Exp-Minus-Log (EML) Operator

• EML is a binary operator defined as exp(x)-ln(y) that forms a Sheffer basis for elementary functions via recursive binary trees.
• Its structure enables precise symbolic regression, function approximation, and scientific modeling with proven computability.
• EML unifies arithmetic and transcendental operations, fostering innovative architectures in neuro-symbolic systems and hardware efficiency.

The Exp-Minus-Log (EML) operator is a binary function $\mathrm{eml}(x, y) = \exp(x) - \ln(y)$, which—paired only with the constant $1$—is provably sufficient to generate the entire closed class of elementary functions over the complex numbers. Unlike the standard approach of assembling mathematical functions from a heterogeneous palette of primitives, the EML system encodes all arithmetic, transcendental, and algebraic operations as finite binary trees of this single node type, establishing a Sheffer completeness analogous to how the NAND gate suffices for Boolean logic. This framework underpins a functional, symbolic, and computational unification of mathematical operations and enables new architectures in symbolic regression, function approximation, scientific modeling, and neural-symbolic systems.

1. Formal Definition and Grammar of EML

The EML system is based on a minimal alphabet $$\Sigma = \{\mathrm{“1”},\ \mathrm{“E”},\ \mathrm{(},\ \mathrm{)},\ \mathrm{,}\}$$ and a recursive binary grammar. The set $\mathcal{E}$ of valid, closed EML expressions is inductively generated by:

$$\varphi \in \mathcal{E} ::= 1 \mid E(\varphi, \varphi)$$

Semantically, for $\varphi \in \mathcal{E}$,

$$[\![1]\!] = 1, \quad [\![E(\alpha,\beta)]\!] = \exp([\![\alpha]\!]) - \log([\![\beta]\!])$$

where $\log$ denotes the principal branch of the complex logarithm. This encoding yields the EML number sets:

$$\mathrm{EML} = \{[\![\varphi]\!] : \varphi \in \mathcal{E},\ [\![\varphi]\!] \text{ defined}\} \subseteq \mathbb{C}$$

$$\mathrm{EML}_\mathbb{R} = \mathrm{EML} \cap \mathbb{R}$$

Every standard elementary constant and operation—including $1$0, $1$1, $1$2, $1$3, $1$4, $1$5, $1$6, $1$7, $1$8, exponentiation, roots, trigonometric, and inverse-trigonometric functions—admits an explicit finite EML-expression via syntactic expansion (Carney, 2 May 2026, Odrzywołek, 23 Mar 2026).

2. Compositional Universality and Algebraic Structure

Odrzywołek’s constructive completeness theorem demonstrates that the EML operator, with constant $1$9, forms a Sheffer basis for elementary functions: the context-free grammar $$\Sigma = \{\mathrm{“1”},\ \mathrm{“E”},\ \mathrm{(},\ \mathrm{)},\ \mathrm{,}\}$$0 can express any closed-form function familiar to mathematical analysis. For example,

$$\Sigma = \{\mathrm{“1”},\ \mathrm{“E”},\ \mathrm{(},\ \mathrm{)},\ \mathrm{,}\}$$1

and all arithmetic operations, e.g., $$\Sigma = \{\mathrm{“1”},\ \mathrm{“E”},\ \mathrm{(},\ \mathrm{)},\ \mathrm{,}\}$$2, are recursively constructed as explicit EML trees.

At the abstract level, EML is situated as an operator of the form $$\Sigma = \{\mathrm{“1”},\ \mathrm{“E”},\ \mathrm{(},\ \mathrm{)},\ \mathrm{,}\}$$3 for a bijection $$\Sigma = \{\mathrm{“1”},\ \mathrm{“E”},\ \mathrm{(},\ \mathrm{)},\ \mathrm{,}\}$$4, with $$\Sigma = \{\mathrm{“1”},\ \mathrm{“E”},\ \mathrm{(},\ \mathrm{)},\ \mathrm{,}\}$$5 encoding a (generalized) group law. In the canonical EML case, $$\Sigma = \{\mathrm{“1”},\ \mathrm{“E”},\ \mathrm{(},\ \mathrm{)},\ \mathrm{,}\}$$6, $$\Sigma = \{\mathrm{“1”},\ \mathrm{“E”},\ \mathrm{(},\ \mathrm{)},\ \mathrm{,}\}$$7, $$\Sigma = \{\mathrm{“1”},\ \mathrm{“E”},\ \mathrm{(},\ \mathrm{)},\ \mathrm{,}\}$$8, yielding a structure where primary operations (+, –, ×, /, $$\Sigma = \{\mathrm{“1”},\ \mathrm{“E”},\ \mathrm{(},\ \mathrm{)},\ \mathrm{,}\}$$9, $\mathcal{E}$0) follow from binary-tree recursion and the properties of the underlying abelian group (Stachowiak, 26 Apr 2026). This algebraic perspective enables the extension of single-operator completeness to wider function families, such as trigonometric or elliptic functions, by replacing the exponential/logarithm pairing with other invertible maps.

