Super-Polynomial Versus Half-Exponential Circuit Size in the Exponential Hierarchy

Peter Bro Miltersen, N. V. Vinodchandran, Osamu Watanabe

Abstract: Lower bounds on circuit size were previously established for functions in \( \sum _2^p \), ZPPNP, \( \sum _2^{\exp } \), ZPEXPNP and MAexp. We investigate the general question: Given a time bound f(n). What is the best circuit size lower bound that can be shown for the classes MA-TIME[f], ZP-TIMENP[f],... using the techniques currently known? For the classes MAexp, ZPEXPNP and \( \sum _2^{\exp } \), the answer we get is “half-exponential”. Informally, a function f is said to be half-exponential if f composed with itself is exponential.