Paper overview, section by section

The paper is concise. Here is a map of its sections and why each matters.

§1 Introduction & motivation

Sets up the analogy: Boolean logic has NAND. Continuous math has… nothing, until now. Asks: what is the minimal basis for elementary functions?

§2 The reduction sequence

The core search. Starting from a "scientific calculator" set of 36 primitives, the author iteratively removes redundant operations, showing each removal is safe.

• Calc 3: 6 primitives {exp, ln, negation, reciprocal, +, constants}
• Calc 2: 4 primitives {exp, ln, subtraction, constants}
• Calc 1: 3 primitives {exponentiation, logarithm, e or π}
• Calc 0: 3 primitives {exp function, logarithm, constants}
• EML: 2 primitives {eml, 1}

See The reduction: 36 → 2 for the intuition behind each drop.

§3 The EML operator

Defines eml(x, y) = exp(x) − ln(y) and demonstrates the grammar S → 1 | eml(S, S) produces a class of binary trees that covers every elementary function.

§4 Master formula and symbolic regression

Shows how to set up EML trees with parametric inputs α + β·x + γ·f(...) so that gradient-based optimization can rediscover a formula from its values at sampled points.

§5 Cousin operators

Two operators with the same Sheffer-like property:

• EDL: exp(x) / ln(y) with constant e
• Negated EML: ln(x) − exp(y) with constant −∞

§6 Compiler and analog circuits

A Python compiler converts standard mathematical expressions into pure EML trees. These trees map 1-to-1 onto analog circuits where every node is the same physical element.

§7 Open questions

• Is there a binary Sheffer operator that needs no constant?
• Is there a univariate activation function that is both a good neural-net activation and a function generator?
• Are there continuous families of Sheffer operators?

Reproducibility

Code, Mathematica notebooks, and Rust implementation are deposited in the SymbolicRegressionPackage repository and on Zenodo (DOI: 10.5281/zenodo.19183008).