## Motivation:

Let’s suppose we have a smooth function $$g: \mathbb{R} \rightarrow \mathbb{R}$$ that describes recently-discovered physical phenomena whose behaviour is non-linear in the neighbourhood of $$x=a$$. In order to describe the behaviour of $$g$$ near $$x=a$$, we may define the function space $$F$$ of Taylor polynomials:

$$\lim_{x \to a} \frac{g(x)}{f(x)} = 1 \implies f \in F$$

where $$F$$ contains an Occam approximation of $$g$$ in the neighbourhood of $$x=a$$ if it satisfies two properties:

$$1.$$ Non-linearity in the neighbourhood of $$x=a$$ implies that the lowest-order Taylor polynomial in $$F$$ must be greater than or equal to two. This requirement implies that $$F$$ contains descriptions of the behaviour of $$g$$.

$$2.$$ $$F$$ is constrained to satisfy the prefix property. Intuitively, this means that $$f' \in F$$ if and only if it appears as a prefix of some $$f \in F$$ where $$f \neq f'$$. A more precise definition of this prefix property shall be given in the next paragraph in terms of computation graphs.

### Computation graphs of Taylor polynomials:

In order to define Occam’s approximation of $$g$$ near $$x=a$$, we may represent each polynomial $$f \in F$$ in terms of its computation graph $$G \circ f$$:

$$f = \sum_{n=1}^N a_n(x) \implies G \circ f = \{a_n \}_{n=1}^N$$

where $$\forall n \in [1,N], a_n(x) = \alpha_n \cdot x^{n-1}$$ for some constant $$\alpha_n \in \mathbb{R}$$. Given $$G \circ f$$, we may also readily recover $$f$$:

$$f = G^{-1} \circ (G \circ f)$$

Using (2), we may represent the computation of a Taylor polynomial $$f$$ by a directed Hamiltonian path from the node $$a_1$$ to the node $$a_n$$ where the edge between consecutive nodes is the addition operation. In this formalism, $$f$$ is a sub-computation of $$f' \in F$$ if $$G \circ f$$ appears as a sub-graph of $$G \circ f'$$:

$$G \circ f \subset G \circ f’$$

In fact, (4) captures the prefix property of $$F$$ in the sense that:

$$f’ \in F \implies \exists f \in F, f’ \neq f \land G \circ f \subset G \circ f’$$

and given (5), Occam’s approximation of $$g$$ near $$x=a$$ is none other than:

$$f = G^{-1} \circ \bigcap_{f’ \in F} G \circ f’$$

and for the sake of brevity:

$$f = \mathcal{O}\big(\lim_{x \to a} g(x)\big)$$

## The case where the lowest-order Taylor polynomial is infinite:

If the lowest-order Taylor polynomial in $$F$$ is infinite, we may use a space of special functions $$H$$ instead in order to guarantee that all computation graphs have a finite number of nodes.

By requiring that $$H$$ satisfy the prefix property, we may guarantee that Occam’s approximation of a special function $$g$$ near $$x=a$$ with respect to $$H$$ is well-defined.

## Discussion:

Due to the scope of such a general theory, I think it is normal to expect that such a theory would go through several iterations in order to iron out difficulties which arise in its application. However, what I have presented here is the first iteration of such a theory.

## References:

1. Nikiforov, A. F. and Uvarov, V. B. Special Functions of Mathematical Physics: A Unified Introduction with Applications. Boston, MA: Birkhäuser, 1988.

2. Evgeny Lifshitz and Lev Landau. Statistical Physics. Butterworth-Heinemann. 1980.