Uwe Bubeck

Bounded universal expansion for preprocessing QBF

We present a new approach for preprocessing Quantified Boolean Formulas (QBF) in conjunctive normal form (CNF) by expanding a selection of universally quantified variables with bounded expansion costs. We describe a suitable selection strategy which exploits locality of universals and combines cost estimates with goal orientation by taking into account unit literals which might be obtained. Furthermore, we investigate how Q-resolution can be integrated into this method. In particular, resolution is applied specifically to reduce the amount of copying necessary for universal expansion. Experimental results demonstrate that our preprocessing can successfully improve the performance of state-of-the-art QBF solvers on wellknown problems from the QBFLIB collection.

The seventh QBF solvers evaluation (QBFEVAL’10)

In this paper we report about QBFEVAL’10, the seventh in a series of events established with the aim of assessing the advancements in reasoning about quantified Boolean formulas (QBFs). The paper discusses the results obtained and the experimental setup, from the criteria used to select QBF instances to the evaluation infrastructure. We also discuss the current state-of-the-art in light of past challenges and we envision future research directions that are motivated by the results of QBFEVAL’10.

Dependency quantified horn formulas: models and complexity

Dependency quantified Boolean formulas (DQBF) extend quantified Boolean formulas with Henkin-style partially ordered quantifiers. It has been shown that this is likely to yield more succinct representations at the price of a computational blow-up from PSPACE to NEXPTIME. In this paper, we consider dependency quantified Horn formulas (DQHORN), a subclass of DQBF, and show that the computational simplicity of quantified Horn formulas is preserved when adding partially ordered quantifiers. We investigate the structure of satisfiability models for DQHORN formulas and prove that for both DQHORN and ordinary QHORN formulas, the behavior of the existential quantifiers depends only on the cases where at most one of the universally quantified variables is zero. This allows us to transform DQHORN formulas with free variables into equivalent QHORN formulas with only a quadratic increase in length. An application of these findings is to determine the satisfiability of a dependency quantified Horn formula Φ with |∀| universal quantifiers in time O(|∀|·|Φ|), which is just as hard as QHORN-SAT.

Models and quantifier elimination for quantified Horn formulas

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.

Resolution and Expressiveness of Subclasses of Quantified Boolean Formulas and Circuits

We present an extension of Q-Unit resolution for formulas that are not completely in clausal form. This b-unit resolution is applied to different classes of quantified Boolean formulas in which the existential and universal variables satisfy the Horn property. These formulas are transformed into propositional equivalents consisting of only polynomially many subformulas. We obtain compact encodings as Boolean circuits and show that both representations have the same expressive power.

A new 3-CNF transformation by parallel-serial graphs

For propositional formulas we present a new transformation into satisfiability equivalent 3-CNF formulas of linear length. The main idea is to represent formulas as parallel-serial graphs. This is a subclass of directed acyclic multigraphs where the edges are labeled with literals and the AND operation (respectively, the OR operation) is expressed as parallel (respectively, serial) connection.

Transformations into normal forms for quantified circuits

We consider the extension of Boolean circuits to quantified Boolean circuits by adding universal and existential quantifier nodes with semantics adopted from quantified Boolean formulas (QBF). The concept allows not only prenex representations of the form ∀x1∃x1...∀xn∃yn c where c is an ordinary Boolean circuit with inputs x1, ..., xn, y1, ..., yn. We also consider more general non-prenex normal forms with quantifiers inside the circuit as in non-prenex QBF, including circuits in which an input variable may occur both free and bound. We discuss the expressive power of these classes of circuits and establish polynomialtime equivalence-preserving transformations between many of them. Additional polynomial-time transformations show that various classes of quantified circuits have the same expressive power as quantified Boolean formulas and Boolean functions represented as finite sequences of nested definitions (NBF). In particular, universal quantification can be simulated efficiently by circuits containing only existential quantifiers if overlapping scopes of variables are allowed.

Quantifier rewriting and equivalence models for quantified horn formulas

In this paper, quantified Horn formulas with free variables (QHORN*) are investigated. The main result is that any quantified Horn formula Φ of length |Φ| with free variables, |∀| universal quantifiers and an arbitrary number of existential quantifiers can be transformed into an equivalent formula of length O(|∀| ·|Φ|) which contains only existential quantifiers. Moreover, it is shown that quantified Horn formulas with free variables have equivalence models where every existential quantifier is associated with a monotone Boolean function. The results allow a simple representation of quantified Horn formulas as purely existentially quantified Horn formulas (∃HORN*). An application described in the paper is to solve QHORN*-SAT in O(|∀| ·|Φ|) by using this transformation in combination with a linear-time satisfiability checker for propositional Horn formulas.

Nested boolean functions as models for quantified boolean formulas

Nested Boolean functions or Boolean programs are an alternative to the quantified Boolean formula (QBF) characterization of polynomial space. The idea is to start with a set of Boolean functions given as propositional formulas and to define new functions as compositions or instantiations of previously defined ones. We investigate the relationship between function instantiation and quantification and present a compact representation of models and countermodels of QBFs with and without free variables as nested Boolean functions. The representation is symmetric with respect to Skolem models and Herbrand countermodels. For a formula with free variables, it can describe both kinds of models simultaneously in one complete equivalence model which can be Skolem or Herbrand depending on actual assignments to the free variables.

Rewriting (dependency-)quantified 2-CNF with arbitrary free literals into existential 2-HORN

We extend quantified 2-CNF formulas by also allowing literals over free variables which are exempt from the 2-CNF restriction. That means we consider quantified CNF formulas with clauses that contain at most two bound literals and an arbitrary number of free literals. We show that these