1. Decision Procedures - An algorithmic point of view
Quantified formulas
Decision procedures –
An algorithmic point of view
Daniel Kroening and Ofer Strichman
Decision Procedures - An algorithmic point of view

2. Why do we need quantifiers ?
As always: more modeling power
Examples of quantifiers usage:
“Everyone in the room has a friend”
“There is a person in the room that all of his cars are red”
“There is not more than one person in the room that earns more than $1M”
Decision Procedures - An algorithmic point of view

3. Decision Procedures - An algorithmic point of view
Quantifiers in Math…
For any integer x there is a smaller integer y
8x2Z 9y2Z. y < x X
Reverse claim: There exists an integer y such that any integer x is greater than y
9y2Z 8x2Z. y < x £
(Bertrand’s postulate) For any natural number greater than 1 there is a prime number p such that n < p < 2n
8n2 N. 9p2 N. n >1 ! (isprime(p) Æ n < p < 2n)
Decision Procedures - An algorithmic point of view

4. Decision Procedures - An algorithmic point of view
Actually…
Satisfiability of (x1,,xn) = does there exist an interpretation of x1,,xn that satisfies  ?
Validity of (x1,,xn) = does it hold that all interpretation of x1,,xn satisfy ?
Conclusion: what we did so far (satisfiability, validity) is non-alternating quantification.
Decision Procedures - An algorithmic point of view

5. Example: Quantified Propositional Logic
Better known as Quantified Boolean Formulas (QBF)
formula: var | : formula | formula Ç formula | ( formula ) | T | F| 8 var. (formula) | 9 var. (formula)
8x. (x Ç 9y. (y ! x))
8x. (9y. ((x Ç :y) Æ (:x Çy)) Æ 9y. ((:y Ç :x) Æ (x Ç y))) X X
Binding scope of y
Decision Procedures - An algorithmic point of view

6. Prenex Normal-Form (PNF)
Formulas in PNF look like this:
’: Q[n]V[n]. .Q[1]V[1]. Quantifier-free formula
where Q[i] 2 {8,9} and V[i] is a variable.
Every quantified formula can be transformed to PNF while preserving validity. How ?
prefix
Decision Procedures - An algorithmic point of view

7. Prenex Normal Form (PNF)
Eliminate ! and $ (transform to Ç Æ :)
Push negations inside using: : 8 x.  $ 9 x. :  : 9 x.  $ 8 x. : 
If there are name conflicts across scopes, solve with renaming.
Move quantifiers out by using recursively rules such as:
Q1 x. 1(x) Æ Q2 y. 2(y) $ Q1 x. Q2 y. (1(x) Æ 2(y)) Qi2{8,9}
Q1 x. 1(x) Ç Q2 y. 2(y) $ Q1 x. Q2 y. (1(x) Ç 2(y)) Qi2{8,9}
1 Æ 9 x. 2(x) $ 9 x. (1 Æ 2(x)) where x does not appear in 1
1 Æ 8 x. 2(x) $ 8 x. (1 Æ 2(x)) where x does not appear in 1
8 x. 1(x) Æ 8 x. 2(x) $ 8 x. (1(x) Æ 2(x))
9 x. 1(x) Ç 9 x. 2(x) $ 9 x. (1(x) Ç 2(x))
Decision Procedures - An algorithmic point of view

8. Prenex Normal Form (PNF): example
:9x. : (9y. ((y ! x) Æ (: x Ç y)) Æ :8y. ((y Æ x) Ç (: x Æ : y)))
1,2. Eliminate !, push negations inside:
8x. (9y. ((:y Ç x) Æ (: x Ç y)) Æ 9y. ((:y Ç : x) Æ (x Ç y)))
3. Renaming:
8x. (9y1. ((:y1 Ç x) Æ (: x Ç y1)) Æ 9y2. ((:y2 Ç : x) Æ (x Ç y2)))
4. Move quantifiers to front:
8x. 9y1. 9y2. (x Ç :y1) Æ (: x Ç y1) Æ (:y2 Ç : x) Æ (x Ç y2)
Decision Procedures - An algorithmic point of view

9. Why eliminating 9x. ÆiLi is enough
A procedure for eliminating an existential quantifier applied to a conjunction of literals is enough, because:
Given a formula , write it in DNF.
Use the fact that
Eliminate universal quantifiers using the fact x.  $ :9x. :
Decision Procedures - An algorithmic point of view

