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
Threshold phenomena for SAT problemsQuantified satisfiability problems
Threshold phenomena for (a,e)-QXOR-SATThreshold phenomenon for (1,2)-QSAT
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
Threshold phenomena for SAT problemsQuantified satisfiability problems
Threshold phenomena for (a,e)-QXOR-SATThreshold phenomenon for (1,2)-QSAT
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
Threshold phenomena for SAT problemsQuantified satisfiability problems
Threshold phenomena for (a,e)-QXOR-SATThreshold phenomenon for (1,2)-QSAT
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
Threshold phenomena for SAT problemsQuantified satisfiability problems
Threshold phenomena for (a,e)-QXOR-SATThreshold phenomenon for (1,2)-QSAT
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).
Threshold phenomena for SAT problemsQuantified satisfiability problems
Threshold phenomena for (a,e)-QXOR-SATThreshold phenomenon for (1,2)-QSAT
An example of a sharp transition : 3-XOR-SAT
0
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0.75 0.8 0.85 0.9 0.95 1
no clauses/no exvars
Threshold phenomena for SAT problemsQuantified satisfiability problems
Threshold phenomena for (a,e)-QXOR-SATThreshold phenomenon for (1,2)-QSAT
An example of a coarse transition : 2-XOR-SAT
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SATf1(x)=exp(x/2)*(1-2*x)**0.25
5k10k20k40k80k
120k160k
Threshold phenomena for SAT problemsQuantified satisfiability problems
Threshold phenomena for (a,e)-QXOR-SATThreshold phenomenon for (1,2)-QSAT
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.
Threshold phenomena for SAT problemsQuantified satisfiability problems
Threshold phenomena for (a,e)-QXOR-SATThreshold phenomenon for (1,2)-QSAT
Probabilistic model : (a,e)-QSAT
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
Threshold phenomena for SAT problemsQuantified satisfiability problems
Threshold phenomena for (a,e)-QXOR-SATThreshold phenomenon for (1,2)-QSAT
Probabilistic model : (a,e)-QSAT
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.
Threshold phenomena for SAT problemsQuantified satisfiability problems
Threshold phenomena for (a,e)-QXOR-SATThreshold phenomenon for (1,2)-QSAT
Probabilistic model : (a,e)-QSAT
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
Threshold phenomena for SAT problemsQuantified satisfiability problems
Threshold phenomena for (a,e)-QXOR-SATThreshold phenomenon for (1,2)-QSAT
Probabilistic model : (a,e)-QSAT
(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.
Threshold phenomena for SAT problemsQuantified satisfiability problems
Threshold phenomena for (a,e)-QXOR-SATThreshold phenomenon for (1,2)-QSAT
Probabilistic model : (a,e)-QSAT
(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.
Threshold phenomena for SAT problemsQuantified satisfiability problems
Threshold phenomena for (a,e)-QXOR-SATThreshold phenomenon for (1,2)-QSAT
Probabilistic model : (a,e)-QSAT
(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)
Threshold phenomena for SAT problemsQuantified satisfiability problems
Threshold phenomena for (a,e)-QXOR-SATThreshold phenomenon for (1,2)-QSAT
Probabilistic model : (a,e)-QSAT
(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.
Threshold phenomena for SAT problemsQuantified satisfiability problems
Threshold phenomena for (a,e)-QXOR-SATThreshold phenomenon for (1,2)-QSAT
The case e ≥ 3The case e = 2
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
Threshold phenomena for SAT problemsQuantified satisfiability problems
Threshold phenomena for (a,e)-QXOR-SATThreshold phenomenon for (1,2)-QSAT
The case e ≥ 3The case e = 2
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.
Threshold phenomena for SAT problemsQuantified satisfiability problems
Threshold phenomena for (a,e)-QXOR-SATThreshold phenomenon for (1,2)-QSAT
The case e ≥ 3The case e = 2
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.
Threshold phenomena for SAT problemsQuantified satisfiability problems
Threshold phenomena for (a,e)-QXOR-SATThreshold phenomenon for (1,2)-QSAT
The case e ≥ 3The case e = 2
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.
Threshold phenomena for SAT problemsQuantified satisfiability problems
Threshold phenomena for (a,e)-QXOR-SATThreshold phenomenon for (1,2)-QSAT
The case e ≥ 3The case e = 2
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.
Threshold phenomena for SAT problemsQuantified satisfiability problems
Threshold phenomena for (a,e)-QXOR-SATThreshold phenomenon for (1,2)-QSAT
The case e ≥ 3The case e = 2
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
Threshold phenomena for SAT problemsQuantified satisfiability problems
Threshold phenomena for (a,e)-QXOR-SATThreshold phenomenon for (1,2)-QSAT
The case e ≥ 3The case e = 2
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.
Threshold phenomena for SAT problemsQuantified satisfiability problems
Threshold phenomena for (a,e)-QXOR-SATThreshold phenomenon for (1,2)-QSAT
The case e ≥ 3The case e = 2
Bad cycles and (a,2)-QXOR-SAT
Pr((a,2)-QXOR-SAT) = Pr(Ga(s) has no bad cycle)
Threshold phenomena for SAT problemsQuantified satisfiability problems
Threshold phenomena for (a,e)-QXOR-SATThreshold phenomenon for (1,2)-QSAT
The case e ≥ 3The case e = 2
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.
