arXiv subjects

Paolo Pistone

Publications and source records attributed to Paolo Pistone.

At least 19 recordsLinked to original sources

arXiv2026

On Higher-Order Probabilistic Verification via the Weighted Relational Model of Linear Logic

The problem of determining whether a probabilistic program terminates almost surely (i.e.~with probability one) is undecidable, and actually $\Pi^0_2$-complete. For this reason, a growing literature has explored classes of programs for which this and related problems can be shown (semi-)decidable. In this work we consider the termination problem for the language of Probabilistic Higher-Order Recursion Schemes (PHORS). Using the weighted relational semantics of linear logic, we translate this problem into the computation of suitable generating functions associated with the program interpreted. This way, we establish the decidability of almost sure termination for a class of programs that extends Li et al.'s affine PHORS via a type discipline with bounded exponentials. To achieve this, we show that the generating functions for such programs are always algebraic, that is, solutions of polynomial equations, yielding an effective method to answer the termination problem.

cs.LO↗

arXiv2026

On the Metric Nature of (Differential) Logical Relations

Differential logical relations are methods to measure distances between higher-order programs where distances between functional programs are themselves \emph{functions}, relating errors in inputs with errors in outputs. This way, differential logical relations provide a more fine-grained and contextual information of program distances. This paper aims to clarify the metric nature of differential logical relations. We introduce the notion of quasi-quasi-metrics and observe that the cartesian closed category of quasi-quasi-metric spaces reflects the construction of differential logical relations in the literature. The cartesian closed structure induces a fundamental lemma, which can be seen as a compositional reasoning principle for program distances. Furthermore, we investigate the quasi-quasi-metric spaces arising from the interpretation of types, and we prove that they satisfy variants of the strong transitivity condition and indistancy condition, as well as a weak form of the symmetry condition. In the last part of this paper, we introduce a notion of differential prelogical relations arising as a quantitative counterpart of the framework of prelogical relations. Roughly speaking, differential prelogical relations are quasi-quasi-metrics on the collection of programs. The poset of differential prelogical relations has the finest differential prelogical relation presented as a formal quantitative equational theory, while the poset lacks a coarsest differential prelogical relation. The absence of a coarsest differential prelogical relation contrasts with the situations of typed lambda calculi, where the contextual equivalences serve as the coarsest program equivalences.

cs.LO↗

arXiv2026

Compiling Quantum Lambda-Terms into Circuits via the Geometry of Interaction

We present an algorithm turning any term of a linear quantum $λ$-calculus into a quantum circuit. The essential ingredient behind the proposed algorithm is Girard's geometry of interaction, which, differently from its well-known uses from the literature, is here leveraged to perform as much of the classical computation as possible, at the same time producing a circuit that, when evaluated, performs all the quantum operations in the underlying $λ$-term. We identify higher-order control flow as the primary obstacle towards efficient solutions to the problem at hand. Notably, geometry of interaction proves sufficiently flexible to enable efficient compilation in many cases, while still supporting a total compilation procedure. Finally, we characterize through a type system those $λ$-terms for which compilation can be performed efficiently.

cs.LO↗

arXiv2025

Tropical Mathematics and the Lambda-Calculus II: Tropical Geometry of Probabilistic Programming Languages

In the last few years there has been a growing interest towards methods for statistical inference and learning based on computational geometry and, notably, tropical geometry, that is, the study of algebraic varieties over the min-plus semiring. At the same time, recent work has demonstrated the possibility of interpreting higher-order probabilistic programming languages in the framework of tropical mathematics, by exploiting algebraic and categorical tools coming from the semantics of linear logic. In this work we combine these two worlds, showing that tools and ideas from tropical geometry can be used to perform statistical inference over higher-order probabilistic programs. Notably, we first show that each such program can be associated with a degree and a n-dimensional polyhedron that encode its most likely runs. Then, we use these tools in order to design an intersection type system that estimates most likely runs in a compositional and efficient way.

cs.LO↗

arXiv2025

On The Metric Nature of (Differential) Logical Relations

Differential logical relations are a method to measure distances between higher-order programs. They differ from standard methods based on program metrics in that differences between functional programs are themselves functions, relating errors in input with errors in output, this way providing a more fine grained, contextual, information. The aim of this paper is to clarify the metric nature of differential logical relations. While previous work has shown that these do not give rise, in general, to (quasi-)metric spaces nor to partial metric spaces, we show that the distance functions arising from such relations, that we call quasi-quasi-metrics, can be related to both quasi-metrics and partial metrics, the latter being also captured by suitable relational definitions. Moreover, we exploit such connections to deduce some new compositional reasoning principles for program differences.

cs.LO↗

arXiv2024

The Lambda Calculus is Quantifiable

In this paper we introduce several quantitative methods for the lambda-calculus based on partial metrics, a well-studied variant of standard metric spaces that have been used to metrize non-Hausdorff topologies, like those arising from Scott domains. First, we study quantitative variants, based on program distances, of sensible equational theories for the $λ$-calculus, like those arising from Böhm trees and from the contextual preorder. Then, we introduce applicative distances capturing higher-order Scott topologies, including reflexive objects like the $D_\infty$ model. Finally, we provide a quantitative insight on the well-known connection between the Böhm tree of a $λ$-term and its Taylor expansion, by showing that the latter can be presented as an isometric transformation.

