EML Expressibility Framework

• EML Expressibility is a formal algebraic system that represents elementary functions using the constant 1 and the binary operation E(α,β)=exp(α)-log(β).
• It unifies symbolic representation, computable analysis, and universal approximation through a recursive tree grammar that constructs a broad class of functions.
• EML has practical applications in causal discovery, dynamics modeling, and reduced biological models by providing closed-form, interpretable expressions.

The Exp-Minus-Log (EML) framework is a mathematically rigorous system for representing elementary functions and constants using a minimal algebraic foundation: the constant $1$ and a single binary operation $E(\alpha,\beta) = \exp(\alpha) - \log(\beta)$. EML has garnered attention for unifying symbolic, computational, and approximation-theoretic aspects of expressibility, with repercussions for algebraic theory, computable analysis, universal approximation, and practical applications in structure learning and dynamics modeling.

1. Formal System and Algebraic Structure

The EML system is defined syntactically and semantically by a precise grammar over the alphabet $$\Sigma = \{\text{‘1’}, \text{‘E’}, \text{‘(’}, \text{‘)’}, \text{‘,’}\}$$, where:

• The set $\mathfrak{E}$ of closed EML-expressions is the smallest collection of strings such that:
  • “1” $\in \mathfrak{E}$
  • If $\alpha,\beta \in \mathfrak{E}$, then “E($\alpha$,$\beta$)” $\in \mathfrak{E}$
• Each $\varphi \in \mathfrak{E}$ represents a complex number $E(\alpha,\beta) = \exp(\alpha) - \log(\beta)$0 by:

$E(\alpha,\beta) = \exp(\alpha) - \log(\beta)$1

where $E(\alpha,\beta) = \exp(\alpha) - \log(\beta)$2 is the principal branch on $E(\alpha,\beta) = \exp(\alpha) - \log(\beta)$3 with imaginary part in $E(\alpha,\beta) = \exp(\alpha) - \log(\beta)$4.

The operator formalism generalizes to a binary function $E(\alpha,\beta) = \exp(\alpha) - \log(\beta)$5, with $E(\alpha,\beta) = \exp(\alpha) - \log(\beta)$6 satisfying the axioms:

• $E(\alpha,\beta) = \exp(\alpha) - \log(\beta)$7 (neutral element)
• $E(\alpha,\beta) = \exp(\alpha) - \log(\beta)$8 (self-cancellation)
• $E(\alpha,\beta) = \exp(\alpha) - \log(\beta)$9 (anti-associativity)

For the classical EML, $$\Sigma = \{\text{‘1’}, \text{‘E’}, \text{‘(’}, \text{‘)’}, \text{‘,’}\}$$0, $$\Sigma = \{\text{‘1’}, \text{‘E’}, \text{‘(’}, \text{‘)’}, \text{‘,’}\}$$1, and $$\Sigma = \{\text{‘1’}, \text{‘E’}, \text{‘(’}, \text{‘)’}, \text{‘,’}\}$$2 (Stachowiak, 26 Apr 2026).

This recursive tree grammar enables the constructive generation of all elementary functions and the closure of the system under the operations of addition, multiplication, exponentiation, logarithm, and their compositions, provided the branches of $$\Sigma = \{\text{‘1’}, \text{‘E’}, \text{‘(’}, \text{‘)’}, \text{‘,’}\}$$3 are respected in the complex domain (Carney, 2 May 2026, Stachowiak, 26 Apr 2026).

