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∃y1...∀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.