res-1-2-movinggt-alea.math.cnrs.fr/alea2008/exposes/creignou.pdf · title: res-1-2-moving.ps...

Threshold phenomena for SAT problemsQuantified satisfiability problems

Threshold phenomena for (a,e)-QXOR-SATThreshold phenomenon for (1,2)-QSAT

Phase transition for random quantified BooleanFormulas

N. Creignou1 H. Daudé2 Uwe Egly3 R. Rossignol4

1LIF, Marseille

2LATP, Marseille

3TU Wien

4Orsay

Alea, 2008



Outline

1 Threshold phenomena for SAT problems

2 Quantified satisfiability problemsProbabilistic model : (a,e)-QSAT

3 Threshold phenomena for (a,e)-QXOR-SATThe case e ≥ 3The case e = 2

4 Threshold phenomenon for (1,2)-QSATComplexity of (1,2)-QSATExperimental resultsRepresentation of (1,2)-QCNF-formulas as labeled digraphsLower and upper bounds for the threshold



A threshold phenomenon for 3-SAT

Pr(SATn,L) = probability that a 3-CNF formula over n variableswith L = c · n clauses is satisfiable.

Pr(SATn,c·n) → 1 for c ≤ 3.52

(Kaporis, Kirousis, Lalas, 2003)

Pr(SATn,c·n) → 0 for c ≥ 4.506

(Dubois, Boufkhad, Mandler, 2000)The critical ratio is estimated at around 4.25



Nature of the transition

Given a constraint satisfaction problem, depending on the sizeof the scaling window the transition SAT/UNSAT is either sharpor coarse.

The transition for 3-SAT is sharp (Friedgut, 1998).

The transition for 2-SAT is sharp, the critical ratio is 1(Chvatal, Reed, Goerdt, 1992).



An example of a sharp transition : 3-XOR-SAT

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.75 0.8 0.85 0.9 0.95 1

no clauses/no exvars



An example of a coarse transition : 2-XOR-SAT

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1


SATf1(x)=exp(x/2)*(1-2*x)**0.25

5k10k20k40k80k

120k160k



To sum up . . .

Problem Scale Nature Critical ratio Complexity3-SAT L = θ(n) Sharp 3.52 ≤ c3 ≤ 4.506 NP-complete2-SAT L = θ(n) Sharp c2 = 1 P

3-XOR-SAT L = θ(n) Sharp cXOR,3 ∼ 0.918 P2-XOR-SAT L = θ(n) Coarse P

Random instances are useful to evaluate the performance ofSAT solvers : hard instances are at the threshold.



Probabilistic model : (a,e)-QSAT

Outline








QSAT

Input : Φ a closed formula of the formQ1x1Q2x2 . . . Qnxnϕ, where Q1, . . . , Qn arearbitrary quantifiers and ϕ is a CNF-formula

Question : Is Φ true ?

QSAT is PSPACE-complete (ranges over the fullpolynomial hierarchy depending on the number ofquantifiers alternation).

QSAT allows the modelling of various problems (games,model checking, verification, etc.).

QSAT is a monotone property.




QSAT and random instances

Is there a phase transition for QSAT ?

What is a "good" probabilistic model ?

Is there an easy-hard-easy pattern for random instances ?

Cadoli et al. 97, Gent and Walsh 99, Chen and Interian 05




(a,e)-QSAT

We will first restrict our attention to formulas with only onealternation of quantifiersAn (a,e)-QCNF-formula is a closed quantified formula of thefollowing type

∀X∃Yϕ(X , Y ),

X and Y denote distinct set of variables,

ϕ(X , Y ) is an (a + e)-CNF-formula such that each clausecontains exactly a variables from X and exactly e variablesfrom Y .

(a,e)-QSAT the property for an (a,e)-QCNF-formula ofbeing true.