Threshold phenomena for SAT problemsQuantified satisfiability problems
Threshold phenomena for (a,e)-QXOR-SATThreshold phenomenon for (1,2)-QSAT
The case e ≥ 3The case e = 2
The distribution functions H0, Ha and H∞
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Ha(x) = ex√
1 − 2x (1 − 4x2)−1/8
H0(x) = ex/2(1 − 2x)0.25
H∞(x) = ex√
1 − 2x
Threshold phenomena for SAT problemsQuantified satisfiability problems
Threshold phenomena for (a,e)-QXOR-SATThreshold phenomenon for (1,2)-QSAT
Complexity of (1,2)-QSATExperimental resultsRepresentation of (1,2)-QCNF-formulas as labeled digraphsLower and upper bounds for the threshold
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
Threshold phenomena for SAT problemsQuantified satisfiability problems
Threshold phenomena for (a,e)-QXOR-SATThreshold phenomenon for (1,2)-QSAT
Complexity of (1,2)-QSATExperimental resultsRepresentation of (1,2)-QCNF-formulas as labeled digraphsLower and upper bounds for the threshold
(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)
Threshold phenomena for SAT problemsQuantified satisfiability problems
Threshold phenomena for (a,e)-QXOR-SATThreshold phenomenon for (1,2)-QSAT
Complexity of (1,2)-QSATExperimental resultsRepresentation of (1,2)-QCNF-formulas as labeled digraphsLower and upper bounds for the threshold
(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.
Threshold phenomena for SAT problemsQuantified satisfiability problems
Threshold phenomena for (a,e)-QXOR-SATThreshold phenomenon for (1,2)-QSAT
Complexity of (1,2)-QSATExperimental resultsRepresentation of (1,2)-QCNF-formulas as labeled digraphsLower and upper bounds for the threshold
When m(n) = n : the threshold occurs at c = 1
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SA
T
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(1,2)-(5k,5k)-QCNF(1,2)-(10k,10k)-QCNF(1,2)-(20k,20k)-QCNF(1,2)-(40k,40k)-QCNF
Threshold phenomena for SAT problemsQuantified satisfiability problems
Threshold phenomena for (a,e)-QXOR-SATThreshold phenomenon for (1,2)-QSAT
Complexity of (1,2)-QSATExperimental resultsRepresentation of (1,2)-QCNF-formulas as labeled digraphsLower and upper bounds for the threshold
When m(n) = 2 : the threshold occurs at c = 2
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SA
T
no clauses/no exvars
(1,2)-(2,10k)-QCNF(1,2)-(2,20k)-QCNF(1,2)-(2,40k)-QCNF
Threshold phenomena for SAT problemsQuantified satisfiability problems
Threshold phenomena for (a,e)-QXOR-SATThreshold phenomenon for (1,2)-QSAT
Complexity of (1,2)-QSATExperimental resultsRepresentation of (1,2)-QCNF-formulas as labeled digraphsLower and upper bounds for the threshold
Intermediate regime when m = α⌈log n⌉
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SA
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(1,2)-(80, 12365)-QCNF(1,2)-(23, 11222)-QCNF(1,2)-(15, 12445)-QCNF
Threshold phenomena for SAT problemsQuantified satisfiability problems
Threshold phenomena for (a,e)-QXOR-SATThreshold phenomenon for (1,2)-QSAT
Complexity of (1,2)-QSATExperimental resultsRepresentation of (1,2)-QCNF-formulas as labeled digraphsLower and upper bounds for the threshold
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 .
Threshold phenomena for SAT problemsQuantified satisfiability problems
Threshold phenomena for (a,e)-QXOR-SATThreshold phenomenon for (1,2)-QSAT
Complexity of (1,2)-QSATExperimental resultsRepresentation of (1,2)-QCNF-formulas as labeled digraphsLower and upper bounds for the threshold
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).
Threshold phenomena for SAT problemsQuantified satisfiability problems
Threshold phenomena for (a,e)-QXOR-SATThreshold phenomenon for (1,2)-QSAT
Complexity of (1,2)-QSATExperimental resultsRepresentation of (1,2)-QCNF-formulas as labeled digraphsLower and upper bounds for the threshold
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)
Threshold phenomena for SAT problemsQuantified satisfiability problems
Threshold phenomena for (a,e)-QXOR-SATThreshold phenomenon for (1,2)-QSAT
Complexity of (1,2)-QSATExperimental resultsRepresentation of (1,2)-QCNF-formulas as labeled digraphsLower and upper bounds for the threshold
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.
Threshold phenomena for SAT problemsQuantified satisfiability problems
Threshold phenomena for (a,e)-QXOR-SATThreshold phenomenon for (1,2)-QSAT
Complexity of (1,2)-QSATExperimental resultsRepresentation of (1,2)-QCNF-formulas as labeled digraphsLower and upper bounds for the threshold
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).
Threshold phenomena for SAT problemsQuantified satisfiability problems
Threshold phenomena for (a,e)-QXOR-SATThreshold phenomenon for (1,2)-QSAT
Complexity of (1,2)-QSATExperimental resultsRepresentation of (1,2)-QCNF-formulas as labeled digraphsLower and upper bounds for the threshold
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.
Threshold phenomena for SAT problemsQuantified satisfiability problems
Threshold phenomena for (a,e)-QXOR-SATThreshold phenomenon for (1,2)-QSAT
Complexity of (1,2)-QSATExperimental resultsRepresentation of (1,2)-QCNF-formulas as labeled digraphsLower and upper bounds for the threshold
The lower and upper bounds
FIG.: a(α) and b(α)
Threshold phenomena for SAT problemsQuantified satisfiability problems
Threshold phenomena for (a,e)-QXOR-SATThreshold phenomenon for (1,2)-QSAT
Complexity of (1,2)-QSATExperimental resultsRepresentation of (1,2)-QCNF-formulas as labeled digraphsLower and upper bounds for the threshold
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.