On Higher-Order Probabilistic Subrecursion

Abstract

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