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. © Springer-Verlag Berlin Heidelberg 2007.
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Friedman, J. (2007). Linear transformations in Boolean complexity theory. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 4497 LNCS, pp. 307–315). https://doi.org/10.1007/978-3-540-73001-9_32
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