Vector Analysis of Threshold Functions

13] M. Krause and S. Waack. Variation ranks of communication matrices and lower bounds for depth two circuits having symmetric gates with unbounded fan-in. 22 Acknowledgments The authors would like to thank the referees for their detailed comments, and for strengthening the result presented in Theorem 6. A plc of the rows of the array that does not contain any row from the bottom N=16 rows will always have at least one positive coordinate. This is because: (1) if the rows are chosen only from the top 3N=4 rows then it follows from the previous discussion that at least one coordinate in the rst four coordinates of the plc will have a positive coordinate, (2) if the plc has at least one row from among other rows (i.e., from row number 3N=4 + 1 to row number (N ?N=16)) then consider the last four coordinates: it is a plc of rows that contains at least one row from among r 1 ; r 2 , and r 3 and does not contain r 4 ; hence, by property P4, at least one coordinate will be positive. Continue the procedure by adding log 4 N (= n=2) layers, each of which is 4 columns wide; thus constructing 2n columns. By then, any subset S that does not contain the very bottom row has the following property: any plc of the rows has at least one positive coordinate. Now change the ?1 in the 4th coordinate of the last row to a +1; this completes the construction of A. It follows from property P3 that any plc that contains the last row of A must have at least one positive coordinate in the rst 4 columns. As for the column sums, properties P1 and P2 ensure that they are all 0, except for the 4th column that we modiied in the last row, which sums to 2. 1. The matrix A in the above construction has identical rows. If one wants to construct an A with distinct rows, then one can add n extra columns to obtain an 2 n 3n matrix as follows: add n extra columns, such that each row has a unique identiier in these columns. Since we use all 2 n identiiers, each of these n columns sums to 0. 2. Since any plc of the rows of A has at least one positive coordinate, it implies that …