10. Quantifier Elimination
Examples first, generalization later.
Example #1: Quantified Boolean Formulas (QBF)
Example #2: Quantified Linear Arithmetic (QLA)
Decision Procedures - An algorithmic point of view

11. Example #1: QBF Examples of Quantified Boolean Formula
: u e.(uÇ :e)(:uÇ e)
: e4e5 u1u2u3 e1e2e3. f(e1,e2,e3,e4,e5,u1,u2,u3)
QBF Problem: is  valid?
P-Space Complete, theoretically harder than NP-Complete problems such as SAT.
Decision Procedures - An algorithmic point of view

12. Decision Procedures - An algorithmic point of view
Motivations
QBF has practical applications:
AI Planning
Sequential circuit verification …
Decision Procedures - An algorithmic point of view

13. Solving QBF with projection: 9
Eliminate 9x. by projecting x on variables in higher quantification levels (their scope includes x’s scope).
In Propositional Logic projection can be done with Resolution.
Resolution example:
a Ç b Ç c’ Ç f
g Ç h’ Ç c Ç f
a Ç b Ç g Ç h’
a Ç b Ç g Ç h’Ç f
Decision Procedures - An algorithmic point of view

14. Solving QBF with projection: 8
Transform 8 to 9 via: (8x. ) $ (:9x. :)
CNF is easier than general formulas:
8u1u2 9e1 8u3(u1Ç:e1)(:u1Çe1)(u2Ç:u3Ç:e1)
8u1u2 9e1:9u3 :((u1Ç:e1)(:u1Çe1)(u2Ç:u3Ç:e1))
8u1u2 9e1:9u3 ((:u1Æ e1)Ç(u1Æ :e1)Ç (:u2Æu3Æe1))
8u1u2 9e1: ((:u1Æ e1)Ç(u1Æ :e1)Ç (:u2Æ(9u3. u3)Æe1))
8u1u2 9e1 : ((:u1Æ e1)Ç(u1Æ :e1)Ç (:u2Æe1))
8u1u2 9e1 (u1Ç:e1)(:u1Çe1)(u2Ç:e1)
Suffix is DNF
Replace with true
Back to CNF
Decision Procedures - An algorithmic point of view
Shortcut for CNF formulas: simply erase universally quantified variables!

15. Resolution Based QBF Algorithm
8u1u29e18u39e3e2(u1Ç:e1)(:u1Ç:e2Çe3)(u2Ç:u3Ç:e1)(e1Çe2)(e1Ç:e3)
8u1u29e18u39e3 (u1Ç:e1)(:u1Çe3Çe1)(u2Ç:u3Ç:e1)(e1Ç:e3)
8u1u29e18u3 (u1Ç:e1)(:u1Çe1)(u2Ç:u3Ç:e1)
8u1u29e1(u1Ç:e1)(:u1Çe1)(u2Ç:e1)
8u1u2(:u1Ç u2)
FALSE
Decision Procedures - An algorithmic point of view

16. Example #2: Quantified Linear Arithmetic
formula = predicate | formula Ç formula | :formula | (formula) | 8 var. formula | 9 var. formula
predicate = i ai xi · c
8x.9y.9z. (y+1 · x Æ z+1 · y Æ 2x+1 · z)
Decision Procedures - An algorithmic point of view

17. Solving QLA with projection
Eliminate 9x. by projecting x.
In Linear Arithmetic over R projection can be done with Fourier-Motzkin elimination.
Fourier-Motzkin method to eliminate a variable xn: - for each pair of constraints: i=1..n-1ai’xi < xn < i=1..n-1aixi
add a constraint i=1..n-1ai’xi < i=1..n-1aixi
- in the end remove all constraints involving xn.
Decision Procedures - An algorithmic point of view

18. Solving QLA with projection
Fourier Motzkin: example.
Eliminate y:
x + 1 · y Æ
y · 3z + 3 Æ
2y · 2z + 4
x + 1 · z Æ
x + 1 · 3z + 3
Decision Procedures - An algorithmic point of view

