Model Theory and Complexity of
Quantified Boolean Formulas
Xishun Zhao
Institute of Logic and Cognition, Sun Yat-sen University
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Outline
Quantified Boolean Formulas.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Outline
Quantified Boolean Formulas.
Solving QBF.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Outline
Quantified Boolean Formulas.
Solving QBF.
Deficiency of QBF.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Outline
Quantified Boolean Formulas.
Solving QBF.
Deficiency of QBF.
Models of QCNF.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
1. Quantified Boolean formulas
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Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Quantified Boolean Formulas
Propositional variables: x, y, z, x1 , y1 , z2 .
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Quantified Boolean Formulas
Propositional variables: x, y, z, x1 , y1 , z2 .
Literals: Variables or their negation: x or ¬x.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Quantified Boolean Formulas
Propositional variables: x, y, z, x1 , y1 , z2 .
Literals: Variables or their negation: x or ¬x.
Clauses: Disjunctions of literals: L1 ∨ L2 ∨ · · · ∨ Lk .
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Quantified Boolean Formulas
Propositional variables: x, y, z, x1 , y1 , z2 .
Literals: Variables or their negation: x or ¬x.
Clauses: Disjunctions of literals: L1 ∨ L2 ∨ · · · ∨ Lk .
CNF formulas: Conjunctions of clauses

 (or multi-set of clauses)
x
y1 ¬y1 ¬x3 y2 ¬x1
 2



 x4 x1
x3 ¬x4 ¬x1 ¬x5  .


¬x2
x5
Here Each column represents a clause.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Quantified Boolean Formulas
Given a CNF formula ϕ. A model of ϕ is a truth assignment
t : var(ϕ) → {0, 1} such that t(ϕ) = 1.
Here var(ϕ) is the set of variables occurring in ϕ.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Quantified Boolean Formulas
Given a CNF formula ϕ. A model of ϕ is a truth assignment
t : var(ϕ) → {0, 1} such that t(ϕ) = 1.
Here var(ϕ) is the set of variables occurring in ϕ.
ϕ is called satisfiable if it has a model.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Quantified Boolean Formulas
Given a CNF formula ϕ. A model of ϕ is a truth assignment
t : var(ϕ) → {0, 1} such that t(ϕ) = 1.
Here var(ϕ) is the set of variables occurring in ϕ.
ϕ is called satisfiable if it has a model.
Examples:


¬a ¬a a
,

¬b b ¬b
Sun Yat-sen University


¬a ¬a ¬a a
¬b
¬b
b
b

.
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Quantified Boolean Formulas
Given a CNF formula ϕ. A model of ϕ is a truth assignment
t : var(ϕ) → {0, 1} such that t(ϕ) = 1.
Here var(ϕ) is the set of variables occurring in ϕ.
ϕ is called satisfiable if it has a model.
Examples:


¬a ¬a a
,

¬b b ¬b


¬a ¬a ¬a a
¬b
¬b
b
b

.
The Satisfiability Problem (SAT, for short), of determining
whether a CNF formula is satisfiable, is NP-complete [Cook,
1971].
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Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Quantified Boolean Formulas
A QCNF formula Φ has the form Φ = Q1 x1 Q2 x2 · · · Qn xn ϕ
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Quantified Boolean Formulas
A QCNF formula Φ has the form Φ = Q1 x1 Q2 x2 · · · Qn xn ϕ
Qi ∈ {∃, ∀} and,
ϕ is a CNF formula over variables x1 , x2 , · · · , xn .
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Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Quantified Boolean Formulas
A QCNF formula Φ has the form Φ = Q1 x1 Q2 x2 · · · Qn xn ϕ
Qi ∈ {∃, ∀} and,
ϕ is a CNF formula over variables x1 , x2 , · · · , xn .
Q1 x1 Q2 x2 · · · Qn xn is the prefix of Φ and ϕ is called the matrix
of Φ.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Quantified Boolean Formulas
A QCNF formula Φ has the form Φ = Q1 x1 Q2 x2 · · · Qn xn ϕ
Qi ∈ {∃, ∀} and,
ϕ is a CNF formula over variables x1 , x2 , · · · , xn .
Q1 x1 Q2 x2 · · · Qn xn is the prefix of Φ and ϕ is called the matrix
of Φ.
xi is called an existential (resp. universal) variable, if its
quantifier Qi is ∃ (resp. ∀).
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Quantified Boolean Formulas
Examples:
∀x∃y((¬x ∨ ¬y) ∧ (x ∨ y)).
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Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Quantified Boolean Formulas
Examples:
∀x∃y((¬x ∨ ¬y) ∧ (x ∨ y)).
∀x1 ∀x2 ∃y1 ((x1 ∨ y1 ) ∧ (x2 ∨ ¬y1 )).
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Quantified Boolean Formulas
Examples:
∀x∃y((¬x ∨ ¬y) ∧ (x ∨ y)).
∀x1 ∀x2 ∃y1 ((x1 ∨ y1 ) ∧ (x2 ∨ ¬y1 )).
∃x
 1 ∀y1 ∀y2 ∃x2 ∃x3 ∃x4 ∃x5

x
y1 ¬y1 ¬x3 y2 ¬x1
 2



 x4 x1
x3 ¬x4 ¬x1 ¬x5 


¬x2
x5
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Quantified Boolean Formulas
Examples:
∀x∃y((¬x ∨ ¬y) ∧ (x ∨ y)).
∀x1 ∀x2 ∃y1 ((x1 ∨ y1 ) ∧ (x2 ∨ ¬y1 )).
∃x
 1 ∀y1 ∀y2 ∃x2 ∃x3 ∃x4 ∃x5

