Abstract
In this paper, we prove that almost all members of a certain natural class of n-input, n-output Boolean sums of cardinality 2 n have monotone circuit complexity n 2 2 O((log log n) 2) . As a corollary, it is shown that there is a linear space computable Boolean sum whose monotone complexity is n 2 2 O((log log n) 2) The main combinatorial achievement in the paper is as follows. For a subset D of [ n], 1 1 We use [ n] to denote the set 0,…, n - 1. denote by s( D) the largest integer k such that EA, B∥A∥ = ∥B∥ = k & A + B ⊆ D. We prove that s( D) = 2 O(( log log n) 2) for almost all D.
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