19. Quantifier elimination - example
8x.9y.9z. (y+1 · x Æ z+1 · y Æ 2x+1 · z)
8x.9y. (y+1 · x Æ 2x+1 · y-1 )
8x. (2x+2 · x-1) // transform to 9
:9x. : (2x+2 · x-1)
:9x. x > -3
:true
false
Decision Procedures - An algorithmic point of view

20. Quantifier elimination by projection: summary
Given a PNF formula f = Q[n]V[n]Q[1]V[1] 
For i = 1 .. n {
if Q[i] = 9 then  = project(,V[i])
else  = :project(:,V[i]) }
Return 
Decision Procedures - An algorithmic point of view

21. Decision Procedures - An algorithmic point of view
More about QBF
Example of using QBF (the diameter problem)
A search-based procedure for QBF.
Acknowledgement: QBF slides borrowed from S. Malik
Decision Procedures - An algorithmic point of view

22. The State Space Diameter Problem
initial state: S0 S0 S2 S1 S3 S5 S4 S2 S1
step 1: S1, S2 S4 S3
step 2: S3, S4 S5
step 3: S5
diameter = 3
Start from the initial states, the minimum number
of steps needed to visit every reachable state
Decision Procedures - An algorithmic point of view

23. Why is the Diameter Problem important?
Bounded model checking (BMC): search for a ‘bad’ state up to k steps from an initial step.
BMC can be formulated as SAT. Increasing k makes is harder.
Q: how deep should we go ?
A: as deep as the diameter
The diameter can be found by solving a QBF problem
Decision Procedures - An algorithmic point of view

24. Why is the Diameter Problem important?
Bounded model checking (BMC)
Circuit state space diameter completes BMC
Can be formulated as QBF instances
Provides insights to sequential verification problems in general Sj Si
Si, Sj: sets of states
state transition relation T
S0: the set of initial states
Does property P hold for the system? S0 S0 S1 S1 S2 S2 S3 S3
useful for falsification, but incomplete for verification
Decision Procedures - An algorithmic point of view

25. Circuit Constructed for the Diameter Problem
The idea: prove that for every state reachable in k+1 steps, there exists inputs that drive the model to this state earlier.
Combinational
Logic I1 O1 In On
In+1
On+1
I1’
O1’
In’
On’
Decision Procedures - An algorithmic point of view

26. Some Terminology for the Formulations
Combinational
Logic I1 O1 In On
In+1
On+1
I1’
O1’
In’
On’
Variables: V
Circuit consistency condition: C(V)
Decision Procedures - An algorithmic point of view

27. Some Terminology for the Formulations
Combinational
Logic I1 O1 In On
In+1
On+1
I1’
O1’
In’
On’
Variables: V’
Circuit consistency condition: C(V’)
Decision Procedures - An algorithmic point of view

28. Decision Procedures - An algorithmic point of view
QBF Formulation
Combinational
Logic I1 O1 In On
In+1
On+1
I1’
O1’
In’
On’
C(V)
C(V’)
V inputs
Other V
variables
V’ variables, incl. inputs
Decision Procedures - An algorithmic point of view

29. Another way to project Boolean variables
Shannon expansion: 9x.  = |x=0 Ç |x=1 8x.  = |x=0 Æ |x=1 // can be derived from 8x. = :9x.:
The same applies for all finite-range variables.
Applying 9x., where  in CNF $ resolution
But:  does not need to be in CNF, and there is no need to transform the formula to DNF.
Decision Procedures - An algorithmic point of view

30. Projection for non-CNF formulas: example
9y 8z 9x. (y Ç (x Æ z))
9y 8z. (y Ç (x Æ z))|x=0 Ç (y Ç (x Æ z))|x=1
9y 8z. ((y) Ç (y Ç z))
9y :9z. (:y Æ :z)
9y. : ((:y Æ :z)|z=0 Ç (:y Æ :z)|z=1)
9y. : (:y)
True
Decision Procedures - An algorithmic point of view

31. Search Based QBF Algorithms
Work by gradually assigning variables
A partial assignment 
[KGS98] M. Cadoli, A. Giovanardi, M. Schaerf. An Algorithm to Evaluate Quantified Boolean Formulae. In Proc. of 16th National Conference on Artificial Intelligence (AAAI-98)
Decision Procedures - An algorithmic point of view

