Subspace-invariant AC0 formulas

We consider the action of a linear subspace U of {0, 1}ⁿ on the set of AC⁰ formulas with inputs labeled by literals in the set (formula presented), where an element u ∈ U acts on formulas by transposing the ith pair of literals for all i ∈ [n] such that ui = 1. A formula is U-invariant if it is fixed by this action. For example, there is a well-known recursive construction of depth d + 1 formulas of size O(n·2^dn1/d) computing the n-variable parity function; these formulas are easily seen to be P-invariant where P is the subspace of even-weight elements of {0, 1}ⁿ. In this paper we establish a nearly matching 2^d(n1/d−1) lower bound on the P-invariant depth d + 1 formula size of parity. Quantitatively this improves the best known (formula presented) lower bound for unrestricted depth d + 1 formulas [Ros15], while avoiding the use of the switching lemma. 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(m1/d−1) where m is the minimum Hamming weight of a vector in U^⊥\V^⊥.

Duke Scholars

Author Benjamin Rossman Computer Science