27 research outputs found
Measurable Cones and Stable, Measurable Functions
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
Formal verification of higher-order probabilistic programs
Probabilistic programming provides a convenient lingua franca for writing
succinct and rigorous descriptions of probabilistic models and inference tasks.
Several probabilistic programming languages, including Anglican, Church or
Hakaru, derive their expressiveness from a powerful combination of continuous
distributions, conditioning, and higher-order functions. Although very
important for practical applications, these combined features raise fundamental
challenges for program semantics and verification. Several recent works offer
promising answers to these challenges, but their primary focus is on semantical
issues.
In this paper, we take a step further and we develop a set of program logics,
named PPV, for proving properties of programs written in an expressive
probabilistic higher-order language with continuous distributions and operators
for conditioning distributions by real-valued functions. Pleasingly, our
program logics retain the comfortable reasoning style of informal proofs thanks
to carefully selected axiomatizations of key results from probability theory.
The versatility of our logics is illustrated through the formal verification of
several intricate examples from statistics, probabilistic inference, and
machine learning. We further show the expressiveness of our logics by giving
sound embeddings of existing logics. In particular, we do this in a parametric
way by showing how the semantics idea of (unary and relational) TT-lifting can
be internalized in our logics. The soundness of PPV follows by interpreting
programs and assertions in quasi-Borel spaces (QBS), a recently proposed
variant of Borel spaces with a good structure for interpreting higher order
probabilistic programs
Double Glueing over Free Exponential: with Measure Theoretic Applications
This paper provides a compact method to lift the free exponential
construction of Mellies-Tabareau-Tasson over the Hyland-Schalk double glueing
for orthogonality categories. A condition "reciprocity of orthogonality" is
presented simply enough to lift the free exponential over the double glueing in
terms of the orthogonality. Our general method applies to the monoidal category
TsK of the s-finite transition kernels with countable biproducts. We show (i)
TsK^{op} has the free exponential, which is shown to be describable in terms of
measure theory. (ii) The s-finite transition kernels have an orthogonality
between measures and measurable functions in terms of Lebesgue integrals. The
orthogonality is reciprocal, hence the free exponential of (i) lifts to the
orthogonality category O_I(TsK^{op}), which subsumes Ehrhard et al's
probabilistic coherent spaces as the full subcategory of countable measurable
spaces. To lift the free exponential, the measure-theoretic uniform convergence
theorem commuting Lebesgue integral and limit plays a crucial role. Our
measure-theoretic orthogonality is considered as a continuous version of the
orthogonality of the probabilistic coherent spaces for linear logic, and in
particular provides a two layered decomposition of Crubille et al's direct free
exponential for these spaces
Density-Based Semantics for Reactive Probabilistic Programming
Synchronous languages are now a standard industry tool for critical embedded
systems. Designers write high-level specifications by composing streams of
values using block diagrams. These languages have been extended with Bayesian
reasoning to program state-space models which compute a stream of distributions
given a stream of observations. However, the semantics of probabilistic models
is only defined for scheduled equations -- a significant limitation compared to
dataflow synchronous languages and block diagrams which do not require any
ordering.
In this paper we propose two schedule agnostic semantics for a probabilistic
synchronous language. The key idea is to interpret probabilistic expressions as
a stream of un-normalized density functions which maps random variable values
to a result and positive score. The co-iterative semantics interprets programs
as state machines and equations are computed using a fixpoint operator. The
relational semantics directly manipulates streams and is thus a better fit to
reason about program equivalence. We use the relational semantics to prove the
correctness of a program transformation required to run an optimized inference
algorithm for state-space models with constant parameters
Tropical Mathematics and the Lambda Calculus I: Metric and Differential Analysis of Effectful Programs
We study the interpretation of the lambda-calculus in a framework based on
tropical mathematics, and we show that it provides a unifying framework for two
well-developed quantitative approaches to program semantics: on the one hand
program metrics, based on the analysis of program sensitivity via Lipschitz
conditions, on the other hand resource analysis, based on linear logic and
higher-order program differentiation. To do that we focus on the semantic
arising from the relational model weighted over the tropical semiring, and we
discuss its application to the study of "best case" program behavior for
languages with probabilistic and non-deterministic effects. Finally, we show
that a general foundation for this approach is provided by an abstract
correspondence between tropical algebra and Lawvere's theory of generalized
metric spaces
Probabilistic Programming Semantics for Name Generation
We make a formal analogy between random sampling and fresh name generation.
We show that quasi-Borel spaces, a model for probabilistic programming, can
soundly interpret Stark's -calculus, a calculus for name generation.
Moreover, we prove that this semantics is fully abstract up to first-order
types. This is surprising for an 'off-the-shelf' model, and requires a novel
analysis of probability distributions on function spaces. Our tools are diverse
and include descriptive set theory and normal forms for the -calculus.Comment: 29 pages, 1 figure; to be published in POPL 202