2. EML Expressibility and Computability

A number $$\Sigma = \{\text{‘1’}, \text{‘E’}, \text{‘(’}, \text{‘)’}, \text{‘,’}\}$$4 (or $$\Sigma = \{\text{‘1’}, \text{‘E’}, \text{‘(’}, \text{‘)’}, \text{‘,’}\}$$5) is EML-expressible if there exists $$\Sigma = \{\text{‘1’}, \text{‘E’}, \text{‘(’}, \text{‘)’}, \text{‘,’}\}$$6 such that $$\Sigma = \{\text{‘1’}, \text{‘E’}, \text{‘(’}, \text{‘)’}, \text{‘,’}\}$$7. A fundamental result is the computability of all EML values:

Theorem (Computability of EML values): Every $$\Sigma = \{\text{‘1’}, \text{‘E’}, \text{‘(’}, \text{‘)’}, \text{‘,’}\}$$8 with $$\Sigma = \{\text{‘1’}, \text{‘E’}, \text{‘(’}, \text{‘)’}, \text{‘,’}\}$$9 is a computable complex number. In particular,

$\mathfrak{E}$0

where $\mathfrak{E}$1 is the set of EML-expressible real numbers and $\mathfrak{E}$2 is the set of computable reals (Carney, 2 May 2026). This computability is established by induction on the tree structure of $\mathfrak{E}$3, leveraging effective procedures for approximating $\mathfrak{E}$4 and $\mathfrak{E}$5 on computable arguments.

Consequently, any non-computable real number is formally inexpressible in EML:

Theorem (Inexpressibility in EML): If $\mathfrak{E}$6 is non-computable, there does not exist a closed EML-expression $\mathfrak{E}$7 with $\mathfrak{E}$8.

Canonical inexpressible constants include Chaitin’s $\mathfrak{E}$9 (halting probability of a universal prefix-free Turing machine), left-c.e. non-computable reals, Martin–Löf random reals, and reals encoding halting set characteristic sequences (Carney, 2 May 2026).

3. Functional Universality and Symbolic Grammar

The expressiveness of EML is equivalent to the field $\in \mathfrak{E}$0 of Chow's EL-numbers, the smallest subfield of $\in \mathfrak{E}$1 closed under $\in \mathfrak{E}$2 and $\in \mathfrak{E}$3. EML captures all numbers constructible by finite compositions of $\in \mathfrak{E}$4 and $\in \mathfrak{E}$5.

In function space, EML trees provide a complete symbolic grammar for the classical ring of real elementary functions:

$\in \mathfrak{E}$6

There exists a finite EML tree representing any such function using this operator and leaf constant $\in \mathfrak{E}$7 (Stachowiak, 26 Apr 2026, Erez, 3 May 2026, Asanuma, 4 Jun 2026). For example, $\in \mathfrak{E}$8, $\in \mathfrak{E}$9, multiplication and powers are encoded as compositions using $\alpha,\beta \in \mathfrak{E}$0 and $\alpha,\beta \in \mathfrak{E}$1. Trigonometric and further transcendental functions are obtained via Euler's formula and analytic continuation (Stachowiak, 26 Apr 2026).

A recursive grammar $\alpha,\beta \in \mathfrak{E}$2 or extensions with affine or sum leaves supports algorithmic construction of closed-form expressions (Erez, 3 May 2026, Asanuma, 4 Jun 2026).

4. Universal Approximation and Complexity Scaling

EML trees, defined as directed acyclic graphs whose nodes are compositions of $\alpha,\beta \in \mathfrak{E}$3, exhibit a universal approximation property over Sobolev spaces $\alpha,\beta \in \mathfrak{E}$4:

• For any function $\alpha,\beta \in \mathfrak{E}$5 and error $\alpha,\beta \in \mathfrak{E}$6, an EML tree $\alpha,\beta \in \mathfrak{E}$7 of size $\alpha,\beta \in \mathfrak{E}$8 and depth $\alpha,\beta \in \mathfrak{E}$9 can be constructed such that

$\alpha$0

and with similar bounds in Sobolev norms $\alpha$1 for $\alpha$2 (Germany et al., 22 Jun 2026).

Construction uses:

• Partition of domain, local polynomial approximation (Bramble–Hilbert lemma), smooth partition of unity via tanh-based bump functions, and explicit symbolic realization with EML atoms.
• Each univariate monomial $\alpha$3 is built from two EML nodes; multivariate monomials and sums are assembled recursively.
• Complexity scales optimally as in polynomial or tanh network approximators.