x
y1 ¬y1 ¬x3 y2 ¬x1
 2



 x4 x1
x3 ¬x4 ¬x1 ¬x5 


¬x2
x5
The evaluation problem (QSAT, for short) of QCNF formulas is
PSPACE-complete.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Quantified Boolean Formulas
Examples:
∀x∃y((¬x ∨ ¬y) ∧ (x ∨ y)).
∀x1 ∀x2 ∃y1 ((x1 ∨ y1 ) ∧ (x2 ∨ ¬y1 )).
∃x
 1 ∀y1 ∀y2 ∃x2 ∃x3 ∃x4 ∃x5

x
y1 ¬y1 ¬x3 y2 ¬x1
 2



 x4 x1
x3 ¬x4 ¬x1 ¬x5 


¬x2
x5
The evaluation problem (QSAT, for short) of QCNF formulas is
PSPACE-complete.
For a CNF formula ϕ over y1 , · · · , yn ,
ϕ is satisfiable iff ∃y1 · · · ∃yn ϕ is true.
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Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
2. Solving QCNF
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Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Solving QCNF
As a matter of fact, the most effective solving tools for
application are SAT solvers.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Solving QCNF
As a matter of fact, the most effective solving tools for
application are SAT solvers.
For a large class of industrial-scale problems such as Planning,
Model Checking for dynamic systems, Scheduling,
Non-monotonic reasoning, and Cryptography, we can
transform these problem to CNF formulas and then apply
SAT-solvers to solve the problems.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Solving QCNF
As a matter of fact, the most effective solving tools for
application are SAT solvers.
For a large class of industrial-scale problems such as Planning,
Model Checking for dynamic systems, Scheduling,
Non-monotonic reasoning, and Cryptography, we can
transform these problem to CNF formulas and then apply
SAT-solvers to solve the problems.
Please note that the above-mentioned problems are harder
that SAT. Thus, the transformations to CNF formulas should
cost too much time, or even too much space. However, it is
easy to translate them to QCNF formulas.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Solving QCNF
As a matter of fact, the most effective solving tools for
application are SAT solvers.
For a large class of industrial-scale problems such as Planning,
Model Checking for dynamic systems, Scheduling,
Non-monotonic reasoning, and Cryptography, we can
transform these problem to CNF formulas and then apply
SAT-solvers to solve the problems.
Please note that the above-mentioned problems are harder
that SAT. Thus, the transformations to CNF formulas should
cost too much time, or even too much space. However, it is
easy to translate them to QCNF formulas.
Then we need effective QBF-slovers.
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Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Solving QCNF: Semantic Tree
The most natural approach to decide the truth of a QCNF
formula is by a direct implementation of the semantics. That is,
we can split iteratively a formula of the form QxΦ into two
shorter formulas Φ[x/1] and Φ[x/0].
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Solving QCNF: Semantic Tree
The most natural approach to decide the truth of a QCNF
formula is by a direct implementation of the semantics. That is,
we can split iteratively a formula of the form QxΦ into two
shorter formulas Φ[x/1] and Φ[x/0].
If Q is ∃, then QxΦ is true iff either Φ[x/1] or Φ[x/0] is true.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Solving QCNF: Semantic Tree
The most natural approach to decide the truth of a QCNF
formula is by a direct implementation of the semantics. That is,
we can split iteratively a formula of the form QxΦ into two
shorter formulas Φ[x/1] and Φ[x/0].
If Q is ∃, then QxΦ is true iff either Φ[x/1] or Φ[x/0] is true.
If Q is ∀, then QxΦ is true iff both Φ[x/1] and Φ[x/0] are
true..
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Solving QCNF: Semantic Tree
The most natural approach to decide the truth of a QCNF
formula is by a direct implementation of the semantics. That is,
we can split iteratively a formula of the form QxΦ into two
shorter formulas Φ[x/1] and Φ[x/0].
If Q is ∃, then QxΦ is true iff either Φ[x/1] or Φ[x/0] is true.
If Q is ∀, then QxΦ is true iff both Φ[x/1] and Φ[x/0] are
true..
Then the resulting proof structure is a binary tree, called
semantic tree.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Solving QCNF: Semantic Tree
The most natural approach to decide the truth of a QCNF
formula is by a direct implementation of the semantics. That is,
we can split iteratively a formula of the form QxΦ into two
shorter formulas Φ[x/1] and Φ[x/0].
If Q is ∃, then QxΦ is true iff either Φ[x/1] or Φ[x/0] is true.
If Q is ∀, then QxΦ is true iff both Φ[x/1] and Φ[x/0] are
true..
Then the resulting proof structure is a binary tree, called
semantic tree.
To simplify the tree size, some heuristics like Intelligent
Backtracking have been proposed.
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Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Solving QCNF: Q-Resolution
Q-resolution [H. Kleine Büning] is a generalization of resolution to
QCNF formulas.
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Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Solving QCNF: Q-Resolution
Q-resolution [H. Kleine Büning] is a generalization of resolution to
QCNF formulas. Φ = ∀x1 ∃y1 ∃y2 ∀x2 ∃y3 ((y1 ∨ x2 ∨ y3 ) ∧ (y2 ∨ ¬y3 ) ∧
(¬x2 ∨ ¬y3 ) ∧ (x1 ∨ ¬y1 ) ∧ (x1 ∨ ¬y2 )).
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Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Solving QCNF: Q-Resolution
Q-resolution [H. Kleine Büning] is a generalization of resolution to
QCNF formulas. Φ = ∀x1 ∃y1 ∃y2 ∀x2 ∃y3 ((y1 ∨ x2 ∨ y3 ) ∧ (y2 ∨ ¬y3 ) ∧
(¬x2 ∨ ¬y3 ) ∧ (x1 ∨ ¬y1 ) ∧ (x1 ∨ ¬y2 )).
y1 ∨ x 2 ∨ y 3
x1 ∨ ¬y1
x 1 ∨ x 2 ∨ y3
x 1 ∨ y2 ∨ x 2 = x 1 ∨ y2
y2 ∨ ¬y3
x1 ∨ ¬y2
x1 = t
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Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Solving QCNF: Q-Resolution
Q-resolution [H. Kleine Büning] is a generalization of resolution to
QCNF formulas. Φ = ∀x1 ∃y1 ∃y2 ∀x2 ∃y3 ((y1 ∨ x2 ∨ y3 ) ∧ (y2 ∨ ¬y3 ) ∧
(¬x2 ∨ ¬y3 ) ∧ (x1 ∨ ¬y1 ) ∧ (x1 ∨ ¬y2 )).
y1 ∨ x 2 ∨ y 3
x1 ∨ ¬y1
x 1 ∨ x 2 ∨ y3
x 1 ∨ y2 ∨ x 2 = x 1 ∨ y2
y2 ∨ ¬y3
x1 ∨ ¬y2
x1 = t
A QCNF formula Φ is false iff the empty clause can be derive from
Φ by Q-resolution refutation.
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Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Solving QCNF: Hard Examples
We can define a sequence Φ1 , Φ2 , · · · , Φn , · · · of QCNF formulas.
Let Φ1 = ∃x1 ∀y1 ∃x2 ∀y2 ∃x3 ϕ1 ,
Φ2 = ∃x1 ∀y1 ∃x2 ∀y2 ∃x3 ∀y3 ∃x4 ∀y4 ∃x5 ϕ2 with

