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Probabilistic coherence spaces (PCoh) yield a semantics of higherorder probabilistic computation, interpreting types as convex sets and programs as power series. We prove that the equality of interpretations in PCoh characterizes the operational indistinguishability of programs in PCF with a random primitive. This is the first result of full abstraction for a semantics of probabilistic PCF. The key ingredient relies on the regularity of power series.Along the way to the theorem, we design a weighted intersection type assignment system giving a logical presentation of PCoh.

We define a notion of stable and measurable map between cones endowed with measurability tests and show that it forms a cpo-enriched cartesian closed category. This category gives a denotational model of an extension of PCF supporting the main primitives of probabilistic functional programming, like continuous and discrete probabilistic distributions, sampling, conditioning and full recursion. We prove the soundness and adequacy of this model with respect to a call-by-name operational semantics and give some examples of its denotations.Panangaden [1999] reframed the work by Kozen in a categorical setting, using the category Kern of stochastic kernels. This category has been presented as the Kleisli category of the so-called Giry's monad [Giry 1982] over the category Meas of measurable spaces and measurable functions. One can precisely state the issue for higher-order types in this framework -both Meas and Kern are cartesian categories but not closed.The quest for a formal syntactic semantics of higher-order probabilistic programming had more success. We mention in particular Park et al. [2008], proposing a probabilistic functional language λ ⃝ based on sampling functions. This language has a type R of sub-probabilistic distributions over the set of real numbers 1 , i.e. measures over the Lebesgue σ -algebra on R with total mass at most 1. Using the usual functional primitives (in particular recursion) together with the uniform distribution over [0, 1] and a sampling construct, the authors encode various methods for generating distributions (like the inverse transform method and rejection sampling) and computing properties about them (such as approximations for expectation values, variances, etc). The amazing feature of λ ⃝ is its rich expressiveness as witnessed by the number of examples and applications detailed in [Park et al. 2008], showing the relevance of the functional paradigm for probabilistic programming.Until now, λ ⃝ lacked a denotation model, [Park et al. 2008] sketching only an operational semantics. In particular, the correctness proof of the encodings follows a syntactic reasoning which is not compositional. Our paper fills this gap, giving a denotational model to a variant of λ ⃝ . As a byproduct, we can check program correctness in a straight way by applying to program denotations the standard laws of calculus (Example 7.3,7.4), even for recursive programs (Example 7.9). This method is justified by the Adequacy Theorem 7.12 stating the correspondence between the operational and the denotational semantics.If we restrict the language to countable data types (like booleans and natural numbers, excluding the real numbers), then the situation is much simpler. Indeed, any distribution over a countable set is discrete, i.e. it can be described as a linear combination of its possible outcomes and there is no need of a notion of measurable space. In previous papers [Ehrhard et al. , 2014Ehrhard and Tasson 2016], we have shown that the category PCoh ! of probabilistic coherence spaces and entire functions...