Hilbert's Thirteenth Problem and Circuit Complexity

We study the following question, communicated to us by Miklós Ajtai: Can all explicit (e.g., polynomial time computable) functions f : ({0, 1}) → {0, 1} be computed by word circuits of constant size? Here, a word circuit is an acyclic circuit where each wire holds a word (i.e., an element of {0, 1}) and each gate G computes some binary operation gG : ({0, 1}) → {0, 1}, defined for all word lengths w. As our main result, we present an explicit function so that its w’th slice for any w ≥ 8 cannot be computed by word circuits with at most 4 gates. Also, we formally relate Ajtai’s question to open problems concerning ACC circuits.