x ¬x1 x2
 1

 y1 ¬y1 y2

x3 x3 ¬x3
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
x ¬x1 x2
  1
 y ¬y
¬x2
y2
1
1
 
 
¬y2  , 
x3 x3 ¬x3
 

 y3 y3 ¬y3
¬x3

x5 x5
x5

¬x2
¬y2
¬x3
x4
¬y3
y4
x5
¬x5




¬x4 
.

¬y4 

¬x5
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Solving QCNF: Hard Examples
We can define a sequence Φ1 , Φ2 , · · · , Φn , · · · of QCNF formulas.
Let Φ1 = ∃x1 ∀y1 ∃x2 ∀y2 ∃x3 ϕ1 ,
Φ2 = ∃x1 ∀y1 ∃x2 ∀y2 ∃x3 ∀y3 ∃x4 ∀y4 ∃x5 ϕ2 with

x ¬x1 x2
 1

 y1 ¬y1 y2

x3 x3 ¬x3

x ¬x1 x2
  1
 y ¬y
¬x2
y2
1
1
 
 
¬y2  , 
x3 x3 ¬x3
 

 y3 y3 ¬y3
¬x3

x5 x5
x5

¬x2
¬y2
¬x3
x4
¬y3
y4
x5
¬x5




¬x4 
.

¬y4 

¬x5
Φn is hard for semantic tree and Q-resolution.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Solving QCNF: Hard Examples
We can define a sequence Φ1 , Φ2 , · · · , Φn , · · · of QCNF formulas.
Let Φ1 = ∃x1 ∀y1 ∃x2 ∀y2 ∃x3 ϕ1 ,
Φ2 = ∃x1 ∀y1 ∃x2 ∀y2 ∃x3 ∀y3 ∃x4 ∀y4 ∃x5 ϕ2 with

x ¬x1 x2
 1

 y1 ¬y1 y2

x3 x3 ¬x3

x ¬x1 x2
  1
 y ¬y
¬x2
y2
1
1
 
 
¬y2  , 
x3 x3 ¬x3
 

 y3 y3 ¬y3
¬x3

x5 x5
x5

¬x2
¬y2
¬x3
x4
¬y3
y4
x5
¬x5




¬x4 
.

