We'll say $P$ is a symmetric property if $\forall x\in \{0,1\}^n:x\in P\iff \forall \pi \in S(n): f_{\pi }(x)\in P$ where $\forall i\in [n]:f_\pi (x)_i=x_{\pi(i)}$.
Given a symmetric property $P$ we want to find an algorithm which tests $P$, meaning given a vector $x\in \{0,1\}^n$ we want to say if $x\in P$ in sublinear time.
I know how to calculate an $\epsilon-$approximation of hamming weight in probability $\frac{2}{3}$ (can make it higher if I want to but can't get to probability $1$, just choose more indexes) in $O(\log \frac{1}{\epsilon^2})$ but can't find any way to use it.
Thought of trying to calculate the hamming weight with small enough $\epsilon$ such that I'll know if $x\in P$ or not, but can't find small enough $\epsilon$ which doesn't depend on $n$.
Any ideas?
