We study the communication complexity of symmetric XOR functions, namely functions $f: \{0,1\}^n \times \{0,1\}^n \rightarrow \{0,1\}$ that can be formulated as $f(x,y)=D(|x\oplus y|)$ for some predicate $D: \{0,1,...,n\} \rightarrow \{0,1\}$, where $|x\oplus y|$ is the Hamming weight of the bitwise XOR of $x$ and $y$. We give a public-coin ... more >>>

Let $f:\{0,1\}^n \to \{0,1\}$ be a boolean function. Its associated XOR function is the two-party function $f_\oplus(x,y) = f(x \oplus y)$.
We show that, up to polynomial factors, the deterministic communication complexity of $f_{\oplus}$ is equal to the parity decision tree complexity of $f$.
This relies on a novel technique ... more >>>

The sum of the absolute values of the Fourier coefficients of a function $f:\mathbb{F}_2^n \to \mathbb{R}$ is called the spectral norm of $f$. Green and Sanders' quantitative version of Cohen's idempotent theorem states that if the spectral norm of $f:\mathbb{F}_2^n \to \{0,1\}$ is at most $M$, then the support of ... more >>>

Boolean function $F(x,y)$ for $x,y \in \{0,1\}^n$ is an XOR function if $F(x,y) = f(x\oplus y)$ for some function $f$ on $n$ input bits, where $\oplus$ is a bit-wise XOR. XOR functions are relevant in communication complexity, partially for allowing Fourier analytic technique. For total XOR functions it is known ... more >>>

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 >>>