Fixed Deficiency and QCNF
QHorn and Satisfiability
Expressive Power
Open Problems for Quantified Boolean Formulas
Hans Kleine Büning
University of Paderborn
Institute for Computer Science
Knowledge-Based Systems Group
Bordeaux, July 4, 2016
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Fixed Deficiency and QCNF
QHorn and Satisfiability
Expressive Power
Introduction
Fixed Deficiency for QCNF
QHorn and Satisfiabilty
Related Horn Problems
1
2
Equivalence Problem
Literal Problem
Expressive Power
1
2
3
Equivalence Models and Propositional Formulas
QHorn and Q2-CNF
Model size and Deficiency
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Formula α = α1 ∧ . . . αm ∈CNF over the variables x1 , . . . , xn .
α is minimal unsatisfiable (MU) iff α ∈ SAT and α \ αi ∈ SAT
for every i.
The deficiency is defined as d(α) = m − n.
MU(k) is the set of MU-formulas with deficiency k
Theorem
1
MU is D P -complete.
(SAT, SAT, Papadimitriou, Wolf)
2
Every formula in MU has deficiency greater than 0 (Lional at
all).
3
Every minimal unsatisfiable Horn formula has deficiency 1.
4
MU(k) is solvable in polynomial time. (Kullmann, Szeider)
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Fixed Deficiency and QCNF
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QCNF: Quantified Boolean formulas with kernel in CNF
Example: Φ = ∀x ∃y : (x ∨ y ) ∧ (x ∨ ¬y )
Φ is false, but
Φ \ (x ∨ y ) = ∀x ∃y : (x ∨ ¬y ) and
Φ \ (x ∨ ¬y ) = ∀x ∃y : (x ∨ ¬y ) are true.
Φ is minimal false.
Deficiency: number of clauses - number of existential variables
d(Φ) = 2 − 1 = 1
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Extension to closed QCNF (minimal falsity and deficiency)
Let Φ = Q 1≤i≤n ϕi ∈ QCNF with universal variables x1 , . . . , xt
and existential variables y1 , . . . , yr .
V
Definition
1. The formula Φ is minimal false (MF) iff Φ is false and for
V
every j the formula Q 1≤i6=j≤n ϕi is true.
2. The deficieny is defined as d(Φ) = n − |var (ϕ|∃ )|.
(number of clauses minus the number the existential variables)
3. For fixed k we define MF(k) = {Φ : Φ ∈ MF and d(Φ) = k}.
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Theorem
(KB, Zhao)
1
The minimal falsity problem MF is PSPACE-complete.
2
If Φ ∈ MF , then d(Φ) ≥ 1.
3
MF (1) is solvable in polynomial time.
4
For fixed k ≥ 1 : MF (k) is in D P .
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Theorem
(KB, Zhao)
1
The minimal falsity problem MF is PSPACE-complete.
2
If Φ ∈ MF , then d(Φ) ≥ 1.
3
MF (1) is solvable in polynomial time.
4
For fixed k ≥ 1 : MF (k) is in D P .
Open Problem:The computational complexity of MF (k) for fixed
k ≥ 2.
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Fixed Deficiency and QCNF
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DHorn: conjunction of implications (a1 , . . . , an → b) and facts (a)
Horn: DHorn ∪ negative clauses (¬a1 ∨ . . . ∨ ¬an )
QHorn: set of quantified Boolean formulas in prenex normal form
with matrix in Horn.
Let Φ = Qφ ∈ QHorn without free variables and prefix Q.
Let k be the number of universal quantifiers.
Theorem
(KB at all)
The satisfiability problem is solvable in time O(k · |Φ|).
Open Problem: Can we solve the satisfiability problem for QHorn
in linear time?