(a,e)-QCNF((m,n),L)-formulas

m the number of universal variables, {x1, . . . , xm}

n for the number of existential variables, {y1, . . . , yn}

Random formulas ∀X∃Yϕ(X , Y ) obtained by choosinguniformly independently and with replacement L clausesamong the N = 2a+e

(ma

)(ne

)

the possible clauses.




(a,e)-QSAT

We are interested in the probability that a randomly chosen(a,e)-QCNF((m,n),L)-formula is true.

Pr(m,n,L)((a,e)-QSAT)

Non-quantified case : e-SAT, i.e., a=0,

Prn,L(e-SAT)




(a,e)-QXOR-SAT a natural variant to initiate aninvestigation

QXOR-SAT is in P (linear algebra framework)

Well studied in the non-quantified version : random graphtools.



The case e ≥ 3The case e = 2

Outline








A sharp threshold

Prn,L(e-Max-rank) ≤ Pr(m,n,L)((a,e)-QXOR-SAT) ≤ Prn,L(e-XOR-SAT)

2Prn,L(e-XOR-SAT) − 12

≤ Prn,L((a,e)-QXOR-SAT) ≤ Prn,L(e-XOR-SAT)

e-XOR-SAT exhibits a sharp threshold when L is Θ(n) and so does(a,e)-QXOR-SAT, with a critical value c3 ≈ 0.918 for e = 3.




Representation of (a,2)-QXOR-formulas as labeledgraphs

Let X = {x1, x2, x3} and let Y = {y1, . . . , y7}. The formula∀X∃Y ϕ(X , Y ) with ϕ(X , Y ) being a conjunction of the followingequations

y1 ⊕ y2 = x1 y1 ⊕ y7 = x2

y2 ⊕ y3 = x3 y2 ⊕ y6 = x2 ⊕ 1y3 ⊕ y4 = x2 ⊕ 1 y3 ⊕ y5 = x3

y4 ⊕ y5 = x3 ⊕ 1 y6 ⊕ y7 = x1 ⊕ 1




Bad cycles

y1

y2

x1

y7

x2

y3

x3

y6

x2+ 1

y4

x2 + 1

y5

x3

x1 + 1

x3 + 1

Cycle is bad if it has a nonzero weight, and good otherwise.




Bad cycles and (a,2)-QXOR-SAT

Pr((a,2)-QXOR-SAT) = Pr(Ga(s) has no bad cycle)




The distribution function for (a,2)-QXOR-SAT

(a,2)-QXOR-SAT has a coarse threshold.

Pr(m,n,cn)((a,2)-QXOR-SAT) −→n→+∞ Hm(c).

The distribution function Hm depends on the number ofuniversal variables.




The distribution functions H0, Ha and H∞

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1no clauses/no exvars

Ha(x) = ex√

1 − 2x (1 − 4x2)−1/8

H0(x) = ex/2(1 − 2x)0.25

H∞(x) = ex√

1 − 2x



Complexity of (1,2)-QSATExperimental resultsRepresentation of (1,2)-QCNF-formulas as labeled digraphsLower and upper bounds for the threshold

Outline








(1,2)-QSAT

m the number of universal variables, {x1, . . . , xm}

n for the number of existential variables, {y1, . . . , yn}

Random formulas ∀X∃Yϕ(X , Y ) obtained by choosinguniformly independently and with replacement L = cnclauses among the N = 23

(m1

)(n2

)

possible 3-clauseshaving 1 universal and 2 existential literals.

We are interested in the probability that such a randomlychosen formula is true.

Pm,c(n)




(1,2)-QSAT and its complexity

If m is constant, (1,2)-QSAT is solvable in linear time.

If m = α⌈log n⌉, (1,2)-QSAT is solvable in polynomial time.

If m = n, then (1,2)-QSAT is coNP-complete.




