The log-rank conjecture, a longstanding problem in communication complexity, has persistently eluded resolution for decades. Consequently, some recent efforts have focused on potential approaches for establishing the conjecture in the special case of XOR functions, where the communication matrix is lifted from a boolean function, and the rank of ... more >>>

Arithmetic circuits are a natural well-studied model for computing multivariate polynomials over a field. In this paper, we study planar arithmetic circuits. These are circuits whose underlying graph is planar. In particular, we prove an $\Omega(n\log n)$ lower bound on the size of planar arithmetic circuits computing explicit bilinear forms ... more >>>

We reduce the best-known upper bound on the length of a program that enumerates a set in terms of the probability of it being enumerated by a random program. We prove a general result that any linear upper bound for finite sets implies the same linear bound for infinite sets.

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