Boolean-Valued Semantics for the Stochastic λ-Calculus

Abstract

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.

On the Termination Problem for Probabilistic Higher-Order Recursive Programs

“…, x d1 1 · · · x d k k . For example, if k = 2, then Inc 2,(1,2) can be defined by Inc 2, (1,2) x y z = AddPr x (1,2) (AddPr x (1,1) (AddPr x (1,1) x (1,0) )) y z, because…”

“…Let F be the functional λ λ s.α(F (γ( s))). Notice that solutions to (2) are precisely the pre-fixpoints of F , and we will thus call them abstract pre-fixpoints of F . There are at least two degrees of freedom here: 1.…”

“…Probabilistic functional programs Probabilistic functional programs have recently attracted the attention of the programming language community, although probabilistic λ-calculi have been known for forty years now [58,38]. Most of the work in this field is concerned with operational semantics [18], denotational semantics (see, e.g., [39,19,21,63,2]), or program equivalence (see, e.g., [17,15,60]), which sometime becomes decidable (e.g. [54]), but only when higher-order recursion is forbidden.…”

Probabilistic programming semantics for name generation

“…This still leaves several recent models, including the stable cones model [Ehrhard et al 2018], a function analytic model [Dahlqvist and Kozen 2020], game semantics [Paquet and Winskel 2018], geometry of interaction [Dal Lago and Hoshino 2019], boolean-valued sets [Bacci et al 2018], a boolean topos model [Simpson 2017], and an operational bisimulation [Lago and Gavazzo 2019]. There are also recent logics for higher order probability [Sato et al 2019].…”

PλωNK: functional probabilistic NetKAT

“…Probabilistic powerdomains have been extensively investigated (see e.g., Bacci et al [2018]; Battenfeld et al [2007]; Goubault-Larrecq and Varacca [2011]; Jones and Plotkin [1989]; Jung and Tix [1998];Saheb-Djahromi [1980]). Nevertheless, until the work of Vákár et al [2019], a convenient continuous probabilistic powerdomain that supports iteration and is commutative proved elusive.…”