Source-linked AI summary
Diversity of EML-type operators
Andrzej Odrzywołek
TL;DR
The paper addresses the limited systematic understanding of EML-type operator variants and the challenge of bringing exact elementary-function evaluation into neural-network-like architectures. It enumerates operator families, proposes a Möbius layer with complex rational operations, and uses eml(x,1/x) to recover exp and ln separately, thereby supporting evaluation of all elementary functions within that framework.
Problem
EML variants have not been systematically enumerated, while exact elementary-function computation in an architecture resembling neural networks remains an open challenge.
Method
The paper recursively enumerates candidate operators and formulas, then proposes a Möbius layer using complex values, division, and a new activation function.
Results
The activation eml(x,1/x) allows exp and ln to be recovered separately, which is sufficient to evaluate all elementary functions through the EML construction.
Takeaways & Limitations
EML is presented as one member of a broader operator family, while Möbius layers provide a rational generalization of neural networks for exact elementary-function evaluation.
Takeaways & Limitations
For one EML-type variant based on cosine, reciprocal and division remain unreachable.
Abstract
from arXiv · showhide
The discovery of the EML operator, sufficient to evaluate the standard explicit purely transcendental elementary functions, has led to considerable interest and discussion across multiple scientific disciplines. However, most authors have focused on the binary EML itself, while numerous similar variants with slightly different properties are now known. This article attempts to close this gap by enumerating and classifying them. We also take this opportunity to clarify common misconceptions related to the EML operator. The principal goal, symbolic regression within an architecture as close as possible to proven neural networks which combine matrix multiplication with a single univariate non-linear activation function, remains beyond reach. Instead, we propose a Möbius layer, with rational functions replacing matrix operations, and showcase the recently discovered activation function eml(x,1/x), which allows exp(x) and ln(x) to be recovered separately, and hence all elementary functions to be evaluated within a rational generalization of the neural network.
1. Introduction and follow-up studies
EML operators can reconstruct standard calculator operations and have motivated symbolic-regression and other applications. Follow-up experiments show that optimization, rather than expressivity, limits recovery for tested EML architectures.
- EML capabilities: Using EML with input variables and constant 1 reconstructs arithmetic, exponentials, logarithms, and other standard scientific-calculator operations.The reconstruction has been checked numerically, including arbitrary precision, and with Wolfram Mathematica.
- Complex-domain reconstruction: For trigonometric functions and π, reconstruction uses complex intermediate values within the principal logarithm branch, with extended-real conventions shortening some formulas.Formalized Lean4 proofs support the reconstruction with qualifications discussed in the article.
- Symbolic regression: EML can in principle skip symbolic regression’s discrete candidate-generation step by encoding a full binary expression tree in one master formula.Conventional symbolic regression first generates candidate formulas and then optimizes their free parameters.
- Follow-up symbolic-regression studies: 0% to 100% recovery of the same target across over 12,000 runs showed that architecture and training protocol determine symbolic-regression outcomes.Balanced non-chain trees were never recovered, indicating that optimization landscape rather than expressivity was limiting in these experiments.
- Applications: EML trees have also served as building blocks for joint function and antiderivative discovery, biological reduced models, and interpretable causal mechanisms.
2. Relevant mathematical background and common misconceptions
The paper clarifies what EML can express, how complex intermediates and extended-real conventions support reconstruction, and how exhaustive search was combined with formal and symbolic verification. It also distinguishes EML’s scope from broader notions of elementary functions and relates it to universal computation and NAND-like operators.
- 2.1. All elementary functions?: “All elementary functions” is definition-dependent: EML expresses Chow’s elementary numbers, but not noncomputable Chaitin’s constant or the generic quintic as EML-class expressions.The paper contrasts its explicit finite-expression convention with broader mathematical interpretations of elementary functions.
- 2.2. Complex domain and branches: Complex intermediate values are essential because real-domain exp, ln, and arithmetic cannot produce oscillatory functions such as sin x; EML therefore uses C×(C\{0}) with the principal complex logarithm.The reconstructed functions remain real-valued, while complex inputs occur only at internal expression nodes.
- 2.3. Use of extended real axis and logarithm at zero: The extended-real convention and software handling of infinities simplify formulas, but implementations must account for differences such as Python and Maple rejecting ln(0).A clean compiler avoids ln(0) at the cost of longer expressions, while Lean4’s total Log convention creates additional proof constraints.
- 2.4. EML as NAND/NOR equivalent for continuous math: EML has several operator relatives analogous to NAND/NOR, with the shared distinction that EML requires the constant 1 whereas NAND does not.The paper identifies EML, EDL, LDE, PLI, and PLM as multiple sufficient-operator cases.
- 2.5. Use of the brute-force search: Exhaustive search selected candidate operators and formulas, but EML’s status depended on extensive numerical, arbitrary-precision, computer-algebra, and formal verification rather than enumeration alone.The paper presents brute-force search as a candidate-generation step followed by hand-crafted verification.
