# Pointers in Recursion: Exploring the Tropics

## TL;DR

This paper introduces pointer recursion and tropical tiering, a new discipline that characterizes various complexity classes including logspace, polylogtime, P, and NC, by translating recursive functions into a pointer-based framework.

## Contribution

It develops a novel pointer recursion framework and tropical tiering discipline, unifying and capturing multiple complexity classes within a single algebraic setting.

## Key findings

- Pointer recursive functions are equivalent to classical partial/primitive recursive functions.
- Tropical tiering captures classes like logspace, polylogtime, P, and NC.
- Pointer recursion with tropical tiering characterizes key complexity classes.

## Abstract

We translate the usual class of partial/primitive recursive functions to a pointer recursion framework, accessing actual input values via a pointer reading unit-cost function. These pointer recursive functions classes are proven equivalent to the usual partial/primitive recursive functions. Complexity-wise, this framework captures in a streamlined way most of the relevant sub-polynomial classes. Pointer recursion with the safe/normal tiering discipline of Bellantoni and Cook corresponds to polylogtime computation. We introduce a new, non-size increasing tiering discipline, called tropical tiering. Tropical tiering and pointer recursion, used with some of the most common recursion schemes, capture the classes logspace, logspace/polylogtime, ptime, and NC. Finally, in a fashion reminiscent of the safe recursive functions, tropical tiering is expressed directly in the syntax of the function algebras, yielding the tropical recursive function algebras.