1. Semantic Minimization of 3-Valued Propositional Formulas
Thomas Reps Alexey Loginov
University of Wisconsin
Mooly Sagiv
Tel-Aviv University

2. Semantic Minimization
p  p = 1, right?
(A): Value of formula  under assignment A
In 3-valued logic, (A) may equal ?
p  p([p  0]) = 1
p  p([p  ?]) = ?
p  p([p  1]) = 1
However,
1([p  0]) = = p  p([p  0])
1([p  ?]) = 1  ? = p  p([p  ?])
1([p  1]) = = p  p([p  1])

3. Motivation Dataflow analysis Hardware verification Shape analysis
Symbolic trajectory evaluation
Shape analysis
Change example to xy + x’z vs xy + x’z + yz?

4. Questions What does “best” mean? Can one find a best formula? How?
Quote Minato for Floor and ceiling? Generating minimal formulas is simpler and cleaner than it might seem.

5. Two- vs. Three-Valued Logic1
Two-valued logic
{0,1}
{0}
{1}
Three-valued logic
{0}  {0,1}
{1}  {0,1}

6. Two- vs. Three-Valued Logic
Two-valued logic
Three-valued logic
{1}
{0,1}
{0} 1 1

7. Two- vs. Three-Valued Logic1
Two-valued logic
{0}
{1}
Three-valued logic
{0,1}

8. Two- vs. Three-Valued Logic1
Two-valued logic 1
Three-valued logic
0 3½
1 3½

9. Boolean Connectives [Kleene]

10. Three-Valued Logic 1 ½ 0  ½ 1  ½ 1: True 0: False 1/2: Unknown
A join semi-lattice: 0  1 = 1/2 1
Information order
0  ½
1  ½

11. Semantic Minimization
1([p  0]) = = p  p([p  0]) 1([p  ½]) = 1  ½ = p  p([p  ½]) 1([p  1]) = = p  p([p  1])
2-valued logic: 1 is equivalent to p  p
3-valued logic: 1 is better than p  p
For a given , is there a best formula?
Yes!

12. Minimal? x + x’ No! x  x’ Yes! xy + x’z xy + x’y’ xy + x’z+ yz

13. Rewrite Rules?     1     0
Change example to xy + x’z vs xy + x’z + yz?

14. 2-Valued Propositional Meaning
xi(a) = a(xi)
(a) = 1 – (a)
1  2(a) = min(1(a), 2(a))
1  2(a) = max(1(a), 2(a))
Change example to xy + x’z vs xy + x’z + yz?

15. 3-Valued Propositional Meaning
xi(a) = a(xi)
(a) = 1 – (a)
1  2(a) = min(1(a), 2(a))
1  2(a) = max(1(a), 2(a))
Change example to xy + x’z vs xy + x’z + yz?

16. 3-Valued Propositional Meaning
xi(A) = A(xi)
(A) = 1 – (A)
1  2(A) = min(1(A), 2(A))
1  2(A) = max(1(A), 2(A))
Change example to xy + x’z vs xy + x’z + yz?

17. Represented by A A = [ p  ½, q  0, r  1, s  ½ ] [ p  0, q  0,
Change example to xy + x’z vs xy + x’z + yz?
Represented by A

18. The Right Definition of “Best”?
Observation
If for all A,
(A)  (A),
 is better than  1
Change example to xy + x’z vs xy + x’z + yz?

19. The Right Definition of “Best”?
Observation
If for all A,
(A)  (A),
 is better than 
0(A) = 0
 ½
=  ½ (A)
0 is better than ½
1(A) = 1
 ½
=  ½ (A)
1 is better than ½
Change example to xy + x’z vs xy + x’z + yz?

20. “Potentially accepts ”
Acceptance Device
A   iff (A)  1 1 1
Change example to xy + x’z vs xy + x’z + yz?
“Potentially accepts ”

21. “Potentially rejects ”
Acceptance Device
A   iff (A)  0 1 1
Change example to xy + x’z vs xy + x’z + yz?
“Potentially rejects ”

22. Acceptance Device    Suppose that A represents a, and
3-valued
2-valued   
Change example to xy + x’z vs xy + x’z + yz?
Suppose that A represents a, and
a  2-valued assignments. We want:
If a  , then A  
If a  , then A  

23. Acceptance Device ½ Suppose that A represents a, and
3-valued
2-valued
Change example to xy + x’z vs xy + x’z + yz?
Suppose that A represents a, and
a  2-valued assignments. We want:
If a  ½, then A  0
If a  ½, then A  0 
Violated!

24. Acceptance Device ½ Suppose that A represents a, and
3-valued
2-valued ½
Change example to xy + x’z vs xy + x’z + yz?
Suppose that A represents a, and
a  2-valued assignments. We want:
If a  ½, then A  1
If a  ½, then A  1 
Violated!

