All elementary functions from a single binary operator
Basic idea and example constructions
- EML is defined as
eml(x,y) = exp(x) – ln(y); with EML and the constant 1, one can build all elementary functions. - Simple derived forms:
exp(x) = eml(x, 1)ln(x) = eml(1, eml(eml(1, x), 1))
- From exp and ln:
- Subtraction:
x - y = eml(ln x, exp y) - Addition via
x + y = ln(exp(x) * exp(y)) - Multiplication, division, powers, roots, trig and hyperbolic functions are then composed using standard identities.
- Subtraction:
- Expanded EML trees become large; e.g., multiplication can require depth-8 trees with 40+ leaves.
Expressiveness, math context, and edge conventions
- The result is likened to NAND/NOR functional completeness, but for continuous/elementary functions rather than Boolean logic.
- Some note that hypergeometric or multi-argument “selector” functions already encode many functions; the novelty here is a binary operation plus one constant.
- The completeness proof sometimes relies on extended real conventions like
ln(0) = -∞,e^{-∞} = 0; this is called out explicitly in the paper and debated:- Some see this as a non-standard caveat.
- Others argue it is standard when working over the extended reals and IEEE‑754 behavior.
- There is discussion of domain issues (e.g., log not a single-valued function over ℂ), and that some constructions pass through
ln(0)or infinities.
Practicality, efficiency, and hardware
- Consensus: this is mainly a theoretical/symbolic result, not a better way to numerically compute basic functions.
- Using EML to express simple operations like
+or*is far more complex and inefficient than standard primitives. - Analogies are drawn to:
- NAND/NOR as universal logical bases, but rarely used directly in optimized designs.
- Lambda calculus/Iota as minimal universal formalisms with little direct practical use.
- Some speculate on:
- EML-based symbolic regression and function discovery, potentially using gradient descent on EML trees.
- Specialized EML coprocessors or analog EML circuits, though others doubt performance benefits versus existing FPUs and polynomial/rational approximations.
Verification, tooling, and reactions
- Several participants reconstruct or verify EML expressions (e.g., using SymPy or small interpreters) and confirm correctness for many constants and operations.
- Others propose using EML as a benchmark challenge for LLMs (“express 2x+y or sin(x)/x in EML+1”).
- Overall tone mixes excitement at the conceptual elegance with skepticism about real-world impact or novelty relative to existing analytic frameworks.