Subspace-Invariant AC^0 Formulas
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We consider the action of a linear subspace U of {0,1}^n on the set of AC^0 formulas with inputs labeled by literals in {X_1, X_1,...,X_n, X_n}, where an element u ∈ U acts by toggling negations on literals with indices i such that u_i=1. A formula is U-invariant if it is fixed by this action. For example, the smallest known construction of depth d+1 formulas computing PARITY_n, of size O(n2^dn^1/d), are P-invariant where P is the subspace of even-weight elements of {0,1}^n. In this paper, we show a nearly matching 2^d(n^1/d-1) lower bound on the P-invariant depth d+1 formula size of PARITY_n. (Quantitatively this improves the best known Ω(2^1/84d(n^1/d-1)) lower bound for unrestricted depth d+1 formulas.) More generally, for any linear subspaces U ⊂ V, we show that if a Boolean function is U-invariant and non-constant over V, then its U-invariant depth d+1 formula size is at least 2^d(m^1/d-1) where m is the minimum Hamming weight of a vector in U^∖ V^.
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