Empirical results indicate that depth-4 real EML trees with $\alpha$490 parameters achieve sub-percent RMSE on standard function benchmarks; further depth increases the risk of overfitting and instability under fixed computational resources (Germany et al., 22 Jun 2026).

5. Applications in Symbolic Modeling, Causal Discovery, and Dynamics

EML-expressible trees enable interpretable symbolic modeling in multiple domains:

• Causal Discovery (EML-CD): Each edge mechanism is modeled as a (gated) EML tree, ensuring closed-form recovery of functions underlying causal structure. Depth-3 trees recover 10 out of 11 elementary function families in controlled bivariate tests and outperform fixed symbolic dictionaries in multivariate SEM recovery benchmarks (f-MSE mean 3.67 vs. 7644) (Asanuma, 4 Jun 2026).
• Analytical Jacobians: The EML-form makes all mechanism derivatives (Jacobians) computable in closed form, aiding interpretability and interventions in causal graphs.
• Reduced Biological Models: EML generates both monotonic and non-monotonic (overshooting) kinetic response blocks with significantly fewer parameters than two-block (e.g., Hill-type) combinations. Cascades of EML gates act as temporal bases in ODE reduction, compactly encoding adaptation and delays (Erez, 3 May 2026).

A table summarizing expressibility in EML for relevant number classes:

Family	EML-expressible?	Example
Rationals, algebraics	Yes	$\alpha$5
Elementary transcendentals	Yes	$\alpha$6, $\alpha$7, $\alpha$8
Limits of computable sequences	Yes	Computable reals
Noncomputable random or l.c.e.	No	$\alpha$9, Martin–Löf rand.

6. Expressibility in Broader Theoretical and Applied Contexts

• Measure and Cardinality: EML-expressible numbers form a countable set; hence, the set of EML-expressible reals is measure zero in $\beta$0 (Carney, 2 May 2026).
• Extension and Limitations: The EML formalism reconstructs all field operations, but requires branch-cut handling in $\beta$1 and explicit constants (at least 1 as the identity). True constant-free generation remains unresolved. Extensions to elliptic functions require more than one generator.
• Comparison to Quantum Expressibility: In quantum models, high "expressibility" (in the quantum sense, e.g., $\beta$2-designs) can cause the quantum neural tangent kernel to concentrate, reducing trainability. Here, expressibility, in the EML sense, provides universal approximation without such kernel collapse, provided parameter regularization and structure are maintained (Yu et al., 2023).

7. Practical Trade-offs and Open Directions

• Depth versus Capacity: Increasing EML-tree depth expands function space coverage (full elementary universality at unbounded depth) but increases parameter count and risk of overfitting. Shallow EMLs are favored where interpretability and sample efficiency dominate.
• Symbolic Stability: Monte Carlo or gradient-based EML training can yield different symbolic coefficients on re-initialization. Improvements in parameter identifiability are an active area.
• Interaction Modeling: Current EML-CD and related frameworks primarily handle univariate or pairwise mechanisms; full multivariate EML expressibility is under active investigation.
• Extensions: EML trees can be pruned or parameter-reduced for enhanced symbolic transparency. Analytical properties under gradient clipping and stochastic training require further mathematical analysis (Asanuma, 4 Jun 2026).

EML thus establishes a robust, algebraically principled, and computationally accessible foundation for symbolic expressibility, with full coverage of the field of elementary functions and reals, excepting all non-computable constants. Its compositional universality, closed-form interpretability, and efficient parameterization distinguish EML from classical symbolic and neural approximators, with implications for future advances in symbolic regression, causal inference, and dynamics modeling (Carney, 2 May 2026, Germany et al., 22 Jun 2026, Asanuma, 4 Jun 2026, Erez, 3 May 2026, Stachowiak, 26 Apr 2026).