We describe a mathematical structure that can give extensional denotational semantics to higher-order probabilistic programs. It is not limited to discrete probabilities, and it is compatible with integration in a way the models that have been proposed before are not. It is organised as a model of propositional linear logic in which all the connectives have intuitive probabilistic interpretations. In addition, it has least fixed points for all maps, so it can interpret recursion.
Just like any other branch of mathematics, denotational semantics of programming languages should be formalised in type theory, but adapting traditional domain theoretic semantics, as originally formulated in classical set theory to type theory has proven challenging. This paper is part of a project on formulating denotational semantics in type theories with guarded recursion. This should have the benefit of not only giving simpler semantics and proofs of properties such as adequacy, but also hopefully in the future to scale to languages with advanced features, such as general references, outs…
We study the category Cstabm of measurable cones and measurable stable functions, which is a denotational model of an higher-order language with continuous probabilities and full recursion. We look at Cstabm as a model for discrete probabilities, by showing the existence of a cartesian closed, full and faithful functor which embeds probabilistic coherence spaces (a fully abstract denotational model of an higher-order language with full recursion and discrete probabilities) into Cstabm. The proof is based on a generalization of Bernstein's theorem from real analysis allowing to see stable funct…
We define sound and adequate denotational and operational semantics for the stochastic lambda calculus. These two semantic approaches build on previous work that used similar techniques to reason about higher-order probabilistic programs, but for the first time admit an adequacy theorem relating the operational and denotational views. This resolves the main issue left open in (Bacci et al. 2018).
We develop a theory of probabilistic coherence spaces equipped with an additional extensional structure and apply it to approximating probability of convergence of ground type programs of probabilistic PCF whose free variables are of ground types. To this end we define an adapted version of Krivine Machine which computes polynomial approximations of the semantics of these programs in the model. These polynomials provide approximations from below and from above of probabilities of convergence; this is made possible by extending the language with an error symbol which is extensionally maximal in…
We consider a programming language that can manipulate both classical and quantum information. Our language is type-safe and designed for variational quantum programming, which is a hybrid classical-quantum computational paradigm. The classical subsystem of the language is the Probabilistic FixPoint Calculus (PFPC), which is a lambda calculus with mixed-variance recursive types, term recursion and probabilistic choice. The quantum subsystem is a first-order linear type system that can manipulate quantum information. The two subsystems are related by mixed classical/quantum terms that specify h…
This paper investigates the usage of generating functions (GFs) encoding measures over the program variables for reasoning about discrete probabilistic programs. To that end, we define a denotational GF-transformer semantics for probabilistic while-programs, and show that it instantiates Kozen's seminal distribution transformer semantics. We then study the effective usage of GFs for program analysis. We show that finitely expressible GFs enable checking super-invariants by means of computer algebra tools, and that they can be used to determine termination probabilities. The paper concludes by …
University College London（伦敦大学学院）；Cornell University（康奈尔大学）
We present a denotational semantics for higher-order probabilistic programs in terms of linear operators between Banach spaces. Our semantics is rooted in the classical theory of Banach spaces and their tensor products, but bears similarities with the well-known Scott semantics of higher-order programs through the use ordered Banach spaces which allow definitions in terms of fixed points. Being based on a monoidal rather than cartesian closed structure, our semantics effectively treats randomness as a resource.
We investigate the formal semantics of a simple imperative language that has both classical and quantum constructs. More specifically, we provide an operational semantics, a denotational semantics and two Hoare-style proof systems: an abstract one and a concrete one. The two proof systems are satisfaction-based, as inspired by the program logics of Barthe et al for probabilistic programs. The abstract proof system turns out to be sound and relatively complete, while the concrete one is sound only.
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.
A long-standing open problem in the semantics of programming languages supporting probabilistic choice is to find a commutative monad for probability on the category DCPO. In this paper we present three such monads and a general construction for finding even more. We show how to use these monads to provide a sound and adequate denotational semantics for the Probabilistic FixPoint Calculus (PFPC) -- a call-by-value simply-typed lambda calculus with mixed-variance recursive types, term recursion and probabilistic choice. We also show that in the special case where we consider continuous dcpo's, …
We define a probabilistic programming language for Gaussian random variables with a first-class exact conditioning construct. We give operational, denotational and equational semantics for this language, establishing convenient properties like exchangeability of conditions. Conditioning on equality of continuous random variables is nontrivial, as the exact observation may have probability zero; this is Borel's paradox. Using categorical formulations of conditional probability, we show that the good properties of our language are not particular to Gaussians, but can be derived from universal pr…
We propose an extension with deterministically timed multiactions of discrete time stochastic and immediate Petri box calculus (dtsiPBC), previously presented by I.V. Tarasyuk, H. Macià and V. Valero. In dtsdPBC, non-negative integers specify multiactions with fixed (including zero) time delays. The step operational semantics is constructed via labeled probabilistic transition systems. The denotational semantics is defined on the basis of a subclass of labeled discrete time stochastic Petri nets with deterministic transitions. The consistency of both semantics is demonstrated. In order to eval…
We introduce a formal meta-language for probabilistic programming, capable of expressing both programs and the type systems in which they are embedded. We are motivated here by the desire to allow an AGI to learn not only relevant knowledge (programs/proofs), but also appropriate ways of reasoning (logics/type systems). We draw on the frameworks of cubical type theory and dependent typed metagraphs to formalize our approach. In doing so, we show that specific constructions within the meta-language can be related via bisimulation (implying path equivalence) to the type systems they correspond. …
We give an adequate denotational semantics for languages with recursive higher-order types, continuous probability distributions, and soft constraints. These are expressive languages for building Bayesian models of the kinds used in computational statistics and machine learning. Among them are untyped languages, similar to Church and WebPPL, because our semantics allows recursive mixed-variance datatypes. Our semantics justifies important program equivalences including commutativity. Our new semantic model is based on `quasi-Borel predomains'. These are a mixture of chain-complete partial orde…