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QHorn and Multi-Horn-SAT
Multi-Horn-SAT:
Instance: α ∈ DHorn, r ≥ 1, Y = {y1 , . . . , ym }, S1 , . . . , Sr ⊆ Y ,
N1 , . . . , Nr negative clauses
Query: ∃j : Sj ∧ α ∧ Nj ∈ SAT?
y1
y4
α
N1
y3
y4
α
N2
y2
y3
α
N3
y1
y2
α
N4
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QHorn
Instance: α ∈ DHorn, r ≥ 1, Y = {y1 , . . . , ym }, S1 , . . . , Sr ⊆ Y ,
N1 , . . . , Nr negative clauses
Query: ∃j : Sj ∧ α ∧ Nj ∈ SAT?
y1
y4
α
N1
y3
y4
α
N2
y2
y3
y1
y2
¬x1 y21
¬x1 y31
¬x2 y11
¬x2 ¬y21 y2
¬x3 ¬y11 y1
¬x3 y41
α
N3
α
N4
α
x1 N1
α
x 2 N2
α
x3 N3
¬x4 ¬y31 y3
¬x4 ¬y41 y4
α
x4 N4
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Fixed Deficiency and QCNF
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y1
y4
α
N1
y3
y4
α
N2
y2
y3
α
N3
y1
y2
α
N4
¬x1 y21
¬x1 y31
α
x1 N1
¬x2 y11
¬x2 ¬y21 y2
α
x 2 N2
Expressive Power
¬x3 ¬y11 y1
¬x3 y41
α
x3 N3
¬x4 ¬y31 y3
¬x4 ¬y41 y4
α
x4 N4
∀x1 ∀x2 ∀x3 ∀x4 ∃Y : α, 1≤i≤4 (xi ∨ Ni ),
(¬x2 ∨ y11 ), (¬x3 ∨ ¬y11 ∨ y1 ), (¬x1 ∨ y21 ), (¬x2 ∨ ¬y21 ∨ y2 ),
(¬x1 ∨ y31 ), (¬x4 ∨ ¬y31 ∨ y3 ), (¬x3 ∨ y41 ), (¬x4 ∨ ¬y41 ∨ y4 )
V
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Fixed Deficiency and QCNF
QHorn and Satisfiability
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Related Horn Problems
1
Instance: α, β ∈ Horn
Query: (Equivalence) α ≈ β ?
Solvable in quadratic time
Open problem: Solvable in linear or O(n log(n)) time?
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Fixed Deficiency and QCNF
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Related Horn Problems
1
Instance: α, β ∈ Horn
Query: (Equivalence) α ≈ β ?
Solvable in quadratic time
Open problem: Solvable in linear or O(n log(n)) time?
2
Instance: α ∈ Horn over the variables X = {x1 , . . . , xm }
Query: Compute NL(α) = {¬xi : 1 ≤ i ≤ m, α |= ¬xi }
P(α) := {xi : 1 ≤ i ≤ m, α |= xi } linear time (unit
propagation)
Open problem: Can we compute NL(α) in linear time?
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Fixed Deficiency and QCNF
QHorn and Satisfiability
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Quantified Boolean formulas with free variables.
1
BF: Boolean Functions
2
BC: Boolean Circuits
3
PL: Propositional logic
4
QCNF: QBF with free variables and kernel in CNF
5
QHornb : QBF with CNF kernel where the bound part of a
clause is a Horn clause
6
∃Hornb : QHornb with existential prefix
7
∃2-Hornb
8
∃ ps-graph+
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Fixed Deficiency and QCNF
QHorn and Satisfiability
Expressive Power
1
Every quantified Boolean formula is equivalent to a
propositional formula.
2
There is no polynomial p
such that for every n and every Boolean function
f : {0, 1}n → {0, 1} there exists a QBF
Φ(x1 , . . . , xn ) = f (x1 , . . . , xn ) and |Φ| ≤ p(n)
(simple counting argument)
|{Φ ∈ QBF : |Φ| ≤ k}| ≤ k k
⇒ (k = p(n) polynomial)
|{Φ ∈ QBF : |Φ| ≤ p(n)}| ≤ (p(n)p(n)
BF (n) := |{f : {0, 1}n → {0, 1}| = 22
n
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Fixed Deficiency and QCNF
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Definition
Let A and B be clasess of formulas. A =p B iff there exists a
polynomial q, such for every α ∈ A there is an equivalent formula
β ∈ B with |β| ≤ q(|α|) and vice versa.
A Ap B iff ∃ polynomial q ∀β ∈ B ∃α ∈ A : α ≈ β, |α| ≤ q(|β|).