- 2.6. Relation to Turing machine and universal computation: EML resembles analog computation because exponentials are treated as primitive, while current digital architectures still require conventional discrete algorithms to evaluate exp(x).The paper identifies unconventional non-von Neumann hardware as a possible direction for efficient EML computation.
- 2.7. Prior knowledge: Basic arithmetic can be encoded through a single binary operation and terminal constants, and unary exp-log operations can be embedded using dummy arguments.The cited construction also reports a constant-free diamond operator in an alternative implementation.
3. Diversity of operators
The article identifies multiple EML-type operators beyond binary EML and organizes them through a common structural framework. It also notes that their general classification remains incomplete and that exhaustive search is still needed.
- 3.1. EML variants: At least five EML-type operators are known, including EML, EDL, LDE, PLI, and PLM.The variants use different constructions and distinguished constants.
- 3.1. EML variants: Stachowiak’s scheme represents known operators using a generating function, an operator, a neutral element, and a distinguished constant.All known operators can be cast in this form, although the admissible pairs remain unknown.
- 3.1. EML variants: The operator M(x, y) is the inverse of a Bennett symmetric hyperoperation, while the table includes addition, subtraction, and symmetrized power constructions.These constructions correspond to different hyperoperation orders.
- 3.1. EML variants: Despite the framework, no general understanding or exhaustive classification of EML-type operators exists, leaving brute-force search as the viable discovery method.The authors also report that reciprocal/division remains unreachable in the discussed cosine-based example.
- 3.1. EML variants: A distinguished input constant complicates the master formula and symbolic-regression implementation, motivating ternary alternatives that avoid this requirement.The cited ternary construction satisfies T_i(x, x, x) = 1 without requiring a distinguished constant.
4. Möbius layer neural networks
The proposed Möbius-layer network modifies conventional neural networks to support exact elementary-function expressions. It replaces affine maps with rational maps, uses a complex activation, and recovers exp and ln separately, although efficient optimization remains unresolved.
- 4. Möbius layer neural networks: EML trees can express elementary functions exactly, but their training is difficult, whereas conventional sigmoid or ReLU networks offer scalable optimization and approximation.The paper frames the proposed architecture as an attempt to combine these symbolic and trainable-network properties.
- 4. Möbius layer neural networks: Exact elementary functions require replacing the activation, using complex internal values, and adding division to the network’s arithmetic.These modifications target the limitations of standard networks with a single real activation and no division.
- 4. Möbius layer neural networks: F enables separate recovery of exp and ln, and together with subtraction thereby reconstructs the EML operator and the paper’s elementary-function repertoire.The construction uses the activation F(z) = e^z + ln z and an algebraic recovery of the exponential.
- 4. Möbius layer neural networks: The Möbius architecture replaces affine input and output maps with ratios of affine forms and replaces the standard activation S with F.Both architectures include constant inputs supplying biases, while each depicted layer has four activation units.
- 4. Möbius layer neural networks: Whether the proposed rational-function network can be optimized efficiently is beyond the scope of the article.The architecture is presented as a direction toward combining symbolic capabilities with neural-network scaling properties.
5. Conclusions
The article argues that EML is one member of a broader family of operators and clarifies that its expressive claim concerns a specific class of elementary functions. It also notes that the proposed rational-function approach may require rational functions for the stated goal.
- 5. Conclusions: EML is presented as the first known member of a larger family of operators that may support new approaches to evaluating elementary functions and enhancing machine learning.The article emphasizes that systematic theory and exhaustive search are still lacking.
- 5. Conclusions: If rational functions are required to achieve the stated neural-network goal, this aligns with the known use of Padé approximation as a strong approximation method.The passage presents this as a conditional observation rather than a demonstrated result of the proposed architecture.
Wolfram Mathematica implementation of a Goldstern-type single operator
The appendix describes a Goldstern-type single-operator implementation that encodes real numbers by a level and payload, then uses level cases to realize arithmetic. The construction avoids distinguished constants and is implemented in Wolfram Mathematica.
- Encoding and operator construction: Every positive integer factorization assigns each real x a floor number l(x) and payload u(x), providing a bijective encoding of real values.The encoding uses β(floor(x)) + 1 = 2^l(x)(2a + 1).
- Operator semantics: The single operator moves values one floor upward while preserving payload, then applies addition, subtraction, multiplication, or division according to the operand levels.Division is defined when the second payload is nonzero.
- Operator semantics: The resulting arithmetic expressions are valid for all real x and y, with the stated nonzero condition for the division case.The construction uses no distinguished constant in the operator definitions.
- Implementation: Constants can also be generated from pure-operator terms, and the complete procedure is implemented in Wolfram Mathematica.The implementation includes the level, payload, successor, encoding, and arithmetic operations.