¬y4 

¬x5
Φn is hard for semantic tree and Q-resolution.
Are these formulas are really hard to solve?
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Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
3. Deficiency
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Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Deficiency of QCNF
There are two ways to define the deficiency of a QCNF formula
Φ = Qϕ.
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Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Deficiency of QCNF
There are two ways to define the deficiency of a QCNF formula
Φ = Qϕ.
dall (Φ) := the difference between the number of clauses of
ϕ and the number of all variables occurring in ϕ.
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Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Deficiency of QCNF
There are two ways to define the deficiency of a QCNF formula
Φ = Qϕ.
d(Φ) := the difference between the number of clauses of ϕ
and the number of existential variables occurring in ϕ.
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Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Deficiency of QCNF
There are two ways to define the deficiency of a QCNF formula
Φ = Qϕ.
d(Φ) := the difference between the number of clauses of ϕ
and the number of existential variables occurring in ϕ.
Maximal deficiency: d∗ (Φ) := max{d(Φ0 ) | Φ0 ⊆ Φ}.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Deficiency of QCNF
There are two ways to define the deficiency of a QCNF formula
Φ = Qϕ.
d(Φ) := the difference between the number of clauses of ϕ
and the number of existential variables occurring in ϕ.
Maximal deficiency: d∗ (Φ) := max{d(Φ0 ) | Φ0 ⊆ Φ}.
Stable Formulas: Φ is stable if ∀Φ0 ⊆ Φ(d(Φ0 ) < d(Φ)).
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Deficiency of QCNF
There are two ways to define the deficiency of a QCNF formula
Φ = Qϕ.
d(Φ) := the difference between the number of clauses of ϕ
and the number of existential variables occurring in ϕ.
Maximal deficiency: d∗ (Φ) := max{d(Φ0 ) | Φ0 ⊆ Φ}.
Stable Formulas: Φ is stable if ∀Φ0 ⊆ Φ(d(Φ0 ) < d(Φ)).
For every Φ, there is a Φ0 ⊆ Φ such that
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Deficiency of QCNF
There are two ways to define the deficiency of a QCNF formula
Φ = Qϕ.
d(Φ) := the difference between the number of clauses of ϕ
and the number of existential variables occurring in ϕ.
Maximal deficiency: d∗ (Φ) := max{d(Φ0 ) | Φ0 ⊆ Φ}.
Stable Formulas: Φ is stable if ∀Φ0 ⊆ Φ(d(Φ0 ) < d(Φ)).
For every Φ, there is a Φ0 ⊆ Φ such that
Φ0 is stable, and d(Φ0 ) = d∗ (Φ),
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Deficiency of QCNF
There are two ways to define the deficiency of a QCNF formula
Φ = Qϕ.
d(Φ) := the difference between the number of clauses of ϕ
and the number of existential variables occurring in ϕ.
Maximal deficiency: d∗ (Φ) := max{d(Φ0 ) | Φ0 ⊆ Φ}.
Stable Formulas: Φ is stable if ∀Φ0 ⊆ Φ(d(Φ0 ) < d(Φ)).
For every Φ, there is a Φ0 ⊆ Φ such that
Φ0 is stable, and d(Φ0 ) = d∗ (Φ),
Φ has the same truth as Φ, and
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Deficiency of QCNF
There are two ways to define the deficiency of a QCNF formula
Φ = Qϕ.
d(Φ) := the difference between the number of clauses of ϕ
and the number of existential variables occurring in ϕ.
Maximal deficiency: d∗ (Φ) := max{d(Φ0 ) | Φ0 ⊆ Φ}.
Stable Formulas: Φ is stable if ∀Φ0 ⊆ Φ(d(Φ0 ) < d(Φ)).
For every Φ, there is a Φ0 ⊆ Φ such that
Φ0 is stable, and d(Φ0 ) = d∗ (Φ),
Φ has the same truth as Φ, and
Φ0 can be computed in polynomial-time.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Deficiency of QCNF
There are two ways to define the deficiency of a QCNF formula
Φ = Qϕ.
d(Φ) := the difference between the number of clauses of ϕ
and the number of existential variables occurring in ϕ.
Maximal deficiency: d∗ (Φ) := max{d(Φ0 ) | Φ0 ⊆ Φ}.
Stable Formulas: Φ is stable if ∀Φ0 ⊆ Φ(d(Φ0 ) < d(Φ)).
For every Φ, there is a Φ0 ⊆ Φ such that
Φ0 is stable, and d(Φ0 ) = d∗ (Φ),
Φ has the same truth as Φ, and
Φ0 can be computed in polynomial-time.
QCNF(k) := {Φ | Φ is stable and d(Φ) = k}.
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Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
QCNF(1): A Lemma
Lemma [H. Kleine Büning, Xishun Zhao, 2006] Let
Φ = ∃X1 ∀Y1 · · · ∃Xm ∀Ym ∃Xm+1 ϕ be a formula in QCNF(1).
Then,
Φ is false
⇔
for all variables y ∈ Yi , 1 ≤ i ≤ m, the formula
∃X1 · · · ∃Xi ∀y∃Xi+1 · · · ∃Xm+1 ϕred is false.
Here, ϕred is the result of removing all universal literals from ϕ
except y and ¬y.
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Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
QCNF(1): A Lemma
Lemma [H. Kleine Büning, Xishun Zhao, 2006] Let
Φ = ∃X1 ∀Y1 · · · ∃Xm ∀Ym ∃Xm+1 ϕ be a formula in QCNF(1).
Then,
Φ is false
⇔
for all variables y ∈ Yi , 1 ≤ i ≤ m, the formula
∃X1 · · · ∃Xi ∀y∃Xi+1 · · · ∃Xm+1 ϕred is false.
Here, ϕred is the result of removing all universal literals from ϕ
except y and ¬y.
If QSAT is poly-time solvable for QCNF(1) formulas with form
∃X∀y∃Zϕ, then QSAT for QCNF(1) is polynomial-time
solvable.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
QCNF(1): Connectivity and Falsity
Let Φ := ∃X∀y∃Zϕ. f, g ∈ ϕ.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
QCNF(1): Connectivity and Falsity
Let Φ := ∃X∀y∃Zϕ. f, g ∈ ϕ.
We say f and g are directly connected without X if f = g or
there is a existential variable z 6∈ X such that (z ∈ f and
¬z ∈ g) or (¬z ∈ f and z ∈ g)
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
QCNF(1): Connectivity and Falsity
Let Φ := ∃X∀y∃Zϕ. f, g ∈ ϕ.
We say f and g are directly connected without X if f = g or
there is a existential variable z 6∈ X such that (z ∈ f and
¬z ∈ g) or (¬z ∈ f and z ∈ g)
We say f and g are connected without X if there are in ϕ
clauses f = f1 , f2 , · · · , fn = g such that fi and fi+1 are
directly connected without X.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
QCNF(1): Connectivity and Falsity
Let Φ := ∃X∀y∃Zϕ. f, g ∈ ϕ.
We say f and g are directly connected without X if f = g or
there is a existential variable z 6∈ X such that (z ∈ f and
¬z ∈ g) or (¬z ∈ f and z ∈ g)
We say f and g are connected without X if there are in ϕ
clauses f = f1 , f2 , · · · , fn = g such that fi and fi+1 are
directly connected without X.
Let Φ = ∃X∀y∃Zϕ be a formula in QCNF(1) with ϕ|∃
unsatisfiable. Then,
Φ is false ⇔ ∀f, g ∈ ϕ :
(y ∈ f, ¬y ∈ g) ⇒ f and g are not connected without X in ϕ.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
QCNF(1): Connectivity and Falsity
Let Φ := ∃X∀y∃Zϕ. f, g ∈ ϕ.
We say f and g are directly connected without X if f = g or
there is a existential variable z 6∈ X such that (z ∈ f and
¬z ∈ g) or (¬z ∈ f and z ∈ g)
We say f and g are connected without X if there are in ϕ
clauses f = f1 , f2 , · · · , fn = g such that fi and fi+1 are
directly connected without X.
Let Φ = ∃X∀y∃Zϕ be a formula in QCNF(1) with ϕ|∃
unsatisfiable. Then,
Φ is false ⇔ ∀f, g ∈ ϕ :
(y ∈ f, ¬y ∈ g) ⇒ f and g are not connected without X in ϕ.
QSAT for QCNF(1) is poly-time solvable [H. KB, X. Zhao].
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
QCNF(1): Examples
Example 1. Let Φ := ∃x1 ∀y∃x2 ∃x3 ∃x4 ∃x5 ϕ, where ϕ is the
following formula