32. Search Based QBF Algorithms
Work by gradually assigning variables
A partial assignment 
Undetermined
Continue search
[KGS98] M. Cadoli, A. Giovanardi, M. Schaerf. An Algorithm to Evaluate Quantified Boolean Formulae. In Proc. of 16th National Conference on Artificial Intelligence (AAAI-98)
Decision Procedures - An algorithmic point of view

33. Search Based QBF Algorithms
Work by gradually assigning variables
A partial assignment 
Undetermined
Conflict
Backtrack
Record the reason
[KGS98] M. Cadoli, A. Giovanardi, M. Schaerf. An Algorithm to Evaluate Quantified Boolean Formulae. In Proc. of 16th National Conference on Artificial Intelligence (AAAI-98)
Decision Procedures - An algorithmic point of view

34. Search Based QBF Algorithms
Work by gradually assigning variables
A partial assignment 
Undetermined
Conflict
Satisfied
Backtrack
Determine the covered satisfying space
[KGS98] M. Cadoli, A. Giovanardi, M. Schaerf. An Algorithm to Evaluate Quantified Boolean Formulae. In Proc. of 16th National Conference on Artificial Intelligence (AAAI-98)
Decision Procedures - An algorithmic point of view

35. Search Based QBF Algorithms
Work by gradually assigning variables
A partial assignment 
Undetermined
Conflict
Satisfied
The majority of QBF solvers are search based, the DPLL algorithm is an example of this
Decision Procedures - An algorithmic point of view

36. Decision Procedures - An algorithmic point of view
Basic DPLL Flow for QBF
eu (e Ç u)(:e Ç :u)
Unknown
True (1)
False(0)
Decision Procedures - An algorithmic point of view

37. Decision Procedures - An algorithmic point of view
Basic DPLL Flow for QBF
eu (e Ç u)(:e Ç :u)
e = 0
Unknown
True (1)
False(0)
Decision Procedures - An algorithmic point of view

38. Basic DPLL Flow for QBF eu (e Ç u)(:e Ç :u)
Existential quantification
eu (e Ç u)(:e Ç :u)
Universal quantification
e = 0
Satisfying Node
Unknown
True (1)
u = 1
False(0)
Decision Procedures - An algorithmic point of view

39. Decision Procedures - An algorithmic point of view
Basic DPLL Flow for QBF
eu (e Ç u)(:e Ç :u)
e = 0
Backtrack
Unknown
True (1)
u = 1
False(0)
Decision Procedures - An algorithmic point of view

40. Decision Procedures - An algorithmic point of view
Basic DPLL Flow for QBF
eu (e Ç u)(:e Ç :u)
e = 0
Unknown
True (1)
u = 1
u = 0
False(0)
Decision Procedures - An algorithmic point of view

41. Decision Procedures - An algorithmic point of view
Basic DPLL Flow for QBF
eu (e Ç u)(:e Ç :u)
e = 0
Unknown
True (1)
u = 1
u = 0
False(0)
Decision Procedures - An algorithmic point of view

42. Decision Procedures - An algorithmic point of view
Basic DPLL Flow for QBF
eu (e Ç u)(:e Ç :u)
e = 1
e = 0
Unknown
True (1)
u = 1
u = 0
False(0)
Decision Procedures - An algorithmic point of view

43. Decision Procedures - An algorithmic point of view
Basic DPLL Flow for QBF
eu (e Ç u)(:e Ç :u)
e = 1
e = 0
Unknown
True (1)
u = 1
u = 1
u = 0
False(0)
Decision Procedures - An algorithmic point of view

44. Decision Procedures - An algorithmic point of view
Basic DPLL Flow for QBF
eu (e Ç u)(:e Ç :u)
e = 1
e = 0
Conflicting Node
Unknown
True (1)
u = 1
u = 1
u = 0
False(0)
Decision Procedures - An algorithmic point of view

45. Decision Procedures - An algorithmic point of view
Basic DPLL Flow for QBF
eu (e Ç u)(:e Ç :u)
e = 1
e = 0
Unknown
True (1)
u = 1
u = 1
u = 0
False(0)
Decision Procedures - An algorithmic point of view

46. Decision Procedures - An algorithmic point of view
Basic DPLL Flow for QBF
False
eu (e Ç u)(:e Ç :u)
e = 1
e = 0
Unknown
True (1)
u = 1
u = 1
u = 0
False(0)
Decision Procedures - An algorithmic point of view

