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Propositional Proof Systems SS 2010
Quantified Boolean Formulas
Uwe Bubeck
University of Paderborn
11.6.2010
Outline
• Introduction
• Syntax and Semantics
• Normal Forms
Literature
H. Kleine Büning, T. Lettmann,
Propositional Logic: Deduction and Algorithms,
Cambridge University Press 1999
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Quantified Boolean Formulas
Introduction
Informal Introduction 1/2
Quantified Boolean Formulas:
propositional formulas
with additional
quantifiers over atoms
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Informal Introduction 2/2
Informal QBF Semantics:
∀x φ(x) is true if and only if
φ[x/0] is true and φ[x/1] is true
∃x φ(x) is true if and only if
φ[x/0] is true or φ[x/1] is true
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Verification and QBF 1/3
Interesting application example:
Bounded Model Checking
Idea: state transition graph of a system is checked for
reachable bad states.
s4 r
Are all reachable states with r
always followed by states with q
(r → X q) ?
s2 q
s0
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s3 p,r
p
Quantified Boolean Formulas
s1
q,r
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Verification and QBF 1/3
Interesting application example:
Bounded Model Checking
Idea: state transition graph of a system is checked for
reachable bad states.
s4 r
Are all reachable states with r
always followed by states with q
(r → X q) ?
No, there is a path to the bad state s3 !
Uwe Bubeck
s3 p,r
s2 q
s0
p
Quantified Boolean Formulas
s1
q,r
7
Verification and QBF 2/3
Problem: state space explosion
Idea: symbolic encoding for paths of fixed length k
z0
z1
z2
z3
Remaining Problem: many copies of δ
Improvement: logic which allows abbreviations
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Verification and QBF 3/3
Even more compact: iterative squaring
zk
z0
z0
∃z‘
∀u‘
∃z‘
∀v‘
zk
→ prefix ∃∀∀∃∀∀...
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Quantified Boolean Formulas
Syntax and Semantics
QBF Syntax 1/2
Definition
The set of quantified Boolean formulas is inductively defined
as follows:
1.
Propositional formulas and Boolean constants are
quantified Boolean formulas.
2.
If Φ is a quantified Boolean formula and x a variable,
∃x Φ and ∀x Φ are quantified Boolean formulas.
3.
If Φ1 and Φ2 are quantified Boolean formulas,
¬Φ1, Φ1∨Φ2 and Φ1∧Φ2 are quantified Boolean formulas.
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QBF Syntax 2/2
Further Definitions and Notation
• universally quantified variables are usually named x1,...,xn.
• existentially quantified variables are usually y1,...,ym.
• variables which are not in the scope of a quantifier are
called free variables and typically named z1,...,zr.
• a formula without free variables is said to be closed.
• QBF is the class of closed quantified Boolean formulas.
• QBF* is the class of QBF formulas with free variables.
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Evaluating QBFs 1/2
Definition
Let Φ be a QBF* formula with free variables z1,...,zr.
Then a truth assignment ℑ is a mapping ℑ:{z1,...,zr} → {0,1}
which satisfies the following conditions:
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Evaluating QBFs 2/2
Observations
• a closed formula is either true or false.
• the truth of a QBF* formula
depends on the values of the free variables.
A QBF* formula Φ is called satisfiable if and only if there
exists a truth assignment ℑ to the free variables such that
ℑ(Φ) = 1, and unsatisfiable otherwise.
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Use of free Variables
Why do we need free variables?
Given: propositional formula φ(z1,...,zr)
Wanted:
shorter QBF* Φ(z1,...,zr) with free variables z1,...,zr
such that ℑΦ(z1,...,zr) = ℑφ(z1,...,zr) („equivalence“)
Example:
(A ∨ ¬B ∨ C ∨ D) ∧ (A ∨ ¬B ∨ C ∨ ¬E) ∧ (A ∨ ¬B ∨ C ∨ F)
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Use of free Variables
Why do we need free variables?
Given: propositional formula φ(z1,...,zr)
Wanted:
shorter QBF* Φ(z1,...,zr) with free variables z1,...,zr
such that ℑΦ(z1,...,zr) = ℑφ(z1,...,zr) („equivalence“)
Example:
(A ∨ ¬B ∨ C ∨ D) ∧ (A ∨ ¬B ∨ C ∨ ¬E) ∧ (A ∨ ¬B ∨ C ∨ F)
≈ ∃y (y → (A ∨ ¬B ∨ C) ∧ (y ∨ D) ∧ (y ∨ ¬E) ∧ (y ∨ F)
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Consequence and Equivalence
Definition
A QBF* formula Φ2 is a consequence of Φ1 (Φ1 |= Φ2)
if and only if for all truth assignments ℑ we have
ℑ(Φ1) = 1 → ℑ(Φ2) = 1.
Φ1 is logically equivalent to Φ2 (Φ1 ≈ Φ2) if and only if
Φ1 |= Φ2 and Φ2 |= Φ1.
Observation
For consequence and equivalence, quantified variables are
not directly taken into account. They can be seen as local
within the respective formula.
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Quantified Boolean Formulas
Normal Forms
Prenex Formulas 1/2
The syntax definition of QBF* allows quantifiers within the
whole formula.
Example: Φ = (∀x ∃y φ(x,y)) ∨ (∀x ∃z ψ(x,z))
Problem: difficult to handle such formulas
Solution:
It is usually assumed to have prenex formulas where all
quantifiers are at the beginning:
Φ = Q1v1...Qkvk φ(v1,...,vk,zz)
Prefix
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Matrix (purely propositional)
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Prenex Formulas 2/2
Every QBF* formula can be transformed into a prenex
formula of linear length with these steps:
1. Renaming of quantified variables
to avoid different quantifiers over the same variables
or free and quantified occurrences of a variable.
2. Transformation into Negation Normal Form (NNF)
with De Morgan‘s laws and the negation law, plus the
equivalences ¬(∃v φ) ≈ ∀v ¬φ and ¬(∀v φ) ≈ ∃v ¬φ.
3. Moving quantifiers to the front.
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Transformations on the Matrix
Lemma
Let Q be a prefix and Φ1 and Φ2 two QBF* formulas.
Then Φ1 ≈ Φ2 ⇒ QΦ1 ≈ QΦ2.
Application
Propositional equivalences can be applied to the
matrix of a QBF*.
This allows the transformation of QBF* formulas into
equivalent quantified CNF (QCNF*) of linear length.
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Simplifications
A QBF* formula Φ can be simplified as follows:
• If Φ is not a tautology, delete all tautological clauses.
• If Φ contains a non-tautological clause with only universal
variables, the formula is unsatisfiable.
• Universal Reduction: from a non-tautological clause, we can
delete a literal over a universal x if no existentially quantified
variable yi in the same clause is in the scope of ∀x.
• Pure Literal Detection:
delete all clauses containing pure existential variables and all
literals over pure universal variables.
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