·




 x
2
ϕ=


x4




Sun Yat-sen University
·
·
·
y
¬y ¬x3
x1
x3
¬x2
·
y
¬x4 ¬x1
x5

· 




¬x1 

¬x5 




Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
QCNF(1): Examples
Example 1. Let Φ := ∃x1 ∀y∃x2 ∃x3 ∃x4 ∃x5 ϕ, where ϕ is the
following formula


·




 x
2
ϕ=


x4




Sun Yat-sen University
·
·
·
y
¬y ¬x3
x1
x3
¬x2
·
y
¬x4 ¬x1
x5

· 




¬x1 

¬x5 




Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
QCNF(1): Examples
Example 1. Let Φ := ∃x1 ∀y∃x2 ∃x3 ∃x4 ∃x5 ϕ, where ϕ is the
following formula


·




 x
2
ϕ=


x4




·
·
·
y
¬y ¬x3
x1
x3
¬x2
·
y
¬x4 ¬x1
x5

· 




¬x1 

¬x5 




Thus, Φ is true.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
QCNF(1): Examples
Example 2. Let Φ := ∃x1 ∃x3 ∀y∃x2 ∃x4 ϕ, where ϕ is the
following formula.



·
·
·
·
· 








 x
¬x1 ¬x3 x3 x4 
1
ϕ=




¬x2
y
x2
¬y








¬x2
¬x4
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
QCNF(1): Examples
Example 2. Let Φ := ∃x1 ∃x3 ∀y∃x2 ∃x4 ϕ, where ϕ is the
following formula.



·
·
·
·
· 








 x
¬x1 ¬x3 x3 x4 
1
ϕ=




¬x2
y
x2
¬y








¬x2
¬x4
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
QCNF(1): Examples
Example 2. Let Φ := ∃x1 ∃x3 ∀y∃x2 ∃x4 ϕ, where ϕ is the
following formula.



·
·
·
·
· 








 x
¬x1 ¬x3 x3 x4 
1
ϕ=




¬x2
y
x2
¬y








¬x2
¬x4
Thus, Φ is false.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
QCNF(1): Examples
Example 2. Let Φ := ∃x1 ∃x3 ∀y∃x2 ∃x4 ϕ, where ϕ is the
following formula.



·
·
·
·
· 








 x
¬x1 ¬x3 x3 x4 
1
ϕ=




¬x2
y
x2
¬y








¬x2
¬x4
Thus, Φ is false.
Qustion: Is QSAT poly-time solvable for QCNF(k) with k > 1
fixed?
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Structure of CNF(1)
Let ϕ be the following unsatisfiable formula in CNF(1):







ϕ := 





Sun Yat-sen University

x1 ¬x1
x2
¬x2
x3
x
¬x3
¬x
¬x
x4
¬x4
x5
x5
¬x5












Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Structure of CNF(1)
Let ϕ be the following unsatisfiable formula in CNF(1):







ϕ := 





Sun Yat-sen University

x1 ¬x1
x2
¬x2
x3
x
¬x3
¬x
¬x
x4
¬x4
x5
x5
¬x5












Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Structure of CNF(1)
Let ϕ be the following unsatisfiable formula in CNF(1):