3. Computability and Expressivity Limits

Every value computed by a closed EML expression is a computable complex number. This is established by induction—using the computability of $\mathcal{E}$1 and $\mathcal{E}$2 and the countable syntax $\mathcal{E}$3. For the real EML numbers,

$\mathcal{E}$4

where $\mathcal{E}$5 denotes the set of Turing-computable real numbers. The system is thus incapable of encoding non-computable numbers: notably, Chaitin’s halting probability $\mathcal{E}$6—the canonical left-computably enumerable but non-computable real—cannot be the value of any EML expression. This formally delineates the boundary of “closed-form” in the EML sense: closed-form is exactly computable under this construction (Carney, 2 May 2026).

4. Symbolic and Algorithmic Implementations

The EML grammar naturally supports symbolic algorithms for manipulating and solving broader classes of expressions, including polynomials with EML (exp-log) coefficients. These are handled by recursive expansions, Newton polygon-based asymptotics, and real/complex root identification via Sturm sequences, enabling the generation of explicit closed-form asymptotic root expansions in EML syntax. Symbolic regression workflows leverage the universal and binary-tree structure of EML, constructing “master-formulas” with parameterized trees and employing gradient-based optimization to fit elementary laws from data. Empirical performance demonstrates that exact symbolic recovery is feasible at shallow depth (depth 2: 100% recovery; depth 3–4: $\mathcal{E}$7 from random initializations), but that a “depth penalty” emerges with combinatorial complexity at greater depths (Strzeboński, 2019, Odrzywołek, 23 Mar 2026, Ipek, 15 Apr 2026).

5. Applications in Modeling, Neuro-Symbolic Systems, and Scientific Computing

EML trees are universal approximators: for any $\mathcal{E}$8, EML trees of sufficient size and depth can approximate $\mathcal{E}$9 arbitrarily closely in $$\varphi \in \mathcal{E} ::= 1 \mid E(\varphi, \varphi)$$0 norm. Explicit constructions exist for polynomial and smooth bump functions, with concrete learning procedures based on gradient descent and parameter “snapping.” Empirical studies confirm subpercent RMSE can be achieved for a variety of canonical univariate targets at moderate tree depth (Germany et al., 22 Jun 2026).

In applied settings, EML provides a structurally complete, differentiable, and topology-free basis for system identification in disciplines such as battery modeling and biological dynamics. In battery equivalent-circuit models, parametrization using EML enables topology-agnostic fitting and symbolic certification but is computationally costly for runtime simulation (25–50× overhead per branch), motivating hybrid workflows that retain classical methods for forward simulation but use EML for offline parameter identification. In biological ODE modeling, EML grammars yield parsimonious representations of non-monotone (overshoot) responses, outperforming classical monotone (Hill, sigmoid) blocks and enabling compression of large dynamical systems (Ipek, 15 Apr 2026, Erez, 3 May 2026).

In neuro-symbolic computation, EML-structured symbolic heads atop deep neural network trunks offer a path to closed-form, interpretable outputs with hardware efficiency on custom FPGA or analog platforms. The adoption of a single EML primitive reduces logic heterogeneity, enabling pipeline fusion and potentially lower end-to-end latency, though on commodity hardware this approach is not faster than standard MLPs. Training remains more challenging due to gradient fragility and costly transcendental operations per node (Ipek, 15 Apr 2026).

6. Theoretical and Practical Implications

The EML system marks a singular point in the landscape of functional completeness for continuous mathematics, showing that a homogeneous, binary-tree architecture suffices to recover the full class of elementary functions. The strict computability of EML-expressible values formally couples closed-form expression with computable analysis. The abelian group plus functional-inverse structure provides a recipe for generating new functionally complete “operator” systems beyond the exponential/logarithm case, suggesting broad theoretical and practical impact in symbolic computation, function representation, and analog/digital hardware design. However, EML does not admit the full real continuum or non-computable reals, establishing a rigorous limit on the scope of symbolic representation in mathematics and computation (Carney, 2 May 2026, Odrzywołek, 23 Mar 2026, Stachowiak, 26 Apr 2026).

7. Summary Table: Key EML System Properties

Property/Result	EML System	Reference
Operator	$$\varphi \in \mathcal{E} ::= 1 \mid E(\varphi, \varphi)$$1	(Odrzywołek, 23 Mar 2026)
Completeness	Generates all elementary functions	(Odrzywołek, 23 Mar 2026, Stachowiak, 26 Apr 2026)
Grammar	$$\varphi \in \mathcal{E} ::= 1 \mid E(\varphi, \varphi)$$2	(Odrzywołek, 23 Mar 2026)
Expressible value class	All computable numbers, none non-computable	(Carney, 2 May 2026)
Universal approximation	Yes, for functions in $$\varphi \in \mathcal{E} ::= 1 \mid E(\varphi, \varphi)$$3	(Germany et al., 22 Jun 2026)
Hardware efficiency	High (custom logic/FPGA), low (CPU/GPU)	(Ipek, 15 Apr 2026)
Key application domains	Symbolic regression, neuro-symbolic nets, scientific modeling	(Ipek, 15 Apr 2026, Erez, 3 May 2026, Ipek, 15 Apr 2026)

EML’s identification as both a symbolic and functional analogue of NAND logic for the continuum establishes a fundamental bridge between abstract algebra, computability, and applied scientific computing, with ongoing exploration into further operator systems and hardware-realizable architectures.