Robust Simulations and Significant Separations

TL;DR

This paper introduces a new notion of robust simulations between complexity classes, providing a more natural framework for understanding significant separations and strengthening many known complexity results.

Contribution

It defines robust simulations and significant separations, and demonstrates their implications, extending classical complexity results to this new framework.

Findings

Robust simulations are a natural alternative to infinitely-often notions.

Most known complexity separations can be strengthened to significant separations.

Analogues of classical theorems hold under the new notions.

Abstract

We define and study a new notion of "robust simulations" between complexity classes which is intermediate between the traditional notions of infinitely-often and almost-everywhere, as well as a corresponding notion of "significant separations". A language L has a robust simulation in a complexity class C if there is a language in C which agrees with L on arbitrarily large polynomial stretches of input lengths. There is a significant separation of L from C if there is no robust simulation of L in C. The new notion of simulation is a cleaner and more natural notion of simulation than the infinitely-often notion. We show that various implications in complexity theory such as the collapse of PH if NP = P and the Karp-Lipton theorem have analogues for robust simulations. We then use these results to prove that most known separations in complexity theory, such as hierarchy theorems, fixed polynomial circuit lower bounds, time-space tradeoffs, and the theorems of Allender and Williams, can be strengthened to significant separations, though in each case, an almost everywhere separation is unknown. Proving our results requires several new ideas, including a completely different proof of the hierarchy theorem for non-deterministic polynomial time than the ones previously known.