The Circuit Size Hierarchy CSH$^a_b$ states that if $a > b \geq 1$ then the set of functions on $n$ variables computed by Boolean circuits of size $n^a$ is strictly larger than the set of functions computed by circuits of size $n^b$. This result, which is a cornerstone of circuit ... more >>>

We investigate randomized LEARN-uniformity, which captures the power of randomness and equivalence queries (EQ) in the construction of Boolean circuits for an explicit problem. This is an intermediate notion between P-uniformity and non-uniformity motivated by connections to learning, complexity, and logic. Building on a number of techniques, we establish the ... more >>>

We show that proving mildly super-linear lower bounds on non-commutative arithmetic circuits implies exponential lower bounds on non-commutative circuits. That is, non-commutative circuit complexity is a threshold phenomenon: an apparently weak lower bound actually suffices to show the strongest lower bounds we could desire.

This is part of a recent ... more >>>

We show that popular hardness conjectures about problems from the field of fine-grained complexity theory imply structural results for resource-based complexity classes. Namely, we show that if either k-Orthogonal Vectors or k-CLIQUE requires $n^{\epsilon k}$ time, for some constant $\epsilon > 1/2$, to count (note that these conjectures are significantly ... more >>>

Circuit analysis algorithms such as learning, SAT, minimum circuit size, and compression imply circuit lower bounds. We show a generic implication in the opposite direction: natural properties (in the sense of Razborov and Rudich) imply randomized learning and compression algorithms. This is the first such implication outside of the derandomization ... more >>>

We introduce the Nondeterministic Strong Exponential Time Hypothesis
(NSETH) as a natural extension of the Strong Exponential Time
Hypothesis (SETH). We show that both refuting and proving
NSETH would have interesting consequences.

In particular we show that disproving NSETH would ... more >>>