Subspace-Invariant AC Formulas

The n-variable PARITY function is computable (by a well-known recursive construction) by AC formulas of depth d+ 1 and leafsize n·2dn1/d . These formulas are seen to possess a certain symmetry: they are syntactically invariant under the subspace P of even-weight elements in {0, 1}, which acts (as a group) on formulas by toggling negations on input literals. In this paper, we prove a 2 −1) lower bound on the size of syntactically P -invariant depth d + 1 formulas for PARITY. Quantitatively, this beats the best 2 −1)) lower bound in the noninvariant setting [18]. More generally, if U ⊂ V are linear subspaces of {0, 1}, we show that every depth d + 1 formula that is syntactically U -invariant and non-constant over V has size at least 2 −1) where m = min{|x| : x ∈ U⊥ \ V ⊥}. This raises the question whether a similar lower bound holds under the weaker hypothesis of semantic U -invariance (i.e. for every depth d+ 1 formula which is identically 0 over U and identically 1 over V \ U).