We present a computable algorithm that assigns probabilities to every logical statement in a given formal language, and refines those probabilities over time. For instance, if the language is Peano arithmetic, it assigns probabilities to all arithmetical statements, including claims about the twin prime conjecture, the outputs of long-running ... more >>>

The power symmetric polynomial on $n$ variables of degree $d$ is defined as
$p_d(x_1,\ldots, x_n) = x_{1}^{d}+\dots + x_{n}^{d}$. We study polynomials that are expressible as a sum of powers
of homogenous linear projections of power symmetric polynomials. These form a subclass of polynomials computed by
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Contemplating the recently announced 1-local expanders of Viola and Wigderson (ECCC, TR16-129, 2016), one may observe that weaker constructs are well know. For example, one may easily obtain a 4-regular $N$-vertex graph with spectral gap that is $\Omega(1/\log^2 N)$, and similarly a $O(1)$-regular $N$-vertex graph with spectral gap $1/\tildeO(\log N)$.
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