And ∀ polynomials q ∃α ∈ A ∀β ∈ B: If α ≈ β then |β| > q(|α|).
1
BF (n) Ap QBF wp PL
2
∃CNF wp QHornb =p ∃ Hornb =p BC wp PL
3
∃2−Hornb wp ∃ps−graph =p PL
Note: Independent of the running time computing an equivalent
formula!
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Fixed Deficiency and QCNF
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Existentially quantified CNF with free variables
Φ = ∃x1 . . . ∃xn : φ over free variables Y = y1 , . . . , ym
Definition
F = (f1 , . . . , fn ) (Boolean functions represented as propositional
formulas, fi (y1 , . . . , ym )) is an equivalence model for Φ
iff
Φ ≈ φ[x1 /f1 (Y ), . . . , xn /fn (Y )]
Example: Φ = ∃x : (y1 ∨ x ) ∧ (¬x ∨ y2 ) ≈ (y1 ∨ y2 )
fx (y1 , y2 ) = ¬y1 then
Φ ≈ φ[x /f (y1 , y2 )] ≈ (y1 ∨ ¬y1 ) ∧ (y1 ∨ y2 )
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Fixed Deficiency and QCNF
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Problem: (Φ = ∃x1 . . . ∃xn : φ ∈ ∃CNF, α propositional formulas)
Size of models versus size of equivalent propositional
formulas
Observation: Let F be a model for Φ. Then there is a
propositional formula α: α ≈ Φ and |α| ≤ |F | · |Φ|.
Open problem: Does there exist a polynomial p, such that for
every Φ ∈ ∃ CNF:
if α ≈ Φ then there exists a model F for Φ with |F | ≤ p(|α|)?
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Fixed Deficiency and QCNF
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Problem: Lower and upper bounds for the size of models.
Reduction to formulas (∃MU+ )
Φ = ∃X :
V
1≤i≤n (ϕi
∨ yi )
ϕi clause over variables X , yi free variables
∀i∃α ∈ MU : α ⊆ {ϕ1 , . . . , ϕn } and ϕi ∈ α
( every clause ϕi belongs to a minimal unsatisfiable sub-formula)
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Fixed Deficiency and QCNF
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Examples:
Φ1 = ∃x : (x ∨ y1 ) ∧ (¬x ∨ y2 ) ≈ (y1 ∨ y2 )
MU-subset: {x , ¬x }
Φ2 = ∃a∃b : (a ∨ y1 ) ∧ (¬a ∨ b ∨ y2 ) ∧ (¬a ∨ y3 ) ∧ (¬b ∨ y4 )
MU-subsets: {a, ¬a}, {a, (¬a ∨ b), ¬b}
Φ2 ≈ (y1 ∨ y3 ) ∧ (y1 ∨ y2 ∨ y4 )
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Fixed Deficiency and QCNF
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Notation
Φ ∈ ∃MU+ , Φ = ∃X :
V
1≤i≤n (ϕi
∨ yi ), and ϕ = {ϕ1 , . . . , ϕn }
S(ϕ) := {α ⊆ ϕ : α ∈ MU}
For α ∈ S(ϕ) : Y (α) := {yi : ϕi ∈ α}
Observation:
V
V
Φ = ∃X : 1≤i≤n (ϕi ∨ yi ) ≈ α∈S(ϕ) Y (α)
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Fixed Deficiency and QCNF
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Single MU
Φ = ∃X : 1≤i≤m (ϕi ∨ yi ) ≈ (y1 ∨ . . . ∨ ym ) and ϕ ∈ MU
(S(ϕ) = {ϕ})
Construct a model F = (fx1 , . . . , fxn ) as follows:
Since ϕ is minimal unsatisfiable, for every clause ϕj there is a truth
assignment vj satisfying ϕ \ ϕj .
For j they might be several satisfying truth assignments vj . We
choose an arbitrary, but fixed vj .