When m(n) = n : the threshold occurs at c = 1

0

0.2

0.4

0.6

0.8

1

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2

SA

T


(1,2)-(5k,5k)-QCNF(1,2)-(10k,10k)-QCNF(1,2)-(20k,20k)-QCNF(1,2)-(40k,40k)-QCNF




When m(n) = 2 : the threshold occurs at c = 2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.75 1.8 1.85 1.9 1.95 2 2.05 2.1 2.15 2.2 2.25

SA

T


(1,2)-(2,10k)-QCNF(1,2)-(2,20k)-QCNF(1,2)-(2,40k)-QCNF




Intermediate regime when m = α⌈log n⌉

0

0.2

0.4

0.6

0.8

1

1 1.2 1.4 1.6 1.8 2

SA

T


(1,2)-(80, 12365)-QCNF(1,2)-(23, 11222)-QCNF(1,2)-(15, 12445)-QCNF




Labeled digraphs

Let φ : ∀x1∃y1y2(x1 ∨ y1 ∨ y2) ∧ (x1 ∨ y1 ∨ y2).

y1 y2

y1 y2

x1

x1

x1x1

y1 y2

y1 y2

y1 y2

y1 y2

The quantified formula is satisfiable

if and only if

there is no pure path y y y for any existential variable y .




A necessary condition for unsatisfiability

Every unsatisfiable (1,2)-QCNF formula contains a pure bicycle.

Let B be the number of pure bicycles in a (1,2)-QCNF formula.

1 − Pm,c(n) ≤ Pr(B ≥ 1) ≤ E(B).




A sufficient condition for unsatisfiabilility

Every (1,2)-QCNF formula that contains some simple snake isunsatisfiable.

Let X be the number of simple snakes of size s + 1 = 2t in a(1,2)-QCNF formula.

1 − Pm,c(n) ≥ Pr(X ≥ 1) ≥

(

E(X ))2

E(X 2)




Mean of the number of pure bicycles

Let p be such that N · p ∼ c · n.

E(B) =

n∑

s=2

(n)s2s[(2s)2 − 1]c(m, s + 1)ps+1 ,

where

c(m, s + 1) =

min(m,s+1)∑

k=1

(

mk

)

· 2k · S(s + 1, k) · k!

with S(m, k) denoting the Stirling number of the second kind.




A lower bound for the threshold

Theorem

When 1 < c < 2, and m = ⌈α ln n⌉ with α >1

ln(2),

E(B) ≤ C(ln n)9/2 · nαH(c)−1 + o(1)

where C is a finite constant depending only on α and c, and

H(c) = ln(c) +(2

c− 1

)

ln(2 − c).

Let a(α) be the solution of the equation α · H(c) = 1, then forc < a(α) the above result shows that E(B) = o(1).




Main result

Let m = ⌈α ln n⌉ where α > 0. There exist 1 < a(α) ≤ b(α) ≤ 2such that the following holds :

if c < a(α), then Pm,c(n) −−−−→n→+∞

1,

if c > b(α), then Pm,c(n) −−−−→n→+∞

0.

Moreover :

1 if α ≤1

ln 2, then a(α) = b(α) = 2,

2 if1

ln 2< α ≤

2ln 2 − 1/2

, then a(α) < b(α) = 2 and a is

strictly decreasing,

3 if α >2

ln 2 − 1/2, then a(α) < b(α) < 2, a and b are

strictly decreasing and limα→+∞

a(α) = limα→+∞

b(α) = 1.




The lower and upper bounds

FIG.: a(α) and b(α)




Conclusion

Validation of the probabilistic models proposed for thestudy of the QSAT transition.

For (a,e)-QXOR-SAT, introduction of quantifiers has aneffect at the level of the distribution function.

For (1,2)-QSAT, introduction of quantifiers influences thelocation of the threshold, the number of universal variablesplays a crucial role.

res-1-2-movinggt-alea.math.cnrs.fr/alea2008/exposes/creignou.pdf · title: res-1-2-moving.ps...

Documents