Quantified Formulas - Decision Procedure

1. Quantified Formulas - Decision Procedure • Daniel Kroening, OferStrichman • Presented by ChangkiHong • 07 NOV 08

2. 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 existential quantifiers. • What we considered so far was the decision problems for formulas with nonalternating quantifiers.

3. Contents Introduction Quantifier Elimination Search-Based Algorithms for Quantified Boolean Formulas Conclusion

4. Quantified Propositional Logic (QPL) • Quantified Propositional Logic is propositional logic enhanced with quantifiers. • Sentences in QBF are better known as quantified Boolean formulas (QBFs). • Syntax • Complexity • The validity problem of QBF is PSPACE-complete • theoretically harder to solve than SAT, which is “only” NP-complete

5. 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.

6. 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

7. PNF - example Transform below formula into PNF Eliminate ‘’ Push negations inside : Renaming Move quantifiers to front :

8. 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.

9. 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

10. 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 relying on a specific value of x. • Example

11. Elimination with Expansion Quantifier elimination is based on expansion of quantifiers, according to the following equivalences: Example

12. Contents Introduction Quantifier Elimination Search-Based Algorithms for Quantified Boolean Formulas Conclusion

13. 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 “ ”.

14. QBF search tree • Definition 2. QBF search tree corresponding to a quantified propositional formula • Given a QBF Q in prenex normal form and an ordering of its variables (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.

15. 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 determined 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 quantification 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.

16. Validity of QBF tree - Example Q • 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. e ¬e ¬u ¬u u u • Example of checking validity of QBF tree • Consider the formula

17. Overview of search-based algorithm F Not valid! F F e= 1 e = 0 S F F u= 0 u= 0 u= 1 Same formula

18. Search-based Algorithm for QBF

19. Search-based algorithm - Example Not valid! F F F e= 1 e = 0 S F F u= 0 u= 0 u= 1 Same formula

20. Contents Introduction Quantifier Elimination Search-Based Algorithms for Quantified Boolean Formulas Conclusion

21. Conclusion • Two quantifier elimination scheme • Resolution • Expansion • Search-based algorithm for QBF • useful to check validity of given formula in QBF