| Summary: | 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.
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