# On functions computed on trees

## TL;DR

This paper characterizes which functions can be computed on tree-structured hierarchies by deriving necessary and sufficient conditions using differential equations, with implications for neural networks and computational models.

## Contribution

It introduces a set of differential equation constraints that determine the functions realizable on tree-structured computation graphs, bridging structure and function.

## Key findings

- Necessary constraints via non-linear PDEs for functions on trees
- Sufficient conditions for analytic and bit-valued functions
- Enumeration of discrete functions with few realizable cases

## Abstract

Any function can be constructed using a hierarchy of simpler functions through compositions. Such a hierarchy can be characterized by a binary rooted tree. Each node of this tree is associated with a function which takes as inputs two numbers from its children and produces one output. Since thinking about functions in terms of computation graphs is getting popular we may want to know which functions can be implemented on a given tree. Here, we describe a set of necessary constraints in the form of a system of non-linear partial differential equations that must be satisfied. Moreover, we prove that these conditions are sufficient in both contexts of analytic and bit-valued functions. In the latter case, we explicitly enumerate discrete functions and observe that there are relatively few. Our point of view allows us to compare different neural network architectures in regard to their function spaces. Our work connects the structure of computation graphs with the functions they can implement and has potential applications to neuroscience and computer science.