A structural view on elementary functions, symbolic AI, and the illusion of diversity
This article reflects ideas from the recently published Paper
All elementary functions from a single binary operator by Andrzej Odrzywołek
The intuition we rarely question
In applied work – whether in physics, data science, or economic modeling – we operate with a broad toolbox:
- exponentials
- logarithms
- polynomials
- trigonometric functions
They appear as fundamentally different objects.
Different behaviors.
Different interpretations.
Different domains of application.
This diversity feels natural.
It is also misleading.
The structural collapse
A recent result shows that all elementary functions can be generated from a single operator:
$$
eml(x,y) = \exp(x) – \ln(y)
$$
Together with a single constant, this is sufficient to construct:
- addition
- multiplication
- powers
- trigonometric functions
- inverse functions
In other words:
The entire space of elementary mathematics can be expressed as recursive compositions of one operator.
This is not a computational trick.
It is a structural statement.
From functions to trees
The implication is immediate:
Every function becomes a tree.
A function is no longer:
an equation
a formula
a symbolic expression
It is:
A recursive composition of identical nodes.
Formally:
$$
S→1∣eml(S,S)
$$
This places elementary mathematics into the same category as:
- Boolean logic with NAND
- lambda calculus
- recursive grammars
The diversity of functions is reduced to:
Differences in tree structure.
What this actually means
This result does not introduce new mathematics.
It removes structure we assumed to be fundamental.
The distinction between:
- exponential growth
- oscillation
- polynomial scaling
is not ontological.
It is representational.
These are not different “types” of behavior.
They are:
Different coordinate projections of the same generative process.
The consequence for modeling
This has direct implications for how models should be understood.
In practice, we often ask:
Which functional form fits the data?
Should we use exponential, polynomial, or logistic models?
This question is ill-posed.
Because:
There is no fundamental difference between these classes.
The real question is:
Which structure of composition captures the system?
Symbolic regression changes character
Most approaches to symbolic regression treat the problem as:
a discrete search over expressions
a combinatorial optimization problem
The eml formulation changes this.
It defines:
A continuous, differentiable space of all elementary functions.
This enables:
gradient-based optimization
unified model spaces
direct mapping from data to closed-form expressions
Not approximately.
Not numerically.
But structurally.
Why this matters for AI
Modern machine learning operates in a paradox:
Neural networks are expressive but opaque
Symbolic models are interpretable but brittle
This approach collapses that distinction.
It provides:
A continuous symbolic space that is both expressive and interpretable.
This is not a minor technical improvement.
It is a shift in representation.
The deeper implication
The important point is not the operator.
It is what disappears:
the distinction between function classes
the need for handcrafted functional assumptions
the separation between symbolic and numerical modeling
What remains is:
A unified generative structure.
From a systems perspective, this suggests:
Models are not collections of functions
Models are instantiations of a single generative mechanism
A note on limitations
This representation is minimal, not efficient.
Like NAND in logic:
it is universal
but not practical as a direct implementation
Tree depth grows quickly.
Numerical stability becomes an issue.
This is not a replacement for standard modeling.
It is a reframing.
Conclusion
The result is simple to state:
All elementary functions share the same generative structure.
What changes is not what we can compute.
What changes is how we understand models.
Not as a collection of tools.
But as:
Different structural realizations of a single underlying construction process.