1. Quantified Formulas - Decision Procedure Daniel Kroening, Ofer Strichman Presented by Changki Hong 07 NOV 08

2. 2 / 21 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. Changki Hong @ pswlab Quantified Formulas

3. 3 / 21 Contents Introduction Quantifier Elimination Search-Based Algorithms for Quantified Boolean Formulas Conclusion Changki Hong @ pswlab Quantified Formulas

4. 4 / 21 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 Changki Hong @ pswlab Quantified Formulas

5. 5 / 21 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. Changki Hong @ pswlab Quantified Formulas

6. 6 / 21 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 Changki Hong @ pswlab Quantified Formulas

7. 7 / 21 PNF - example Transform below formula into PNF 1. Eliminate ‘  ’ 2. Push negations inside : 3. Renaming 4. Move quantifiers to front : Changki Hong @ pswlab Quantified Formulas

8. 8 / 21 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. Changki Hong @ pswlab Quantified Formulas

9. 9 / 21 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 Changki Hong @ pswlab Quantified Formulas

10. 10 / 21 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 Changki Hong @ pswlab Quantified Formulas

11. 11 / 21 Elimination with Expansion Quantifier elimination is based on expansion of quantifiers, according to the following equivalences: Example Changki Hong @ pswlab Quantified Formulas

12. 12 / 21 Contents Introduction Quantifier Elimination Search-Based Algorithms for Quantified Boolean Formulas Conclusion Changki Hong @ pswlab Quantified Formulas

13. 13 / 21 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 “ ”. Changki Hong @ pswlab Quantified Formulas

14. 14 / 21 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, x 1, …, x n ), 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 x i+1, and the other with ¬ x i+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. Changki Hong @ pswlab Quantified Formulas

15. 15 / 21 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. Changki Hong @ pswlab Quantified Formulas

16. 16 / 21 Validity of QBF tree - Example Example of checking validity of QBF tree Consider the formula Changki Hong @ pswlab Quantified Formulas 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¬e uu ¬u¬u ¬u¬u

17. 17 / 21 Overview of search-based algorithm Same formula Changki Hong @ pswlab Quantified Formulas e = 0 u = 0 e = 1 u = 0 S u = 1 F F F F F Not valid!

18. 18 / 21 Search-based Algorithm for QBF Changki Hong @ pswlab Quantified Formulas

19. 19 / 21 Search-based algorithm - Example Changki Hong @ pswlab Quantified Formulas Same formula e = 0 u = 0 e = 1 u = 0 S u = 1 F F F F F Not valid!

20. 20 / 21 Contents Introduction Quantifier Elimination Search-Based Algorithms for Quantified Boolean Formulas Conclusion Changki Hong @ pswlab Quantified Formulas

21. 21 / 21 Conclusion Two quantifier elimination scheme Resolution Expansion Search-based algorithm for QBF useful to check validity of given formula in QBF Changki Hong @ pswlab Quantified Formulas