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.
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© 2007 Springer-Verlag Berlin Heidelberg
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Friedman, J. (2007). Linear Transformations in Boolean Complexity Theory. In: Cooper, S.B., Löwe, B., Sorbi, A. (eds) Computation and Logic in the Real World. CiE 2007. Lecture Notes in Computer Science, vol 4497. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73001-9_32
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DOI: https://doi.org/10.1007/978-3-540-73001-9_32
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-73000-2
Online ISBN: 978-3-540-73001-9
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