All elementary functions from a single binary operator
6 hours ago
- #symbolic regression
- #binary operator
- #elementary functions
- A single binary operator eml(x,y)=exp(x)-ln(y), with the constant 1, generates all elementary functions like sin, cos, sqrt, and log, analogous to how a single gate suffices for Boolean logic.
- This operator, found through systematic exhaustive search, enables the construction of constants such as e, pi, i, and arithmetic operations (addition, subtraction, multiplication, division, exponentiation).
- All expressions can be represented as binary trees of identical eml nodes, with a simple grammar S -> 1 | eml(S,S), providing a uniform structure.
- EML trees facilitate gradient-based symbolic regression, allowing exact recovery of closed-form elementary functions from numerical data using optimizers like Adam, particularly at shallow tree depths up to 4.
- The architecture can fit arbitrary data, but when the underlying law is elementary, it may recover the exact formula, demonstrating feasibility for scientific calculator functions.