quantified formulas - decision procedure

Quantified Formulas - Decision Procedure

Daniel Kroening, Ofer StrichmanPresented by Changki Hong

07 NOV 08

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Why do we need Quantifier

More modeling power Examples of quantifiers usage :

“Everyone in the room has a friend” “There exists a person whose age is 26.” Hard to express such examples in propositional logic which has

no quantifier. Example of quantifiers usage in math :

For any integer x, there is an integer y smaller than x:

There is quantifier alternation between the universal and existen-tial quantifiers.

What we considered so far was the decision problems for formu-las with nonalternating quantifiers.

Changki Hong @ pswlab Quantified Formulas

xyZyZx . .

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Contents

Introduction Quantifier Elimination Search-Based Algorithms for Quantified Boolean Formu-

las Conclusion


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Quantified Propositional Logic (QPL)

Quantified Propositional Logic is propositional logic en-hanced with quantifiers. Sentences in QBF are better known as quantified Boolean formu-

las (QBFs). Syntax

Complexity The validity problem of QBF is PSPACE-complete

- theoretically harder to solve than SAT, which is “only” NP-complete


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Prenex Normal Form (PNF)

Definition 1. Prenex normal form A formula is said to be in prenex normal form (PNF) if it is in the

form

where for all and is a variable. Quantification prefix

- Quantification string on the left of the formula Quantification suffix

- Quantifier-free formula We can make quantifier alternating PNF by introducing dummy

variable.- From here on we are considering that input formulas are in PNF.


lafree formuquantifiernn -Q[1]V[1].]]V[Q[

},{][Q },,...,1{ ini ][iV

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Transform an input formula into PNF

Lemma 1. For every quantified formula Q there exist a formula Q’ in prenex normal form such that Q is valid if and only if Q’ is valid.

Algorithm - transformation an input formula into PNF


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PNF - example

Transform below formula into PNF

1. Eliminate ‘’

2. Push negations inside :

3. Renaming

4. Move quantifiers to front :


))( . ))( ) .((( . : xyyyxxyyxQ

))( . ))( ) .((( . : xyyyxxyyxQ

))( . ))( ) .((( . : xyyyxxyyxQ

))( . ))( ) .((( . : 22111 xyyyxxyyxQ

)( )( ) ( . . . : 21121 xyyxxyyyxQ

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Quantifier Elimination

Eliminating an existential quantifier over a conjunction of Boolean literals is trivial If x appears with both phases in the conjunction

- Unsatisfiable otherwise

- x can be removed.- ex)

Eliminating universal quantifier using the fact-

However, we need an algorithm which can directly applicable to CNF

- Since converting formulas to DNF can result in an exponential growth in the formula size.


yyyxxy . ..

. . xx

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Elimination with binary resolution (1/2)

Resolution based QBF algorithm Resolution gives us a method to eliminate a variable x from a pair

of clauses in which x appears with opposite phases. Solving

- apply resolution to all pairs of clauses where x appears with opposite phases.

- Example


)()()()( . . . yzxzyxzxyxzy

)()()( . . yzzyzyzy

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Elimination with binary resolution (2/2)

Solving Transform to via : Eliminating universal quantifiers in CNF

- Simply erase them form the formula : if the formula is evaluated TRUE for all value of x, this means that we can’t satisfy a clause while rely-ing on a specific value of x.

Example


. . xx

)()()()( . . . yzxzyxzxyxzy

)()()()( . . yzzyzyzy

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Elimination with Expansion Quantifier elimination is based on expansion of quantifiers,

according to the following equivalences:

Example


10

10

.

.

xx

xx

x

x

10 ))(( ))(( .. xx zxyzxyzy

))(( ... zxyxzy

)( .. zyzy

10 )( )( . zz zyzyy

)( . yy

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Contents


las Conclusion


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Search-Based Algorithms for QBF

Most competitive QBF solvers are based on an adaptation of DPLL solvers. resembles the basic DPLL algorithm without learning and

nonchronological backtracking. However, QBF solvers required to handle of quantifier alternation

The binary search tree has to be changed Distinguish between universal nodes and existential nodes. Universal node are labeled with a symbol “ ”.


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QBF search tree

Definition 2. QBF search tree corresponding to a quanti-fied propositional formula Given a QBF Q in prenex normal form and an ordering of its vari-

ables (say, x1, … , xn), a QBF search tree corresponding to Q is a binary labeled tree of height n+1 with two types of internal nodes, universal and existential, in which:

- The root node is labeled with Q and associated with depth 0.- One of the children of each node at level i, , is marked with

xi+1, and the other with ¬xi+1 - A node in level i, , is universal if the variable in level i+1 is

universally quantified.- A node in level i, , is existential if the variable in level i+1 is

existentially quantified.


ni 0

ni 0

ni 0

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Validity of QBF tree

Definition 3. validity of a QBF tree A QBF tree is valid if and only if its root is satisfied. This is deter-

mined recursively according to the following rules:- A leaf in a QBF binary tree corresponding to a QBF Q is satisfied if the

assignment corresponding to the path to this leaf satisfies the quan-tification suffix of Q.

- A universal node is satisfied if both of its children are satisfied.- A existential node is satisfied if at least one of its children is satisfied.


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Validity of QBF tree - Example

Example of checking validity of QBF tree Consider the formula


)()( . . : ueueueQ

• The second and third u nodes are the only nodes that are satisfied.

• e and ¬e are not satisfied, because both of their children aren’t satisfied

• Therefore, the root node, representing Q, is not satisfied and hence Q is not valid.

Q

e ¬e

u u ¬u¬u

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Overview of search-based algorithm

Same formula


)()( . . : ueueueQ

e = 0

u = 0

e = 1

u = 0 Su = 1

F

F

F

F

F

Not valid!

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Search-based Algorithm for QBF


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Search-based algorithm - Example


Same formula

e = 0

u = 0

e = 1

u = 0 Su = 1

F

F

F

F

F

Not valid!

)()( . . : ueueueQ

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Contents


las Conclusion


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Conclusion

Two quantifier elimination scheme Resolution Expansion

Search-based algorithm for QBF useful to check validity of given formula in QBF


quantified formulas - decision procedure

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