The Depth Irreducibility Hypothesis

We propose the following computational assumption: in general if we try to compress the depth of a circuit family (parallel time) more than a constant factor we will suffer super-quasipolynomial blowup in the size (number of processors). This assumption is only slightly stronger than the popular assumption about the robustness of NC, and we observe that it has surprising consequences. Note also that the choice of super-quasi-polynomial blowup is the smallest bound that avoids the circuit class collapse of [Vol98]. In this proposal we put our assumption in perspective, discuss its relation to the existing literature, and show that it has two important consequences. The first consequence is NC 6= SC, where NC is the class characterized by uniform circuits of poly-logarithmic depth and polynomial size, and SC is characterized by algorithms that run in poly-logarithmic space and polynomial time. For the second consequence we use an additional but mild complexity assumption to obtain a strong separation between the graph isomorphism GraphIso and the group isomorphism GroupIso problem. In particular, we show that GraphIso is not reducible to GroupIso using circuits of O(log n) depth.