EML-CD recovers causal DAG structure and closed-form mechanisms via gated EML trees, matching PC/GES SHD on Sachs data while recovering 10 of 11 function families in bivariate tests and outperforming SINDy on mechanism f-MSE.
All elementary functions from a single binary operator
7 Pith papers cite this work. Polarity classification is still indexing.
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Pith papers citing it
abstract
A single two-input gate suffices for all of Boolean logic in digital hardware. No comparable primitive has been known for continuous mathematics: computing elementary functions such as sin, cos, sqrt, and log has always required multiple distinct operations. Here I show that a single binary operator, eml(x,y)=exp(x)-ln(y), together with the constant 1, generates the standard repertoire of a scientific calculator. This includes constants such as e, pi, and i; arithmetic operations including addition, subtraction, multiplication, division, and exponentiation as well as the usual transcendental and algebraic functions. For example, exp(x)=eml(x,1), ln(x)=eml(1,eml(eml(1,x),1)), and likewise for all other operations. That such an operator exists was not anticipated; I found it by systematic exhaustive search and established constructively that it suffices for the concrete scientific-calculator basis. In EML (Exp-Minus-Log) form, every such expression becomes a binary tree of identical nodes, yielding a grammar as simple as S -> 1 | eml(S,S). This uniform structure also enables gradient-based symbolic regression: using EML trees as trainable circuits with standard optimizers (Adam), I demonstrate the feasibility of exact recovery of closed-form elementary functions from numerical data at shallow tree depths up to 4. The same architecture can fit arbitrary data, but when the generating law is elementary, it may recover the exact formula.
years
2026 7representative citing papers
A derivative algebra with EML and SOL primitives plus additive atomic forests enables simultaneous symbolic recovery of functions and antiderivatives from data, matching or exceeding XGBoost on 13 of 17 benchmarks with interpretable formulas.
EML operator serves as a grammar to generate single-block activation-suppression modules that produce non-monotone dynamics in reduced biological ODEs, matching data on PKA-R and Rho-GTPase systems.
In the Exp-Minus-Log system every expressible number is computable and Chaitin's Ω_U is inexpressible.
The EML operator forms an abelian group with functional inverses, giving a constructive algebraic route to many distinct families of transcendental elementary functions via binary trees.
Proposes a new exponential-logarithmic descriptor (ELD) for drug-receptor interactions that integrates thermodynamic and probabilistic aspects and exhibits broader dynamic range in numerical simulations.
citing papers explorer
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EML-CD: Causal Mechanism Recovery via EML Symbolic Trees in Structure Learning
EML-CD recovers causal DAG structure and closed-form mechanisms via gated EML trees, matching PC/GES SHD on Sachs data while recovering 10 of 11 function families in bivariate tests and outperforming SINDy on mechanism f-MSE.
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Additive Atomic Forests for Symbolic Function and Antiderivative Discovery
A derivative algebra with EML and SOL primitives plus additive atomic forests enables simultaneous symbolic recovery of functions and antiderivatives from data, matching or exceeding XGBoost on 13 of 17 benchmarks with interpretable formulas.
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Non-Monotone Response Modules and Cascades from the EML Operator for Reduced Models of Biological Dynamics
EML operator serves as a grammar to generate single-block activation-suppression modules that produce non-monotone dynamics in reduced biological ODEs, matching data on PKA-R and Rho-GTPase systems.
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Inexpressibility in Exp-Minus-Log
In the Exp-Minus-Log system every expressible number is computable and Chaitin's Ω_U is inexpressible.
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Algebraic structure behind Odrzywo{\l}ek's EML operator
The EML operator forms an abelian group with functional inverses, giving a constructive algebraic route to many distinct families of transcendental elementary functions via binary trees.
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An exponential logarithmic measure of drug receptor binding and saturation
Proposes a new exponential-logarithmic descriptor (ELD) for drug-receptor interactions that integrates thermodynamic and probabilistic aspects and exhibits broader dynamic range in numerical simulations.
- Architecture-Induced Recoverability Bias in Differentiable Symbolic Regression