The complexity of Boolean functions
The complexity of Boolean functions
Resolution for quantified Boolean formulas
Information and Computation
An algorithm to evaluate quantified Boolean formulae
AAAI '98/IAAI '98 Proceedings of the fifteenth national/tenth conference on Artificial intelligence/Innovative applications of artificial intelligence
Propositional Logic: Deduction and Algorithms
Propositional Logic: Deduction and Algorithms
Bounded universal expansion for preprocessing QBF
SAT'07 Proceedings of the 10th international conference on Theory and applications of satisfiability testing
SAT'04 Proceedings of the 7th international conference on Theory and Applications of Satisfiability Testing
Equivalence models for quantified boolean formulas
SAT'04 Proceedings of the 7th international conference on Theory and Applications of Satisfiability Testing
The second QBF solvers comparative evaluation
SAT'04 Proceedings of the 7th international conference on Theory and Applications of Satisfiability Testing
A branching heuristics for quantified renamable horn formulas
SAT'05 Proceedings of the 8th international conference on Theory and Applications of Satisfiability Testing
Resolution and Expressiveness of Subclasses of Quantified Boolean Formulas and Circuits
SAT '09 Proceedings of the 12th International Conference on Theory and Applications of Satisfiability Testing
Transformations into normal forms for quantified circuits
SAT'11 Proceedings of the 14th international conference on Theory and application of satisfiability testing
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In this paper, quantified Horn formulas (QHORN) are investigated. We prove that the behavior of the existential quantifiers depends only on the cases where at most one of the universally quantified variables is zero. Accordingly, we give a detailed characterization of QHORN satisfiability models which describe the set of satisfying truth assignments to the existential variables. We also consider quantified Horn formulas with free variables (QHORN^*) and show that they have monotone equivalence models. The main application of these findings is that any quantified Horn formula @F of length |@F| with free variables, |@?| universal quantifiers and an arbitrary number of existential quantifiers can be transformed into an equivalent quantified Horn formula of length O(|@?|.|@F|) which contains only existential quantifiers. We also obtain a new algorithm for solving the satisfiability problem for quantified Horn formulas with or without free variables in time O(|@?|.|@F|) by transforming the input formula into a satisfiability-equivalent propositional formula. Moreover, we show that QHORN satisfiability models can be found with the same complexity.