# A Probabilistic and Non-Deterministic Call-by-Push-Value Language

## TL;DR

This paper introduces a probabilistic, non-deterministic call-by-push-value language with a domain-theoretic semantics, enabling rigorous mathematical treatment of higher-order probabilistic programs.

## Contribution

It presents a novel language design combining probabilistic choice and non-determinism within the call-by-push-value framework, with sound, adequate, and fully abstract semantics.

## Key findings

- Semantic models are sound and adequate.
- Language achieves full abstraction with additional primitives.
- Framework handles higher-order probabilistic languages mathematically.

## Abstract

There is no known way of giving a domain-theoretic semantics to higher-order probabilistic languages, in such a way that the involved domains are continuous or quasi-continuous - the latter is required to do any serious mathematics. We argue that the problem naturally disappears for languages with two kinds of types, where one kind is interpreted in a Cartesian-closed category of continuous dcpos, and the other is interpreted in a category that is closed under the probabilistic powerdomain functor. Such a setting is provided by Paul B. Levy's call-by-push-value paradigm. Following this insight, we define a call-by-push-value language, with probabilistic choice sitting inside the value types, and where conversion from a value type to a computation type involves demonic non-determinism. We give both a domain-theoretic semantics and an operational semantics for the resulting language, and we show that they are sound and adequate. With the addition of statistical termination testers and parallel if, we show that the language is even fully abstract - and those two primitives are required for that.