Semantic Minimization of 3-Valued Propositional Formulas

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Semantic Minimizationof 3-Valued Propositional
Formulas

• Thomas Reps Alexey Loginov
• University of Wisconsin
• Mooly Sagiv
• Tel-Aviv University

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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) 1 ?p ? ?p?(p ?
  0)
• ?1?(p ? ?) 1 ? ? ?p ? ?p?(p ?
  ?)
• ?1?(p ? 1) 1 ?p ? ?p?(p ?
  1)

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Motivation

• Dataflow analysis
• Hardware verification
• Symbolic trajectory evaluation
• Shape analysis

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Questions

• What does best mean?
• Can one find a best formula?
• How?

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Two- vs. Three-Valued Logic
0 ? 0,1
1 ? 0,1
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Two- vs. Three-Valued Logic
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Two- vs. Three-Valued Logic
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Two- vs. Three-Valued Logic
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Two- vs. Three-Valued Logic
0 ?3½
1 ?3½
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Boolean Connectives Kleene
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Three-Valued Logic

• 1 True
• 0 False
• 1/2 Unknown
• A join semi-lattice 0 ? 1 1/2

0 ? ½
1 ? ½
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Semantic Minimization

• ?1?(p ? 0) 1 ?p ? ?p?(p ? 0)
  ?1?(p ? ½) 1 ? ½ ?p ? ?p?(p ? ½) ?1?(p
  ? 1) 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!
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Minimal?
x x x ? x xy xz xy xy xy xz
yz xy xz yz
No! Yes! No! Yes! Yes! No!
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Rewrite Rules?
? ? ?? ? 1
? ? ?? ? 0
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2-Valued Propositional Meaning
?0?(a) 0 ?1?(a) 1 ?xi?(a) a(xi) ????(a)
1 ???(a) ??1 ? ?2?(a) min(??1?(a),
??2?(a)) ??1 ? ?2?(a) max(??1?(a), ??2?(a))
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3-Valued Propositional Meaning
?½?(a) ½
?0?(a) 0 ?1?(a) 1 ?xi?(a) a(xi) ????(a)
1 ???(a) ??1 ? ?2?(a) min(??1?(a),
??2?(a)) ??1 ? ?2?(a) max(??1?(a), ??2?(a))
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3-Valued Propositional Meaning
?½?(A) ½
?0?(A) 0 ?1?(A) 1 ?xi?(A) A(xi) ????(A)
1 ???(A) ??1 ? ?2?(A) min(??1?(A),
??2?(A)) ??1 ? ?2?(A) max(??1?(A), ??2?(A))
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A p ? ½, q ? 0, r ? 1, s ? ½
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The Right Definition of Best?
Observation If for all A, ???(A) ? ???(A), ?
is better than ?
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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
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Acceptance Device
A ? ? iff ???(A) ? 1
Potentially accepts ?
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Acceptance Device
A ? ?? iff ???(A) ? 0
Potentially rejects ?
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Acceptance Device
3-valued
2-valued
?

• Suppose that A represents a, and
• a ? 2-valued assignments. We want
• If a ? ?, then A ? ?
• If a ? ??, then A ? ??

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Acceptance Device
3-valued
2-valued

• 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!
?
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Acceptance Device
3-valued
2-valued

• 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!
?
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The Right Definition of Best?
Observation If for all A, ???(A) ? ???(A), ?
is better than ?
Not all better formulas preserve potential
acceptance of 2-valued assignments
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What Does Best Mean?
Supervaluational meaning ?????(A) ?
???(a) a rep. by A
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Semantic Minimization
???(A) ?????(A)
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Example
??p ? ?p??(p ? ½) ? ?p ? ?p?(a)

a?p ? 0,
p ?
1 ?p ?
?p?(p ? 0)
? ?p ? ?p?(p ? 1)
1 ? 1
1 ?1?(p ? ½)
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Example
??½??(p ? ½) ? ?½?(a)
a?p ? 0,

p ? 1 ?½?(p ?
0) ? ?½?(p ? 1)
½ ? ½
½ ?½?(p
? ½)
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Semantic Minimization
???(A) ?????(A)
? For all A, ???(A) ? ???(A) ? is better than
?
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Realization of aMonotonic Boolean
FunctionBlamey 1980
f ? Formula f
b
a
? ab 1b ab a1 ab ? (ab)
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Realization of aMonotonic Boolean
FunctionBlamey 1980
f ? Formula f
b
a
? ab ab a1 ab ? (ab 1b)
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Our Problem
????? ? Formula?????
b
a
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Special Case? contains no occurrences of ½ or ?
? ????? contains no occurrences of ½ in corners
b
? ab 1b ab a1 ab ? (ab)
a
? ab 1b ab a1 ab
? (ab)
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Special Case? contains no occurrences of ½ or ?
? ????? contains no occurrences of ½ in corners
b
b
a
a
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How Do We Obtain ??????
Represent ????? with a pair floor ? ????? ?
?½? 0 ceiling ? ????? ?
?½? 1
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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
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Semantically Minimal Formula

• General case
• ? primes(? ????? ?) ? ?(? primes(? ? ????? ?))
• When ? contains no occurrences of ½ and ?
• ? primes(? ????? ?)

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Example
Original formula (?) xy xz
yz Minimal formula (?) xy xz yz xy xz
yz
A ???(A)
???(A) x ? ½, y ? 0, z ? 0 1
½ x ? 0, y ? 1, z ? ½ 1
½ x ? 1, y ? ½, z ? 1 1
½
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Example
Original formula (? if x then y else z)
xy xz Minimal formula (?) xy xz yz
A ???(A)
???(A) x ? ½, y ? 1, z ? 1 1
½
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Demo
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Related Work

• Blamey 1980, 1986
• 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

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Our Questions

• What does best mean?
• For all A, ???(A) ?????(A)
• Can one find a best formula?
• Yes
• How?
• Create (???????, ???????)
• Return ? primes(? ????? ?) ? ?(? primes(? ?
  ????? ?))

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