47. Decision Procedures - An algorithmic point of view
Basic DPLL Flow for QBF
ue (u Ç e)(:u Ç :e)
Unknown
True (1)
False(0)
Decision Procedures - An algorithmic point of view

48. Decision Procedures - An algorithmic point of view
Basic DPLL Flow for QBF
ue (u Ç e)(:u Ç :e)
u = 1
Unknown
True (1)
False(0)
Decision Procedures - An algorithmic point of view

49. Decision Procedures - An algorithmic point of view
Basic DPLL Flow for QBF
ue (u Ç e)(:u Ç :e)
u = 1
Unknown
True (1)
e = 1
False(0)
Decision Procedures - An algorithmic point of view

50. Decision Procedures - An algorithmic point of view
Basic DPLL Flow for QBF
ue (u Ç e)(:u Ç :e)
u = 1
Unknown
True (1)
e = 1
e = 0
False(0)
Decision Procedures - An algorithmic point of view

51. Decision Procedures - An algorithmic point of view
Basic DPLL Flow for QBF
ue (u Ç e)(:u Ç :e)
u = 1
Unknown
True (1)
e = 1
e = 0
False(0)
Decision Procedures - An algorithmic point of view

52. Decision Procedures - An algorithmic point of view
Basic DPLL Flow for QBF
ue (u Ç e)(:u Ç :e)
u = 1
u = 0
Unknown
True (1)
e = 1
e = 0
e = 1
False(0)
Decision Procedures - An algorithmic point of view

53. Decision Procedures - An algorithmic point of view
Basic DPLL Flow for QBF
True
ue (u Ç e)(:u Ç :e)
u = 1
u = 0
Unknown
True (1)
e = 1
e = 0
e = 1
False(0)
Decision Procedures - An algorithmic point of view

54. Decision Procedures - An algorithmic point of view
What we saw is DPLL - QBF
while (true) {
decide_next_branch() //choose a branch variable
while(true) {
status = deduce(); //unit propagation
if (status == CONFLICT) {
blevel = analyze_conflict(); //find out the reason for conflict
if (blevel < 0) return UNSATISFIABLE;
else backtrack(blevel); }
else if (status == SATISFIABLE) {
blevel = analyze_SAT() //find out the reason for satisfaction
if (blevel < 0) return SATISFIABLE;
else break; }}
Decision Procedures - An algorithmic point of view

55. Naïve DPLL Based Approach
Works on a CNF database
Learning and non-chronological backtracking is much harder – requires a change!
Modern QBF solvers do not work with CNF, rather with a combination of CNF with Cubes. This lets them apply learning efficiently.
Decision Procedures - An algorithmic point of view

56. Decision Procedures - An algorithmic point of view
ACNF
Definition: Augmented CNF (ACNF)
 = C1 Æ C2Æ …Æ CmÇ S1 Ç S2 Ç ……Ç Sk
Where Ci’s are clauses, and Sj ’s are cubes.
Each Sj is contained in the clause term C1 C2…Cm.
i.e.i{1,2…k}, Si  C1 Æ C2 Æ … Æ Cm
In ACNF, cubes are redundant
Example:
(:aÇ:bÇ:c)Æ(:aÇbÇc) Æ(aÇ:bÇc) Æ(aÇbÇ:c) Ç
(:aÆ:bÆ:c) Ç (aÆ :bÆ c)
Decision Procedures - An algorithmic point of view

57. Solving QBF of ACNF formulas
Cubes are necessary for saving information on the (universal) space already covered.
We will see a special case: 2QBF
2QBF = QBF with one quantifier alternation
very useful!
Decision Procedures - An algorithmic point of view

58. Decision Procedures - An algorithmic point of view
A special case: 2QBF
DPLL search based, utilize a standard SAT solver
Algorithm I: Assign universal variables first
Algorithm II: no restriction in decision order w.r.t. variable quantification order
Resolution based
No simplification, just Q-resolution
With complete two-level minimization (using Logic-Minimization) at each resolution step
Decision Procedures - An algorithmic point of view

