# On Higher-Order Probabilistic Subrecursion

## TL;DR

This paper explores how adding probabilistic choice to higher-order calculi like G"odel's T affects their expressive power, showing that some forms do not increase computational capabilities while others do.

## Contribution

It introduces a probabilistic variation of reducibility and analyzes the impact of probabilistic choice on the expressive power of G"odel's T.

## Key findings

- Probabilistic choice operators can alter the class of distributions generated.
- Simple probabilistic choice does not change the expressive power of G"odel's T.
- All studied calculi can represent the same functions as G"odel's T.

## Abstract

We study the expressive power of subrecursive probabilistic higher-order calculi. More specifically, we show that endowing a very expressive deterministic calculus like G\"odel's $\mathbb{T}$ with various forms of probabilistic choice operators may result in calculi which are not equivalent as for the class of distributions they give rise to, although they all guarantee almost-sure termination. Along the way, we introduce a probabilistic variation of the classic reducibility technique, and we prove that the simplest form of probabilistic choice leaves the expressive power of $\mathbb{T}$ essentially unaltered. The paper ends with some observations about the functional expressive power: expectedly, all the considered calculi capture the functions which $\mathbb{T}$ itself represents, at least when standard notions of observations are considered.