1. Automated Theorem Proving
Lecture 2
Propositional Satisfiability

2. Decision procedures Boolean programs Arithmetic programs
Propositional satisfiability
Arithmetic programs
Propositional satisfiability modulo theory of linear arithmetic
Memory programs
Propositional satisfiability modulo theory of linear arithmetic + arrays

3. Case I: Boolean programs
Boolean-valued variables and boolean operations
 Formula := b |  |   
b  SymBoolConst

4. SAT First NP-complete problem (Cook 1972)
Davis-Putnam algorithm (1960)
resolution-based
may use exponential memory
Davis-Logemann-Loveland algorithm (1962)
search-based
basis for all successful modern solvers
Conflict-driven learning and non-chronological backtracking (1996)
resolution strikes back!
Amazing progress
GRASP, SATO, Chaff, ZChaff, BerkMin, …

5. Conjunctive Normal Form
 CNF Formula ::= c1  c2  … cm
c  Clause ::= l1  l2  … ln
l  Literal ::= b | b
b  SymBoolConst
Unit clause ( l )
a clause containing a single literal
Empty clause ( )
a clause containing no literal
equivalent to false

6. Conversion into CNF
In general, converting  into an equivalent CNF formula may result in an exponential blow-up
We are only interested in satisfiability of 
Convert into an equi-satisfiable CNF formula EQCNF()
 is satisfiable iff EQCNF() is satisfiable
size of EQCNF() is polynomial in size of 

7. Conversion into CNF Convert formula  into normal form NF()
NF() is polynomial in 
Convert  = NF() into equisatisfiable CNF formula EQCNF()
EQCNF() is polynomial in 

8. Normal Form Normal form: NF()   Negated normal form: NNF()  
NF(b) = b
NNF(b) = b
NF() = NNF()
NNF() = NF()
NF(1  2) = NF(1)  NF(1)
NNF(1  2) = NNF(1)  NNF(2)

9. Equi-satisfiable CNF Let  be a formula in normal form.
For each subformula  of :
- create a fresh symbol v in SymBoolConst
Identify vb with b and vb with b
Cl(b) = Cl(b) = true
Cl() = Cl()  Cl() 
(v  v  v)  (v  v)  (v  v)
Cl() = Cl()  Cl() 
(v  v  v)  (v  v)  (v  v)
EQCNF() = v  Cl()

10. Resolution (c1  b) (c2  b) (c1  c2) c1, c2 independent of b
clauses
(c1  b) (c2  b)
(c1  c2)
resolvent
resolvent(b, c1  b, c2  b) = c1  c2 = b. (c1  b)  (c2  b)

11.   (c1  b)  (c2  b)  (c1  c2)
Theorem
  (c1  b)  (c2  b)
iff
  (c1  b)  (c2  b)  (c1  c2)
Adding the resolvent to the set of clauses does not
affect the satisfiability of the clause set.

12. Unit resolution One of the clauses being resolved is a unit clause
( b ) (c2  b)
( c2 )
( b ) (c2  b)
( c2 )
Derivation of the empty clause (denoted by )
( b ) ( b ) 

13. Davis-Putnam algorithm (I)
Given clause set C:
Rule 1: If a clause (c  l  l)  C, replace it with (c  l)
Rule 2: If a clause (c  b  b)  C, remove it from C
Rule 3a: If b does not occur in any clause in C,
remove every clause containing b from C
Rule 3b: If b does not occur in any clause in C,
remove every clause containing b from C

14. Davis-Putnam algorithm (II)
Saturate C w.r.t Rules 1, 2, 3a, and 3b
while (C is nonempty) {
Pick a variable b appearing in some clause in C
C’ = { resolvent(b,c1,c2) | c1,c2  C }
Saturate C’ w.r.t. Rules 1, 2, 3a, and 3b
if (  C’) return unsatisfiable
C = C’ }
return satisfiable

15. Satisfiable example (a  b  c) (b  c  f) (b  c) Rule 3a
(c  c  f)
Resolve on b
Rule 2
Clause set is empty

16. Unsatisfiable example
(a  b) (a  b) (a  c) (a  c)
Pick b
( a ) (a  c) (a  c)
Pick a
( c ) ( c )
Pick c 

17. Correctness Saturate C w.r.t Rules 1, 2, 3a, and 3b
while (C is nonempty) {
Pick a variable b appearing in some clause in C
C’ = { resolvent(b,c1,c2) | c1,c2  C }
Saturate C’ w.r.t. Rules 1, 2, 3a, and 3b
if (  C’) return unsatisfiable
C = C’ }
return satisfiable
Two observations:
- Each of the rules 1, 2, 3a, and 3b preserve satisfiability
- C’ = b. C

18. Memory explosion Saturate C w.r.t Rules 1, 2, 3a, and 3b
while (C is nonempty) {
Pick a variable b appearing in some clause in C
C’ = { resolvent(b,c1,c2) | c1,c2  C }
Saturate C’ w.r.t. Rules 1, 2, 3a, and 3b
if (  C’) return unsatisfiable
C = C’ }
return satisfiable
Let n be the number of clauses in the input clause set
Number of clauses after i-th iteration of loop: O(n^(2^i))

19. Davis-Logemann-Loveland algorithm
Slides of sat_course1.pdf
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20. Davis-Logemann-Loveland algorithm
Eliminates exponential memory requirement
Might still need exponential time

21. Conflict-driven learning and non-chronological backtracking
Slides 2-20 of sat_course2.pdf
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