We present two new positive results for reliable computation using formulas over physical alphabets of size$q \gt 2$. First, we show that for logical alphabets of size$\ell = q$the threshold for denoising using gates subject to q-ary symmetric noise with error probability$\varepsilon $is strictly larger than that for Boolean computation, and we show that reliable computation is possible as long as signals remain distinguishable, i.e.$\epsilon \lt (q - 1) / q$, in the limit of large fan-in$k \rightarrow \infty $. We also determine the point at which generalized majority gates with bounded fan-in fail, and show in particular that reliable computation is possible for$\epsilon \lt (q - 1) / (q (q + 1))$in the case of q prime and fan-in$k = 3$. Secondly, we provide an example where$\ell \lt q$, showing that reliable Boolean computation,$\ell = 2$, can be performed using 2-input ternary,$q = 3$, logic gates subject to symmetric ternary noise of strength$\varepsilon \lt 1/6$by using the additional alphabet element for error signaling.

We present an improved method for symbolic regression that seeks to fit data to formulas that are Pareto-optimal, in the sense of having the best accuracy for a given complexity. It improves on the previous state-of-the-art by typically being orders of magnitude more robust toward noise and bad data, and also by discovering many formulas that stumped previous methods. We develop a method for discovering generalized symmetries (arbitrary modularity in the computational graph of a formula) from gradient properties of a neural network fit. We use normalizing flows to generalize our symbolic regression method to probability distributions from which we only have samples, and employ statistical hypothesis testing to accelerate robust brute-force search.