25. The Right Definition of “Best”?
Observation
If for all A,
(A)  (A),
 is better than 
Change example to xy + x’z vs xy + x’z + yz?
Not all “better” formulas preserve
potential acceptance of 2-valued assignments

26. Supervaluational meaning
What Does “Best” Mean?
Supervaluational meaning
(A) =  (a)
a rep. by A
Change example to xy + x’z vs xy + x’z + yz?

27. Semantic Minimization
(A) = (A)
Non-truth-functional
semantics
Truth-functional
Minimization
Change example to xy + x’z vs xy + x’z + yz?

28. Example p  p([p  ½]) =  p  p(a) = p  p([p  0])
= 1  1
= 1
= 1([p  ½])
Change example to xy + x’z vs xy + x’z + yz?

29. Example ½([p  ½]) =  ½(a) = ½([p  0])  ½([p  1])
= ½  ½
= ½
= ½([p  ½])
Change example to xy + x’z vs xy + x’z + yz?

30. Semantic Minimization
(A) = (A)
Non-truth-functional
semantics
Truth-functional
Minimization
Change example to xy + x’z vs xy + x’z + yz?
 For all A, (A)  (A)
“ is better than ”

31. Realization of a Monotonic Boolean Function [Blamey 1980]
f  Formula[ f ] b  1
Change example to xy + x’z vs xy + x’z + yz? a
 a’b + 1b + ab + a1 + ab’
 (a’b’)’

32. Realization of a Monotonic Boolean Function [Blamey 1980]
f  Formula[ f ] b 1
Change example to xy + x’z vs xy + x’z + yz? a
 a’b + ab + a1 + ab’
 (a’b’ + 1b)’

33.   Formula[]
Our Problem
  Formula[] b
([½, 1]) =  (a)
a{[0,1], [1,1]}
= ([0,0])
 ([1,1])
= 1  1
= 1 1
Change example to xy + x’z vs xy + x’z + yz? a

34. Special Case:  contains no occurrences of ½ or 
  contains no occurrences of ½ in corners b  1
 a’b + 1b + ab + a1 + ab’
 (a’b’)’
Change example to xy + x’z vs xy + x’z + yz? a
 a’b + 1b + ab + a1 + ab’
 (a’b’)’

35. Special Case:  contains no occurrences of ½ or 
  contains no occurrences of ½ in corners b b 1  1
Change example to xy + x’z vs xy + x’z + yz? a a

36. How Do We Obtain ? Represent  with a pair
floor:    ½ = 0
ceiling:    ½ = 1
Change example to xy + x’z vs xy + x’z + yz?

37. How Do We Obtain (, )?
0  (a.0, a.0)
1  (a.1, a.1)
½  (a.0, a.1)
xi  (a.a(xi), a.a(xi))
( f ,  f )  ( f ,  f )
( f 1,  f1 )  ( f2 ,  f2 )  ( f 1   f2 ,  f1    f2 )
( f 1,  f1 )  ( f2 ,  f2 )  ( f 1   f2 ,  f1    f2 )
BDD operations
Change example to xy + x’z vs xy + x’z + yz?

38. Semantically Minimal Formula
General case
 primes(  )  ( primes(   ))
When  contains no occurrences of ½ and 
 primes(  )
Quote Minato for Floor and ceiling? Generating minimal formulas is simpler and cleaner than it might seem.

39. Example Original formula () xy’+ x’z’+ yz Minimal formula ()
x’y + x’z’+ yz + xy’+ xz + y’z’
A (A) (A)
[x  ½, y  0, z  0] ½
[x  0, y  1, z  ½] ½
[x  1, y  ½, z  1] ½
Change example to xy + x’z vs xy + x’z + yz?

40. Example Original formula ( = if x then y else z) xy + x’z
Minimal formula ()
xy+ x’z+ yz
A (A) (A)
[x  ½, y  1, z  1] ½
Change example to xy + x’z vs xy + x’z + yz?

41. Demo
Change example to xy + x’z vs xy + x’z + yz?

42. Related Work [Blamey 1980, 1986] [Godefroid & Bruns 2000]
Realization of a monotonic Boolean function
[Godefroid & Bruns 2000]
Supervaluational (“thorough”) semantics for model checking partial Kripke structures
For propositional formulas
Deciding “(A)  1?” is NP-complete
Quote Minato for Floor and ceiling? Generating minimal formulas is simpler and cleaner than it might seem.

43. Our Questions What does “best” mean? Can one find a best formula? How?
For all A, (A) = (A)
Can one find a best formula?
Yes
How?
Create (, )
Return  primes(  )  ( primes(   ))
Quote Minato for Floor and ceiling? Generating minimal formulas is simpler and cleaner than it might seem.

44. Change example to xy + x’z vs xy + x’z + yz?