#
Subspace-invariant AC^{0}
formulas

Published

Journal Article

© B. Rossman. 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·2dn1/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 2d(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 2d(m1/d−1) where m is the minimum Hamming weight of a vector in U⊥\V⊥.

### Full Text

### Duke Authors

### Cited Authors

- Rossman, B

### Published Date

- January 1, 2019

### Published In

### Volume / Issue

- 15 / 3

### Start / End Page

- 3:1-3:12 -

### Electronic International Standard Serial Number (EISSN)

- 1860-5974

### Digital Object Identifier (DOI)

- 10.23638/LMCS-15(3:3)2019

### Citation Source

- Scopus