We define for every variable xi (1 ≤ i ≤ n) a Boolean function
fxi (y1 , . . . , ym ) represented as propositional DNF-formula as
follows:
W
fxi (y1 , . . . , ym ) := 1≤j≤m,vj (xi )=1 (¬y1 ∧ . . . ∧ ¬yj−1 ∧ yj )
V
Then F = (fx1 , . . . , fxn ) is a model for Φ. (|F | ≤ m3 )
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{Upper bound}
Φ = ∃X :
V
1≤i≤m (ϕi
∨ yi ) ∈ ∃MU+ , ϕ :=
V
1≤i≤m
ϕi
Theorem
(k minimal unsatisfiable sub-formulas)
If ϕ contains at most k MU-subformulas,
1. then Φ has a model of size ≤ kmk+2 .
2. then there is a propositionel formula α ≈ Φ with |α| ≤ km
Open Problem: Gap between length of models and equivalent
formulas?
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Fixed Deficiency and QCNF
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{Upper bound}
Φ = ∃X :
V
1≤i≤m (ϕi
∨ yi ) ∈ ∃MU+ , ϕ :=
V
1≤i≤m
ϕi
Theorem
(k minimal unsatisfiable sub-formulas)
If ϕ contains at most k MU-subformulas,
1. then Φ has a model of size ≤ kmk+2 .
2. then there is a propositionel formula α ≈ Φ with |α| ≤ km
Open Problem: Gap between length of models and length of
equivalent formulas?
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Fixed Deficiency and QCNF
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Lower Bounds: Single MU with deficiency 1:
V
V
Φ = ∃X : 1≤i≤m (ϕi ∨ yi ) ∈ ∃MU+ , ϕ := 1≤i≤m ϕi in MU(1)
Theorem
1
2
3
upper bound m3
if ϕ is marginal then a lower bound is
(few satisfying truth assignments)
(m−1)2
4
+
m−1
2
if ϕ is in MAX-MU then a lower bound is m2 · log2 (m)
(max. number of satisfying truth assignments)
1
(improve upper bound)
2
lower bounds for single MU with deficiency k
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Fixed Deficiency and QCNF
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Borderline: minimal unsatisfiable Horn formulas have deficiency 1.
∃Hornb =p BC.
Model size for ∃(2-Horn ∩ MU)+
Case 1: ∃ps-graph+ =P propositional formulas
Figure: ps-graph
∃x1 ∃x2 ∃x3 : x1 , (¬x1 ∨ x2 ∨ a), (¬x2 ∨ x3 ∨ b), (¬x2 ∨ x3 ∨ ¬c), (¬x3 )
≈ (a ∨ (b ∧ ¬c))
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Fixed Deficiency and QCNF
QHorn and Satisfiability
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Borderline: minimal unsatisfiable Horn formulas have deficiency 1.
∃Hornb =p BC.
Model size for ∃(2-Horn ∩ MU)+
Case 1: ∃ps-graph+ =P propositional formulas
Theorem
1. ∃ps-graph+ =p PL
2 Formulas in ∃ps-graph+ have poly-size models.
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Fixed Deficiency and QCNF
QHorn and Satisfiability
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Borderline: minimal unsatisfiable Horn formulas have deficiency 1.
∃Hornb =p BC.
Model size for ∃(2-Horn ∩ MU)+
Case 2: ∃DAG(1)+ wp propositional formluas (Ap or =p open)
Figure: DAG(1)
∃x1 ∃x2 ∃x3 ∃x4 : x1 , labeled edges, ¬x4
≈ (a ∨ c ∨ d), (e ∨ d), (a ∨ b)
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Fixed Deficiency and QCNF
QHorn and Satisfiability
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Borderline: minimal unsatisfiable Horn formulas have deficiency 1.
∃Hornb =p BC.
Model size for ∃(2-Horn ∩ MU)+
Case 2: ∃DAG(1)+ wp propositional formluas (Ap or =p open)
Theorem
Formulas in ∃DAG(1)+ have poly-size models iff
the formulas have poly-size equivalent propositional formulas iff
∃DAG(1)+ =p PL
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Fixed Deficiency and QCNF
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Summary
Fixed Deficiency for QCNF
QHorn and Satisfiabilty
Expressive Power
1
2
3
Equivalence Models and Propositional Formulas
QHorn and Q2-CNF
Model size and Deficiency
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Fixed Deficiency and QCNF
QHorn and Satisfiability
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Thank You for Your Attention!
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