Transformations into Normal Forms for Quantified Circuits

Abstract

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 ($\mbox{\rm QBF}$). The concept allows not only prenex representations of the form $\forall x_{1}\exists y_{1}...\forall x_{n}\exists y_{n}\ c$ where c is an ordinary Boolean circuit with inputs x ₁,...,x _n ,y ₁,...,y _n . We also consider more general non-prenex normal forms with quantifiers inside the circuit as in non-prenex $\mbox{\rm QBF}$, 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 polynomial-time 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 ($\mbox{\rm NBF}$). In particular, universal quantification can be simulated efficiently by circuits containing only existential quantifiers if overlapping scopes of variables are allowed.

Research partially supported by DFG grant KL 529/QBF and NSFC grant 60970040.