arXiv subjects

Pietro Di Gianantonio

Publications and source records attributed to Pietro Di Gianantonio.

6 recordsLinked to original sources

arXiv2026

A Domain-Theoretic Foundation for Imprecise Probability and Credal Sets

We develop a domain-theoretic framework for imprecise probability reasoning and inference on general topological spaces with a countably based continuous lattice of open sets. We address two distinct forms of uncertainty: partial or incomplete event descriptions, and sets of probability distributions as represented by credal sets -- as well as their combination. Within this framework, we construct a theory of conditional probability and derive novel inference rules for performing Bayesian updating in the presence of these two complementary types of imprecision. These results are extended to a theory of conditional independence for imprecise probabilistic events. We also formulate logical predicates for conditional probability, Bayesian updating, and conditional independence, and we obtain the relevant soundness and completeness results. A key contribution is the construction of a Scott-continuous mapping from any credal set to the domain of intervals, providing a domain-theoretic realisation of classical results from capacity theory and Choquet integration. Finally, we introduce and study a new family of credal sets generated by iterated function systems with imprecise probability weights, broadening the scope of computationally tractable imprecise probabilistic models. The resulting computable framework unifies logical, topological, and measure-theoretic perspectives on uncertainty, supporting robust probabilistic inference under partial and set-valued information.

cs.LO↗

arXiv2025

A domain-theoretic framework for conditional probability and Bayesian updating in programming

We present a domain-theoretic framework for probabilistic programming that provides a constructive definition of conditional probability and addresses computability challenges previously identified in the literature. We introduce a novel approach based on an observable notion of events that enables computability. We examine two methods for computing conditional probabilities -- one using conditional density functions and another using trace sampling with rejection -- and prove they yield consistent results within our framework. We implement these ideas in a simple probabilistic functional language with primitives for sampling and evaluation, providing both operational and denotational semantics and proving their consistency. Our work provides a rigorous foundation for implementing conditional probability in probabilistic programming languages.

cs.LO↗

arXiv2024

A Cartesian Closed Category for Random Variables

We present a novel, yet rather simple construction within the traditional framework of Scott domains to provide semantics to probabilistic programming, thus obtaining a solution to a long-standing open problem in this area. Unlike current main approaches that employ some probability measures or continuous valuations on non-standard or rather complex structures, we use the Scott domain of random variables from a standard sample space -- the unit interval or the Cantor space -- to any given Scott domain. The map taking any such random variable to its corresponding probability distribution provides an effectively given, Scott continuous surjection onto the probabilistic power domain of the underlying Scott domain, establishing a new basic result in classical domain theory. We obtain a Cartesian closed category by enriching the category of Scott domains to capture the equivalence of random variables on these domains. The construction of the domain of random variables on this enriched category forms a strong commutative monad, which is suitable for defining the semantics of probabilistic programming.

cs.PL↗

arXiv2023

A Language for Evaluating Derivatives of Functionals Using Automatic Differentiation

We present a simple functional programming language, called Dual PCF, that implements forward mode automatic differentiation using dual numbers in the framework of exact real number computation. The main new feature of this language is the ability to evaluate correctly up to the precision specified by the user -- in a simple and direct way -- the directional derivative of functionals as well as first order functions. In contrast to other comparable languages, Dual PCF also includes the recursive operator for defining functions and functionals. We provide a wide range of examples of Lipschitz functions and functionals that can be defined in Dual PCF. We use domain theory both to give a denotational semantics to the language and to prove the correctness of the new derivative operator using logical relations. To be able to differentiate functionals -- including on function spaces equipped with their compact-open topology that do not admit a norm -- we develop a domain-theoretic directional derivative that is Scott continuous and extends Clarke's subgradient of real-valued locally Lipschitz maps on Banach spaces to real-valued continuous maps on Hausdorff topological vector spaces. Finally, we show that we can express arbitrary computable linear functionals in Dual PCF.

cs.LO↗

arXiv2019

A Definitional Implementation of the Lax Logical Framework LLFP in Coq, for Supporting Fast and Loose Reasoning

The Lax Logical Framework, LLFP, was introduced, by a team including the last two authors, to provide a conceptual framework for integrating different proof development tools, thus allowing for external evidence and for postponing, delegating, or factoring-out side conditions. In particular, LLFP allows for reducing the number of times a proof-irrelevant check is performed. In this paper we give a shallow, actually definitional, implementation of LLFP in Coq, i.e. we use Coq both as host framework and oracle for LLFP. This illuminates the principles underpinning the mechanism of Lock-types and also suggests how to possibly extend Coq with the features of LLFP. The derived proof editor is then put to use for developing case-studies on an emerging paradigm, both at logical and implementation level, which we call fast and loose reasoning following Danielsson et alii [6]. This paradigm trades off efficiency for correctness and amounts to postponing, or running in parallel, tedious or computationally demanding checks, until we are really sure that the intended goal can be achieved. Typical examples are branch-prediction in CPUs and optimistic concurrency control.

cs.LO↗

arXiv2009

RPO, Second-order Contexts, and Lambda-calculus

First, we extend Leifer-Milner RPO theory, by giving general conditions to obtain IPO labelled transition systems (and bisimilarities) with a reduced set of transitions, and possibly finitely branching. Moreover, we study the weak variant of Leifer-Milner theory, by giving general conditions under which the weak bisimilarity is a congruence. Then, we apply such extended RPO technique to the lambda-calculus, endowed with lazy and call by value reduction strategies. We show that, contrary to process calculi, one can deal directly with the lambda-calculus syntax and apply Leifer-Milner technique to a category of contexts, provided that we work in the framework of weak bisimilarities. However, even in the case of the transition system with minimal contexts, the resulting bisimilarity is infinitely branching, due to the fact that, in standard context categories, parametric rules such as the beta-rule can be represented only by infinitely many ground rules. To overcome this problem, we introduce the general notion of second-order context category. We show that, by carrying out the RPO construction in this setting, the lazy observational equivalence can be captured as a weak bisimilarity equivalence on a finitely branching transition system. This result is achieved by considering an encoding of lambda-calculus in Combinatory Logic.

cs.PL↗