Evaluating the Exp-Minus-Log Sheffer Operator for Battery Characterization

Abstract: Odrzywolek (2026) recently introduced the Exp-Minus-Log (EML) operator eml (x, y) = exp(x) - ln(y) and proved constructively that, paired with the constant 1, it generates the entire scientific-calculator basis of elementary functions; in this sense EML is to continuous mathematics what NAND is to Boolean logic. We investigate whether such a uniform single-operator representation can accelerate either the forward simulation or the parameter identification of a six-branch RC equivalent-circuit model (6rc ECM) of a lithium-ion battery cell. We give the analytical EML rewrite of the discretized state-space recursion, derive an exact operation count, and quantify the depth penalty of the master-formula construction used for gradient-based symbolic regression. Our analysis shows that direct EML simulation is slower than the classical exponential-Euler scheme (a ~ 25x instruction overhead per RC branch), but EML-based parametrization offers a structurally complete, gradient-differentiable basis that competes favourably with non-parametric DRT deconvolution and metaheuristic optimisation when the cardinality of RC branches is unknown a priori. We conclude with a concrete recommendation: use EML only on the parametrization side of the 6rc workflow, keeping the classical recursion at runtime.

• The paper introduces the Exp-Minus-Log operator to reformulate the 6rc ECM, enabling symbolic regression for battery parameter identification.
• The EML approach incurs significant computational overhead, rendering it impractical for real-time simulation while ensuring expressive basis completeness.
• EML-based symbolic regression robustly extracts RC branch parameters using gradient optimization, yielding certifiable models for safety-critical battery applications.

Exp-Minus-Log Sheffer Operator in Battery Parameter Identification: A Technical Analysis

Background and Motivation

The Exp-Minus-Log (EML) operator, $\eml(x,y) = \exp(x) - \ln(y)$, was recently formalized as a functionally complete binary operator capable of representing all elementary functions in a manner analogous to the universality of the NAND gate in Boolean logic (Odrzywołek, 23 Mar 2026). This paper probes the engineering utility of EML within the workflow of lithium-ion battery characterization, focusing on the 6-branch RC equivalent circuit model (6rc ECM). The central inquiry is whether an EML-centric formulation can facilitate either accelerated forward simulation or improved symbolic parameter identification relative to traditional techniques.

Classical 6rc ECM and Existing Parametrization Methods

The 6rc ECM represents battery behavior as an open-circuit voltage source, a series resistance, and six parallel RC branches. Increased branch cardinality enables more nuanced modeling of aged or temperature-affected battery cells but exacerbates the parameter identification ill-posedness, with a parameter vector dimension rising to thirteen. Established identification methods include pulse-relaxation extraction, recursive least-squares/extended Kalman filter (RLS/EKF), non-parametric Distribution of Relaxation Times (DRT) deconvolution, and physics-informed neural networks (PINNs). These exhibit known limitations: requirement for explicit topology, sensitivity to measurement noise and regularization, or interpretability gaps.

EML Reformulation: Analytical and Computational Perspective

The paper presents the exact EML rewrite of the discretized state-space recursion governing the 6rc ECM:

• The classical update per branch involves a single exponential evaluation and a few arithmetic operations.
• EML-based reformulation replaces every arithmetic primitive with binary-tree structures of EML nodes, which ultimately compile down to multiple $\exp$ and $\ln$ calls internally.
• The operation count per RC branch in EML form is 67 nodes; for 6 branches, 508 nodes. This results in a $25\times$ instruction count overhead per branch and a $150$-$300\times$ wall-clock penalty for the full model step relative to the classical exponential-Euler scheme.

This computational penalty is inherent and structural. The classical recursion is already at its information-theoretic minimum (one transcendental per state per step), and EML's universality cannot circumvent this constraint. Consequently, the EML approach fails to meet real-time constraints for hardware-in-the-loop (HiL) and ISO 26262 deployment.

EML Symbolic Regression for Parameter Identification

The EML grammar’s completeness becomes advantageous for symbolic parametric identification. The master formula approach constructs an EML tree whose inputs are affine combinations of variables reparametrized via the probability simplex. The number of free parameters in a level-$n$ tree follows $P_n = 5\cdot2^n-6$, admitting smooth, gradient-based optimization compatible with modern autodiff frameworks.

EML-SR identification operates as follows:

• Initialization of a level-4 EML tree with random weights.
• Adam optimization on square-loss plus entropy regularization.
• Weights are snapped to simplex vertices post-optimization, yielding a closed-form symbolic model.
• Parameters $\{R_j, \tau_j\}$ are extracted by pattern-matching.

This method is topologically agnostic and remains robust when the true number of RC branches is unknown—a significant advantage for aged batteries where relaxation modes may emerge unpredictably. Compared to time-window and recursive ARX (RLS/EKF) methods, EML-SR provides structural completeness and gradient differentiation, competing favorably with DRT deconvolution and metaheuristic genetic programming (2604.13873).

Practical and Theoretical Implications

Simulation Versus Identification Asymmetry

Direct EML simulation is inefficient and impractical for deployed battery management systems due to inherent computational overhead. However, EML-SR as a search basis for parameter identification offers unique advantages:

• Guarantees against basis mis-specification due to grammar completeness.
• Native compatibility with gradient-based optimization and symbolic autodiff.
• Robustness in topology-unknown settings, such as in batteries with emergent relaxation times.

Synergy With DRT Prior and Certifiability

DRT distributions can effectively seed EML-SR initializations, mitigating basin-of-attraction problems in symbolic regression. EML-SR’s final snapped models provide interpretable, closed-form algebraic expressions amenable to formal verification—an asset in safety-critical EV powertrain digital twin workflows and ISO 26262 applications. This addresses the principal objection to PINN surrogates' opacity, offering certifiable model evaluation graphs.

Future Directions

The paper raises the question of whether alternative one-operator bases, such as ternary or EDL variants, can achieve lower representational depth for the sum-of-exponentials family found in battery models. Empirical follow-up is warranted to assess whether DRT-seeded EML-SR achieves near-perfect symbolic recovery rates in high-cardinality scenarios.

Conclusion

This study demonstrates that while EML-encoded forward simulation is computationally prohibitive for real-time battery management, the EML master-formula approach provides a structurally complete, differentiable basis for symbolic parameter identification. EML-SR is competitive with established methods, especially in settings where RC topology is unknown. The recommended workflow retains the classical recursion for runtime implementation and leverages EML-SR offline for parameter extraction, ensuring certifiability and real-time performance. Future research may explore more efficient one-operator representations and their integration into battery characterization pipelines (2604.13873).