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In this work, we study the parity complexity measures pCmin[f] and PDT[f]. Pcmin[f] is the parity kill number of f, the fewest number of parities on the input variables one has to fix in order to "kill" f, i.e. To make it constant. PDT[f] is the depth of the shortest emph{parity decision tree} which computes f. These complexity measures have in recent years become increasingly important...
We give a "regularity lemma" for degree-d polynomial threshold functions (PTFs) over the Boolean cube {-1,1}n. Roughly speaking, this result shows that every degree-d PTF can be decomposed into a constant number of subfunctions such that almost all of the subfunctions are close to being regular PTFs. Here a "regular" PTF is a PTF sign(p(x)) where the influence of each variable...
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