ϕ := 





Sun Yat-sen University

x1 ¬x1
x2
¬x2
x3
x
¬x3
¬x
¬x
x4
¬x4
x5
x5
¬x5












Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Structure of CNF(1)
Let ϕ be the following unsatisfiable formula in CNF(1):







ϕ := 






x1 ¬x1
x2
¬x2
x3
x
¬x3
¬x
¬x
x4
¬x4
x5
x5
¬x5












Qustion: Is QSAT poly-time solvable for QCNF(k) with k > 1
fixed?
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
4. Models
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Models of QCNF Formulas
Definition [H. Kleine Büning, Xishun Zhao].
Let Φ = ∀x1 ∃y1 ∀x2 ∃y2 · · · ∀xk ∃yk ϕ be a QCNF formula. For a
sequence of Boolean functions M = (f1 , · · · , fk ) with
fi : {x1 , · · · , xi } → {0, 1}, if ϕ[y1 /f1 (x1 ), · · · , yk /fk (x1 , · · · , xk )]
is tautological, then we say M is a model of Φ.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Models of QCNF Formulas
Definition [H. Kleine Büning, Xishun Zhao].
Let Φ = ∀x1 ∃y1 ∀x2 ∃y2 · · · ∀xk ∃yk ϕ be a QCNF formula. For a
sequence of Boolean functions M = (f1 , · · · , fk ) with
fi : {x1 , · · · , xi } → {0, 1}, if ϕ[y1 /f1 (x1 ), · · · , yk /fk (x1 , · · · , xk )]
is tautological, then we say M is a model of Φ.
If the functions f1 , · · · , fk are in a class K then M is called a
K-model for Φ.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Models of QCNF Formulas
Definition [H. Kleine Büning, Xishun Zhao].
Let Φ = ∀x1 ∃y1 ∀x2 ∃y2 · · · ∀xk ∃yk ϕ be a QCNF formula. For a
sequence of Boolean functions M = (f1 , · · · , fk ) with
fi : {x1 , · · · , xi } → {0, 1}, if ϕ[y1 /f1 (x1 ), · · · , yk /fk (x1 , · · · , xk )]
is tautological, then we say M is a model of Φ.
If the functions f1 , · · · , fk are in a class K then M is called a
K-model for Φ.
Example: The formula Φ = ∀x∃y(x ∨ y) ∧ (¬x ∨ ¬y) is true and
for fy (x) = ¬x, M = (fy ) is a model, because
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Models of QCNF Formulas
Definition [H. Kleine Büning, Xishun Zhao].
Let Φ = ∀x1 ∃y1 ∀x2 ∃y2 · · · ∀xk ∃yk ϕ be a QCNF formula. For a
sequence of Boolean functions M = (f1 , · · · , fk ) with
fi : {x1 , · · · , xi } → {0, 1}, if ϕ[y1 /f1 (x1 ), · · · , yk /fk (x1 , · · · , xk )]
is tautological, then we say M is a model of Φ.
If the functions f1 , · · · , fk are in a class K then M is called a
K-model for Φ.
Example: The formula Φ = ∀x∃y(x ∨ y) ∧ (¬x ∨ ¬y) is true and
for fy (x) = ¬x, M = (fy ) is a model, because
(x ∨ y) ∧ (¬x ∨ ¬y)[y/fy (x)] = (x ∨ ¬x) ∧ (¬x ∨ x) is tautological.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Models of QCNF Formulas
Let B a subclass of QBF, K a class of Boolean functions.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Models of QCNF Formulas
Let B a subclass of QBF, K a class of Boolean functions.
K-Model Checking for B: Determining whether a sequence
of Boolean functions in K is a K-model of a formula Φ ∈ B.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Models of QCNF Formulas
Let B a subclass of QBF, K a class of Boolean functions.
K-Model Checking for B: Determining whether a sequence
of Boolean functions in K is a K-model of a formula Φ ∈ B.
K-Model Existence for B: Determining whether a formula
Φ ∈ B has a K-model.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Models of QCNF Formulas
Let B a subclass of QBF, K a class of Boolean functions.
K-Model Checking for B: Determining whether a sequence
of Boolean functions in K is a K-model of a formula Φ ∈ B.
K-Model Existence for B: Determining whether a formula
Φ ∈ B has a K-model.
Lemma [H. KB, X. Zhao, 2003]
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Models of QCNF Formulas
Let B a subclass of QBF, K a class of Boolean functions.
K-Model Checking for B: Determining whether a sequence
of Boolean functions in K is a K-model of a formula Φ ∈ B.
K-Model Existence for B: Determining whether a formula
Φ ∈ B has a K-model.
Lemma [H. KB, X. Zhao, 2003]
CNF-model checking for QCNF is co-NP-compltete.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Models of QCNF Formulas
Let B a subclass of QBF, K a class of Boolean functions.
K-Model Checking for B: Determining whether a sequence
of Boolean functions in K is a K-model of a formula Φ ∈ B.
K-Model Existence for B: Determining whether a formula
Φ ∈ B has a K-model.
Lemma [H. KB, X. Zhao, 2003]
CNF-model checking for QCNF is co-NP-compltete.
CNF-model existence for QCNF is PSPACE-complete.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Size of Models
Let t : IN → IN be a function and B a subclass of QBF.
Suppose, every true formula Φ in B has a propositional model
M = (f1 , · · · , fm ), such that the size of propositional formula f i
is smaller than or equal to t(|Φ|). Then we say B has t(n)-size
models. If t is a polynomial, then we say B has polynomial size
