The ordinary untyped λ-calculus has a set-theoretic model proposed in two related forms by Scott and Plotkin in the 1970s. Recently Scott showed how to introduce probability by extending these models with random variables. However, to reason about correctness and to add further features, it is useful to reinterpret the construction in a higher-order Boolean-valued model involving a measure algebra. In this paper we develop the semantics of an extended stochastic λ-calculus suitable for modeling a simple higher-order probabilistic programming language. We exhibit a number of key equations satisfied by the terms of our language. The terms are interpreted using a continuation-style semantics with an additional argument, an infinite sequence of coin tosses, which serve as a source of randomness. The construction of the model requires a subtle measure-theoretic analysis of the space of coin-tossing sequences. We also introduce a fixpoint operator as a new syntactic construct, as β-reduction turns out not to be sound for unrestricted terms. Finally, we develop a new notion of equality between terms interpreted in a measure algebra, allowing one to reason about terms that may not be equal almost everywhere. This provides a new framework and reasoning principles for probabilistic programs and their higher-order properties.

The expectation monad is introduced abstractly via two composable adjunctions, but concretely captures measures. It turns out to sit in between known monads: on the one hand the distribution and ultrafilter monad, and on the other hand the continuation monad. This expectation monad is used in two probabilistic analogues of fundamental results of Manes and Gelfand for the ultrafilter monad: algebras of the expectation monad are convex compact Hausdorff spaces, and are dually equivalent to so-called Banach effect algebras. These structures capture states and effects in quantum foundations, and the duality between them. Moreover, the approach leads to a new re-formulation of Gleason's theorem, expressing that effects on a Hilbert space are free effect modules on projections, obtained via tensoring with the unit interval.

Abstract. C * -algebras form rather general and rich mathematical structures that can be studied with different morphisms (preserving multiplication, or not), and with different properties (commutative, or not). These various options can be used to incorporate various styles of computation (set-theoretic, probabilistic, quantum) inside categories of C * -algebras. At first, this paper concentrates on the commutative case and shows that there are functors from several Kleisli categories, of monads that are relevant to model probabilistic computations, to categories of C * -algebras. This yields a new probabilistic version of Gelfand duality, involving the "Radon" monad on the category of compact Hausdorff spaces. We then show that the state space functor from C * -algebras to Eilenberg-Moore algebras of the Radon monad is full and faithful. This allows us to obtain an appropriately commuting state-and-effect triangle for C * -algebras.

Abstract. C * -algebras form rather general and rich mathematical structures that can be studied with different morphisms (preserving multiplication, or not), and with different properties (commutative, or not). These various options can be used to incorporate various styles of computation (set-theoretic, probabilistic, quantum) inside categories of C * -algebras. At first, this paper concentrates on the commutative case and shows that there are functors from several Kleisli categories, of monads that are relevant to model probabilistic computations, to categories of C * -algebras. This yields a new probabilistic version of Gelfand duality, involving the "Radon" monad on the category of compact Hausdorff spaces. We then show that the state space functor from C * -algebras to Eilenberg-Moore algebras of the Radon monad is full and faithful. This allows us to obtain an appropriately commuting state-and-effect triangle for C * -algebras.

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