Institute NPTEL-course](https://reader034.vdocument.in/reader034/viewer/2022050513/5f9d3885e68ef51b6e4410ef/html5/thumbnails/76.jpg)
Final automatonAφÉ Compatible states + q0
É Compatible transitions
É If no Until subformula, then every state is accepting
É Otherwise, accepting set for each Until subformula: {F1,F2, . . . ,Fk}
É This will be a Generalized NBA (GNBA)
Every GNBA can be converted to an equivalent NBA (Unit 6 - Module 4)
In general, this algorithm gives NBA which is exponential in size offormula
15/15