Linear Transformations in Boolean Complexity Theory

Authors:
Joel Friedman
Affiliations:
Department of Computer Science, University of British Columbia, Vancouver, BC V6T 1Z4, Canada and Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada
Venue:
CiE '07 Proceedings of the 3rd conference on Computability in Europe: Computation and Logic in the Real World
Year:
2007

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Abstract

We attempt to understand a cohomological approach to lower bounds in Boolean circuits (of [Fri05]) by studying a very restricted case; in this case Boolean complexity is described via the kernel (or nullspace) of a fairly simple linear transformation and its transpose. We look at this linear transformation approach for Boolean functions where we only allow AND gates, which is essentially the SET COVER problem. These linear transformations can recover the linear programming bound. More importantly, we learn that the optimal linear transformation to use can depend on the Boolean function whose complexity we wish to bound; we also learn that infinite complexity (that can occur in AND complexity over arbitrary sets and "bases") appears as a limits of the linear transformation bounds.