The Quest for Functional Completeness

In digital logic and computer science, the concept of functional completeness is foundational. The NAND gate can be composed to construct any Boolean function, meaning every modern processor is essentially billions of identical NAND gates networked together. But continuous mathematics—calculus, topology, and complex analysis—has historically relied on a sprawling, heterogeneous toolkit of disparate operations: addition, multiplication, exponentiation, logarithms, and trigonometric functions.

To a mathematician, this heterogeneity is aesthetically and structurally displeasing. For decades, theorists have sought a “NAND gate” for continuous mathematics—a single, elegant operator from which all other continuous functions could be derived.

The theoretical computer science and mathematics communities have recently formalized this exact construct: the Exp-Minus-Log (EML) function. As researchers push the boundaries of symbolic regression and neural topologies, understanding the rigorous mathematical formulation of this operator becomes critical.

The Algebraic Mechanism and Core Derivations

The EML function relies on a deceptively simple binary operator combined with the constant $1$. It is defined as:

$$\text{eml}(x, y) = e^x - \ln(y)$$

The EML function in combination with the constant $1$ is universal for all elementary functions. By recursively nesting this single operator, we can extract the entirety of basic arithmetic and continuous algebra.

To prove functional completeness, we must demonstrate that this single operation can span the standard arithmetic basis. We achieve this by mapping between the additive and multiplicative domains.

1. The Identity and Basis Functions: By supplying the constant $1$ as the second argument, the logarithm collapses, yielding pure exponentiation:

$$E(x) = \text{eml}(x, 1) = e^x - \ln(1) = e^x$$

By evaluating the function at $\text{eml}(1, 1)$, we derive Euler’s number trivially: $e^1 - 0 = e$.

2. The Subtraction Operator: The structural genius of the EML function lies in its juxtaposition of the exponential and logarithmic forms. If we pass a logarithm into the first argument and an exponential into the second, the inverse properties of the functions cancel out, yielding pure subtraction:

$$S(x, y) = \text{eml}(\ln(x), e^y) = e^{\ln(x)} - \ln(e^y) = x - y$$

3. Multiplication and Addition (The Isomorphisms): Exponentials translate addition into multiplication ($e^{a+b} = e^a \cdot e^b$), while logarithms translate multiplication into addition ($\ln(a \cdot b) = \ln(a) + \ln(b)$). By carefully nesting $\text{eml}$ functions, these properties act as an algebraic bridge, allowing us to build arbitrary integers and basic arithmetic operations strictly through repeated application.

Multiplication is derived by leveraging the additive property of logarithms:

$$M(x, y) = x \cdot y = e^{\ln(x) + \ln(y)}$$

Using our previously established basis functions, we can construct the entirety of a mathematical field entirely out of nested $\text{eml}$ calls. Through sheer composition, this function acts as a Lego block, allowing mathematicians to exact a purely homogeneous tree of repeated similar operations, rather than a tree containing different building operations.

Branch Cuts, Extended Reals, and Complex Trigonometry

To achieve true universality and maintain closure across highly complex, nested tree structures, the EML function cannot be restricted to positive real numbers. It requires operating within a strictly defined mathematical framework.

The Insight: A universal continuous function must elegantly handle zero, infinity, and negative inputs without undefined states breaking the computational tree.

To achieve this, the EML function utilizes the extended reals, mathematically defining $\ln(0) = -\infty$ and $e^{-\infty} = 0$. This allows edge cases to gracefully decay to zero rather than throwing undefined errors. For example, the pure negation of the natural logarithm can be constructed directly via the infinite boundary:

$$N(x) = \text{eml}(-\infty, x) = e^{-\infty} - \ln(x) = 0 - \ln(x) = -\ln(x)$$

Furthermore, generating subtraction and trigonometric functions necessitates exploring the complex plane. The exponential and logarithm are evaluated as complex, taking the principal branch. When the tree inevitably attempts to evaluate the logarithm of a negative number, the principal branch yields an imaginary component (e.g., $\ln(-1) = i\pi$).

This imaginary leap is what allows the EML tree to native generate trigonometric waveforms (via Euler’s formula) and complex polynomials. Because $\sin(x)$ is defined as $\frac{e^{ix} - e^{-ix}}{2i}$, the EML function can construct the sine wave purely by generating $i$ through the principal logarithm of a negative scalar, and composing it with the basis exponential operator $E(x)$.

Topology and Parametrized Machine Learning

Why does a theoretical curiosity regarding functional completeness matter to modern system architects and AI researchers?

The answer lies in the approximation of continuous functions. A 2026 paper formalized that tree-structured compositions of EML functions enjoy a universal approximation property for functions in Sobolev spaces $W^{k, \infty}$ for $k \in \mathbb{N}$. The authors exploit the ability to explicitly construct EML trees that perfectly mimic polynomial representations.

In a Sobolev space $W^{k, \infty}$, a function and its weak derivatives up to order $k$ have a finite essential supremum norm. Proving universal approximation in this topological space guarantees that an EML tree can closely map not just the values of a target function, but its gradients—a strict requirement for backpropagation.

In practical machine learning, this threatens to overhaul how we design neural architectures.

  • Homogeneous Architecture: Instead of building heterogeneous networks with distinct linear layers, ReLU activations, and softmax functions, a model can be constructed entirely as an EML tree.
  • Parametrized Optimization: Researchers have introduced a generalized EML atom equipped with six learnable parameters ($a_i, b_i, c_i, d_i, e_i, f_i$) to improve practical optimization during model training. The generalized node is defined mathematically as: $$\text{EML}_{node}(x, y) = a_i e^{b_i x + c_i} - d_i \ln(e_i y + f_i)$$ This parametrization smooths the loss landscape, preventing gradient explosion while retaining the function’s universal approximation guarantees.
  • Symbolic Regression: EML trees offer a radically interpretable alternative to traditional neural networks. Because every node is identical, the complexity of a learned model can be strictly measured by the depth and size of its EML tree.

Instead of an AI calculating disparate, mathematically isolated floating-point operations, we are approaching an era where models are represented as vast, uniform, and mathematically beautiful trees of a single continuous operator.

Down the Rabbit Hole