Extensional Uniformity for Boolean Circuits

Extensional Uniformity for Boolean Circuits

Imposing an extensional uniformity condition on a non-uniform circuit complexity class C means simply intersecting C with a uniform class L. By contrast, the usual intensional uniformity conditions require that a resource-bounded machine be able to exhibit the circuits in the circuit family defining C. We say that (C,L) has the Uniformity Duality Property if the extensionally uniform class C∩L ...

Non-cancellative Boolean Circuits: A Generalization of Monotone Boolean Circuits

Cancellations are known to be helpful in e cient algebraic computation of polynomials over elds. We de ne a notion of cancellation in Boolean circuits and de ne Boolean circuits that do not use cancellation to be non-cancellative. Non-cancellative Boolean circuits are a natural generalization of monotone Boolean circuits. We show that in the absence of cancellation, Boolean circuits require sup...

Cyclic Boolean circuits

A Boolean circuit is a collection of gates and wires that performs a mapping from Boolean inputs to Boolean outputs. The accepted wisdom is that such circuits must have acyclic (i.e., loop-free or feed-forward) topologies. In fact, the model is often defined this way – as a directed acyclic graph (DAG). And yet simple examples suggest that this is incorrect. We advocate that Boolean circuits sh...

Lecture Notes: Boolean Circuits

Minimally Unsatisfiable Boolean Circuits

Automated reasoning tasks in many real-world domains involve analysis of redundancies in unsatisfiable instances of SAT. In CNF-based instances, some of the redundancies can be captured by computing a minimally unsatisfiable subset of clauses (MUS). However, the notion of MUS does not apply directly to non-clausal instances of SAT, particularly those that are represented as Boolean circuits. In...