models.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Size of Models
Let t : IN → IN be a function and B a subclass of QBF.
Suppose, every true formula Φ in B has a propositional model
M = (f1 , · · · , fm ), such that the size of propositional formula f i
is smaller than or equal to t(|Φ|). Then we say B has t(n)-size
models. If t is a polynomial, then we say B has polynomial size
models.
Lemma [H. Kleine Büning, Xishun Zhao, 2003]
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Size of Models
Let t : IN → IN be a function and B a subclass of QBF.
Suppose, every true formula Φ in B has a propositional model
M = (f1 , · · · , fm ), such that the size of propositional formula f i
is smaller than or equal to t(|Φ|). Then we say B has t(n)-size
models. If t is a polynomial, then we say B has polynomial size
models.
Lemma [H. Kleine Büning, Xishun Zhao, 2003]
QHORN has linear size CNF-models.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Size of Models
Let t : IN → IN be a function and B a subclass of QBF.
Suppose, every true formula Φ in B has a propositional model
M = (f1 , · · · , fm ), such that the size of propositional formula f i
is smaller than or equal to t(|Φ|). Then we say B has t(n)-size
models. If t is a polynomial, then we say B has polynomial size
models.
Lemma [H. Kleine Büning, Xishun Zhao, 2003]
QHORN has linear size CNF-models.
Here, a QCNF formula is in QHORN if every clause
contains at most one positive literal.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Size of Models
Let t : IN → IN be a function and B a subclass of QBF.
Suppose, every true formula Φ in B has a propositional model
M = (f1 , · · · , fm ), such that the size of propositional formula f i
is smaller than or equal to t(|Φ|). Then we say B has t(n)-size
models. If t is a polynomial, then we say B has polynomial size
models.
Lemma [H. Kleine Büning, Xishun Zhao, 2003]
QHORN has linear size CNF-models.
Q2CNF has 1-size CNF-models.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Size of Models
Let t : IN → IN be a function and B a subclass of QBF.
Suppose, every true formula Φ in B has a propositional model
M = (f1 , · · · , fm ), such that the size of propositional formula f i
is smaller than or equal to t(|Φ|). Then we say B has t(n)-size
models. If t is a polynomial, then we say B has polynomial size
models.
Lemma [H. Kleine Büning, Xishun Zhao, 2003]
QHORN has linear size CNF-models.
Q2CNF has 1-size CNF-models.
Here, a QCNF formula is in Q2CNF if every clause contains
at most two literal.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Size of Models
Let t : IN → IN be a function and B a subclass of QBF.
Suppose, every true formula Φ in B has a propositional model
M = (f1 , · · · , fm ), such that the size of propositional formula f i
is smaller than or equal to t(|Φ|). Then we say B has t(n)-size
models. If t is a polynomial, then we say B has polynomial size
models.
Lemma [H. Kleine Büning, Xishun Zhao, 2003]
QHORN has linear size CNF-models.
Q2CNF has 1-size CNF-models.
For any fixed k > 2, if ΣPk 6= ΣP2 , then QCNF can not have
polynomial size CNF-models.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Models Size of QCNF(k)
For any true QCNF(1) formula Φ = Qϕ, there is a
universal variable x such that Φ has a model in which
each function is either a constant 0 or 1, or can be
represented as x or ¬x.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Models Size of QCNF(k)
For any true QCNF(1) formula Φ = Qϕ, there is a
universal variable x such that Φ has a model in which
each function is either a constant 0 or 1, or can be
represented as x or ¬x.
For any k ≥ 1 and any true QCNF(k ) formula Φ = Qϕ,
there is a set U with at most 24k/3 universal variables
such that Φ has a CNF-model M = (f1 , · · · , fm ) such
that each fi has at most 2k clauses and var(fi ) ⊆ U .
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Complexity of QCNF(k)
For any fixed k , the evaluation problem for QCNF(k ) is
in NP.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Complexity of QCNF(k)
For any fixed k , the evaluation problem for QCNF(k ) is
in NP.
Input ∀x1 ∃y1 ∀x2 ∃ · · · ∀xn ∃yn ϕ ∈QCNF(k).
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Complexity of QCNF(k)
For any fixed k , the evaluation problem for QCNF(k ) is
in NP.
Input ∀x1 ∃y1 ∀x2 ∃ · · · ∀xn ∃yn ϕ ∈QCNF(k).
1. Guess a subset U ⊆ {x1 , · · · , xn } with |U | ≤ k , and
guess a sequence (f1 , · · · , fn ) such that fi ∈CNF
over variables U ∩ {x1 , · · · , xi }.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Complexity of QCNF(k)
For any fixed k , the evaluation problem for QCNF(k ) is
in NP.
Input ∀x1 ∃y1 ∀x2 ∃ · · · ∀xn ∃yn ϕ ∈QCNF(k).
