1 Quantifier structure in search based procedures for QBFs Enrico Giunchiglia, Massimo Narizzano, Armando Tacchella Abstract The best currently available solvers for Quantified Boolean Formulas (QBFs) process their input in prenex form, i.e., all the quantifiers have to appear in the prefix of the formula separated from the purely propositional part representing the matrix. However, in many QBFs deriving from applications, the propositional part is intertwined with the quantifier structure. To tackle this problem, the standard approach is to convert such QBFs in prenex form, thereby loosing structural information about the prefix. In the case of search based solvers, the prenex form conversion introduces additional constraints on the branching heuristic, and reduces the benefits of the learn- ing mechanisms. In this paper we show that conversion to prenex form is not necessary: current search based solvers can be nat- urally extended in order to handle non prenex QBFs and to exploit the original quantifier structure. We highlight the two mentioned drawbacks of the conversion in prenex form with a simple example, and we show that our ideas can be useful also for solving QBFs in prenex form. To validate our claims, we implemented our ideas in the state- of-the-art search based solver QUBE, and conducted an extensive experimental analysis. The results show that very substantial speedups can be obtained. Index Terms—Formal Verification, Satisfiability, Quanti- fied Boolean Formulas Manuscript received April 3, 2006; revised August 8, 2006. This work was partially supported by MIUR. E. Giunchiglia, M. Narizzano and A. Tacchella are with the Dipar- timento di Informatica Sistemistica e Telematica of the Universit´ a di Genova, Viale Causa 13, 16145 Genova, Italy Copyright (c) 2006 IEEE. Personal use of this material is per- mitted. However, permission to use this material for any other pur- poses must be obtained from the IEEE by sending an email to pubs- permissions@ieee.org. I. Introduction The use of Quantified Boolean Formulas (QBFs) to encode problems arising from Formal Verification (see, e.g., [1], [2], [3], [4]) and Artificial Intelligence (see, e.g., [5], [6]), has attracted increasing interest in recent years. The application-driven quest for efficiency has in turn propelled the research on solvers, and each year a comparative evaluation among the different available QBF solvers takes place, see [7] for the report of the last event in the serie. All the best currently available solvers assume that 1) the input QBF is in prenex form, i.e., all the quan- tifiers have to appear in the prefix of the formula separated from the purely propositional part, and 2) the input QBF is in conjunctive normal form (CNF), i.e., the propositional part of the formula (called matrix) consists of a set of clauses. However, in many QBFs deriving from applications in computer aided verification and artificial intelligence, the propositional part is intertwined with the quantifier struc- ture and the matrix is not in CNF. Examples of such applications are diameter calculation of sequential circuits (see, e.g., [3], [8]), model checking of early requirements (see, e.g., [9]) and formal equivalence checking of partial implementations (see, e.g., [1]). The situation is simpler in the propositional satisfiability (SAT) case, corresponding to QBFs in which all the quantifiers are existential: in SAT, the first problem does not show up, and several papers have been dedicated to efficient and effective conversions to CNF and/or to the implementation of SAT solvers able to handle non CNF formulas (see, e.g., [10], [11] for two recent papers on these issues). The solutions devised in SAT to handle non CNF formulas can be easily lifted to the more complex QBF case. Still, in the QBF case we are left with the first issue. Indeed, the standard solution is to convert any non prenex QBF into a prenex one using standard quantifier rewriting rules like (∃xϕ(x) ∧∀yψ(y)) →∃x∀y(ϕ(x) ∧ ψ(y))