This paper makes two primary contributions. First, we introduce the concept of counting martingales and use it to define counting measures, counting dimensions, and counting strong dimensions. Second, we apply these new tools to strengthen previous circuit lower bounds.

Resource-bounded measure and dimension have traditionally focused on deterministic time ... more >>>

We prove that for the bit pigeonhole principle with any number of pigeons and $n$ holes, any depth $D$ proof in resolution over parities must have size $\exp(\Omega(n^3/D^2))$. Our proof uses the random walk with restarts approach of Alekseev and Itsykson [STOC '25], along with ideas from recent simulation theorems ... more >>>

The Forrelation problem is a central problem that demonstrates an exponential separation between quantum and classical capabilities. In this problem, given query access to $n$-bit Boolean functions $f$ and $g$, the goal is to estimate the Forrelation function $\mathrm{forr}(f,g)$, which measures the correlation between $g$ and the Fourier transform of ... more >>>