1. Guess a subset U ⊆ {x1 , · · · , xn } with |U | ≤ k , and
guess a sequence (f1 , · · · , fn ) such that fi ∈CNF
over variables U ∩ {x1 , · · · , xi }.
2. Delete from ϕ all occurrences of universal variables
not in U we get ϕU .
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Complexity of QCNF(k)
For any fixed k , the evaluation problem for QCNF(k ) is
in NP.
Input ∀x1 ∃y1 ∀x2 ∃ · · · ∀xn ∃yn ϕ ∈QCNF(k).
1. Guess a subset U ⊆ {x1 , · · · , xn } with |U | ≤ k , and
guess a sequence (f1 , · · · , fn ) such that fi ∈CNF
over variables U ∩ {x1 , · · · , xi }.
2. Delete from ϕ all occurrences of universal variables
not in U we get ϕU .
3. Replace yi by fi we get ϕ0 := ϕU [y1 /f1 , · · · , yn /fn ].
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Complexity of QCNF(k)
For any fixed k , the evaluation problem for QCNF(k ) is
in NP.
Input ∀x1 ∃y1 ∀x2 ∃ · · · ∀xn ∃yn ϕ ∈QCNF(k).
1. Guess a subset U ⊆ {x1 , · · · , xn } with |U | ≤ k , and
guess a sequence (f1 , · · · , fn ) such that fi ∈CNF
over variables U ∩ {x1 , · · · , xi }.
2. Delete from ϕ all occurrences of universal variables
not in U we get ϕU .
3. Replace yi by fi we get ϕ0 := ϕU [y1 /f1 , · · · , yn /fn ].
4. If ϕ0 is tautological then return YES.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Complexity of QCNF(k)
For any fixed k , the evaluation problem for QCNF(k ) is
in NP.
Input ∀x1 ∃y1 ∀x2 ∃ · · · ∀xn ∃yn ϕ ∈QCNF(k).
1. Guess a subset U ⊆ {x1 , · · · , xn } with |U | ≤ k , and
guess a sequence (f1 , · · · , fn ) such that fi ∈CNF
over variables U ∩ {x1 , · · · , xi }.
2. Delete from ϕ all occurrences of universal variables
not in U we get ϕU .
3. Replace yi by fi we get ϕ0 := ϕU [y1 /f1 , · · · , yn /fn ].
4. If ϕ0 is tautological then return YES.
Open Question: Is QSAT for QCNF(k) poly-time
solvable for any fixed k > 1?
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Some Subclasses
QEHORN:=the class of QCNF formulas in which each
clause contains at most one positive existential literal.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Some Subclasses
QEHORN:=the class of QCNF formulas in which each
clause contains at most one positive existential literal.
QE2CNF:=the class of QCNF formulas in which each
clause contains at most two existential literals.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Some Subclasses
QEHORN:=the class of QCNF formulas in which each
clause contains at most one positive existential literal.
QE2CNF:=the class of QCNF formulas in which each
clause contains at most two existential literals.
QSAT for QEHORN and QE2CNF remains
PSPACE-complete [H. Kleine Büning, et al].
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Some Subclasses
QEHORN:=the class of QCNF formulas in which each
clause contains at most one positive existential literal.
QE2CNF:=the class of QCNF formulas in which each
clause contains at most two existential literals.
QSAT for QEHORN and QE2CNF remains
PSPACE-complete [H. Kleine Büning, et al].
For any fixed k , is QSAT for QEHORN(k) and
QE2CNF(k) poly-time solvable?
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Unfolding
Q∀x∃y1 · · · ∃ym ϕ =⇒
0 (ϕ[x/0] ∧ ϕ[x/1][y /y 0 , · · · , y /y 0 ])
Q∃y1 · · · ∃ym ∃y10 · · · ∃ym
1 1
m m
for new variables y10 , · · · , yk0 .
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Unfolding
Q∀x∃y1 · · · ∃ym ϕ =⇒
0 (ϕ[x/0] ∧ ϕ[x/1][y /y 0 , · · · , y /y 0 ])
Q∃y1 · · · ∃ym ∃y10 · · · ∃ym
1 1
m m
for new variables y10 , · · · , yk0 .
Unfolding preserves the Truth.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Unfolding
Q∀x∃y1 · · · ∃ym ϕ =⇒
0 (ϕ[x/0] ∧ ϕ[x/1][y /y 0 , · · · , y /y 0 ])
Q∃y1 · · · ∃ym ∃y10 · · · ∃ym
1 1
m m
for new variables y10 , · · · , yk0 .
Unfolding preserves the Truth.
QEHORN formulas =⇒ ∃HORN formulas by iterative
unfolding.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Unfolding
Q∀x∃y1 · · · ∃ym ϕ =⇒
0 (ϕ[x/0] ∧ ϕ[x/1][y /y 0 , · · · , y /y 0 ])
Q∃y1 · · · ∃ym ∃y10 · · · ∃ym
1 1
m m
for new variables y10 , · · · , yk0 .
Unfolding preserves the Truth.
QEHORN formulas =⇒ ∃HORN formulas by iterative
unfolding.
QE2CNF formulas =⇒ ∃2CNF formulas by iterative
unfolding.
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Unfolding
Q∀x∃y1 · · · ∃ym ϕ =⇒
0 (ϕ[x/0] ∧ ϕ[x/1][y /y 0 , · · · , y /y 0 ])
Q∃y1 · · · ∃ym ∃y10 · · · ∃ym
1 1
m m
for new variables y10 , · · · , yk0 .
Unfolding preserves the Truth.
QEHORN formulas =⇒ ∃HORN formulas by iterative
unfolding.
QE2CNF formulas =⇒ ∃2CNF formulas by iterative
unfolding.
For any fixed k , QSAT for QEHORN(k) and QE2CNF(k)
is poly-time solvable [H. KB, Xishun Zhao].
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p
Thank You!
Sun Yat-sen University
Model Theory and Complexity of Quantified Boolean FormulasJuly 20, 2007 – p