Hilbert’s Thirteenth Problem and Circuit Complexity

Abstract

We study the following question, communicated to us by Miklós Ajtai: Can all explicit (e.g., polynomial time computable) functions f: ({0,1}^w)³ →{0,1}^w be computed by word circuits of constant size? A word circuit is an acyclic circuit where each wire holds a word (i.e., an element of {0,1}^w) and each gate G computes some binary operation $g_G:(\{0,1\}^w)^2 \rightarrow \{0,1\}^w$, defined for all word lengths w. 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 ACC⁰ circuits.