59. Coverage Cubes and Blocking Clauses
(u1 Ç u2 Ç e1) Æ
(u3 Ç :e2) Æ
(:u1Ç :u2Ç e2) Æ
(u1Ç :e1Ç :e2)
satisfying assignment :
{u1=1, u2=0, u3=X, e1=0, e2=0}
coverage cube for the universal Boolean space:
u1  :u2
select a set of literals
that satisfy all clauses
blocking clause:
:u1 Ç u2
prevents revisiting the already searched space
satisfying cube:
{u1=1, u2=0, e2=0}
Decision Procedures - An algorithmic point of view

60. Example for Algorithm I
8u1u29e1e2. (u1Ç e1)Æ(:u1Ç :e1)Æ(:u1Ç u2Ç:e2) Æ(:u2Ç:e1Ç e2) Æ (e1Ç:e2)
universal assignment: u1=0, u2=0
SAT assignment: u1=0, u2=0, e1=1, e2=1
satisfying cube (cover set): u1=0, e1=1, e2=1
coverage cube: u1=0
universal assignment: u1=1, u2=0
SAT assignment: u1=1, u2=0, e1=0, e2=0
satisfying cube (cover set): u1=1, e1=0, e2=0
coverage cube: u1=1
no more universal assignment left, instance is true
Decision Procedures - An algorithmic point of view

61. Decision Procedures - An algorithmic point of view
universal assignment
SAT assignment 5 1 2
satisfying cube
(cover set) 3 4
coverage cube
universal space
(u1,u2,…,um)
existential space
(e1,e2,…,en)
Decision Procedures - An algorithmic point of view

62. Example for Algorithm II
8u1u29e1e2. (u1Ç e1)Æ(:u1Ç :e1)Æ(:u1Ç u2Ç:e2) Æ(:u2Ç:e1Ç e2) Æ (e1Ç:e2)
SAT assignment: u1=0, e1=1, e2=1, u2=0
blocking clause: (u1 Ç :e1 Ç :e2)
coverage cube: u1=0
SAT assignment: u1=1, e1=0, e2=0 , u2=0
blocking clause: (u1 Ç :e1 Ç :e2)
coverage cube: u1=1
The entire universal space is covered, instance is true
Decision Procedures - An algorithmic point of view

63. universal variable space
Algorithm II
SAT assignment, no need to respect quantification order to get that
coverage cube 3 4 1
blocking clause 2
all variable space
(u1,u2,…,um,e1,e2,…,en)
universal variable space
(u1,u2,…,um)
Decision Procedures - An algorithmic point of view

64. Resolution Based Algorithm
First resolve out existential variables
After resolving out all existential variables
An empty clause (a clause with no literal or consisting only of universal variables)  false
An empty set of clauses  true
Has the memory blowup problem
Alleviate by simplifying the propositional part after each resolution step
Decision Procedures - An algorithmic point of view

65. Example for Resolution Based Algorithm
8u1u29e1e2. (u1Ç e1)Æ(:u1Ç :e1)Æ(:u1Ç u2Ç:e2) Æ(:u2Ç:e1Ç e2) Æ (e1Ç:e2)
resolve out e1
8u1u29e2. (u1Ç :u2 Ç e2)Æ(:u1Ç :e2)Æ(:u1Ç u2Ç:e2)
simplify
8u1u29e2. (u1Ç :u2 Ç e2)Æ(:u1Ç :e2)
resolve out e2
empty set of clause
instance is true
Decision Procedures - An algorithmic point of view

66. Experimental Results (Malik et al.)
# of clauses (100 vars, 5 lits/clause)
Alg. I
Alg. II
Res w/o simp
Res w/ simp
Quaffle
100 (100 instances)
0(100)
16.22(98)
0.01(100)
0.25(100)
>400(0)
200 (10 instances)
160(6)
>400(0)
0.36(10)
169.09(6)
>400(0)
Circuit
Depth
Alg. I
Alg. II
Res w/o simp
Res w/ simp
Quaffle
s1488 1
0.15
0.26
>400
2.19
0.27 3
0.80
294.27
>400
25.04
18.23 5
2.5
>400
>400
208.6
>400
s1423 1
0.15
>400
>400
4.29
0.22 15
2.11
>400
>400
>400
>400
34**
298.86
>400
>400
>400
>400
* The tests were done on an Intel Pentium III 933 MHz machine with 1GB of RAM running linux.
** Improves on previous diameter lower bound of 26
Decision Procedures - An algorithmic point of view