cs.LO↗

arXiv2024

On Randomized Computational Models and Complexity Classes: a Historical Overview

Since their appearance in the 1950s, computational models capable of performing probabilistic choices have received wide attention and are nowadays pervasive in almost every areas of computer science. Their development was also inextricably linked with inquiries about computation power and resource issues. Although most crucial notions in the field are well-known, the related terminology is sometimes imprecise or misleading. The present work aims to clarify the core features and main differences between machines and classes developed in relation to randomized computation. To do so, we compare the modern definitions with original ones, recalling the context in which they first appeared, and investigate the relations linking probabilistic and counting models.

cs.LO↗

arXiv2023

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.

cs.LO↗

arXiv2023

Enumerating Error Bounded Polytime Algorithms Through Arithmetical Theories

We consider a minimal extension of the language of arithmetic, such that the bounded formulas provably total in a suitably-defined theory à la Buss (expressed in this new language) precisely capture polytime random functions. Then, we provide two new characterizations of the semantic class BPP obtained by internalizing the error-bound check within a logical system: the first relies on measure-sensitive quantifiers, while the second is based on standard first-order quantification. This leads us to introduce a family of effectively enumerable subclasses of BPP, called BPP_T and consisting of languages captured by those probabilistic Turing machines whose underlying error can be proved bounded in the theory T. As a paradigmatic example of this approach, we establish that polynomial identity testing is in BPP_T where T=$\mathrm{I}Δ_0+\mathrm{Exp}$ is a well-studied theory based on bounded induction.

cs.LO↗

arXiv2023

On the Lattice of Program Metrics

In this paper we are concerned with understanding the nature of program metrics for calculi with higher-order types, seen as natural generalizations of program equivalences. Some of the metrics we are interested in are well-known, such as those based on the interpretation of terms in metric spaces and those obtained by generalizing observational equivalence. We also introduce a new one, called the interactive metric, built by applying the well-known Int-Construction to the category of metric complete partial orders. Our aim is then to understand how these metrics relate to each other, i.e., whether and in which cases one such metric refines another, in analogy with corresponding well-studied problems about program equivalences. The results we obtain are twofold. We first show that the metrics of semantic origin, i.e., the denotational and interactive ones, lie \emph{in between} the observational and equational metrics and that in some cases, these inclusions are strict. Then, we give a result about the relationship between the denotational and interactive metrics, revealing that the former is less discriminating than the latter. All our results are given for a linear lambda-calculus, and some of them can be generalized to calculi with graded comonads, in the style of Fuzz.

cs.LO↗

arXiv2023

An Arithmetic Theory for the Poly-Time Random Functions

We introduce a new bounded theory RS^1_2 and show that the functions which are Sigma^b_1-representable in it are precisely random functions which can be computed in polynomial time. Concretely, we pass through a class of oracle functions over string, called POR, together with the theory of arithmetic RS^1_2. Then, we show that functions computed by poly-time PTMs are arithmetically characterized by a class of probabilistic bounded formulas.

cs.CC↗

arXiv2022

On Quantitative Algebraic Higher-Order Theories

We explore the possibility of extending Mardare et al. quantitative algebras to the structures which naturally emerge from Combinatory Logic and the lambda-calculus. First of all, we show that the framework is indeed applicable to those structures, and give soundness and completeness results. Then, we prove some negative results which clearly delineate to which extent categories of metric spaces can be models of such theories. We conclude by giving several examples of non-trivial higher-order quantitative algebras.

cs.LO↗

arXiv2022

Curry and Howard Meet Borel

We show that an intuitionistic version of counting propositional logic corresponds, in the sense of Curry and Howard, to an expressive type system for the probabilistic event lambda-calculus, a vehicle calculus in which both call-by-name and call-by-value evaluation of discrete randomized functional programs can be simulated. Remarkably, proofs (respectively, types) do not only guarantee that validity (respectively, termination) holds, but also reveal the underlying probability. We finally show that by endowing the type system with an intersection operator, one obtains a system precisely capturing the probabilistic behavior of lambda-terms.

cs.LO↗

arXiv2021

A New Conjecture About Identity of Proofs

A central problem in proof-theory is that of finding criteria for identity of proofs, that is, for when two distinct formal derivations can be taken as denoting the same logical argument. In the literature one finds criteria which are either based on proof normalization (two derivations denote the same proofs when they have the same normal form) or on the association of formal derivations with graph-theoretic structures (two derivations denote they same proof when they are associated with the same graph). In this paper we argue for a new criterion for identity of proofs which arises from the interpretation of formal rules and derivations as natural transformations of a suitable kind. We show that the naturality conditions arising from this interpretation capture in a uniform and elegant ways several forms of "rule-permutations" which are found in proof-systems for propositional, first- and second-order logic.

math.LO↗

arXiv2021

From Identity to Difference: A Quantitative Interpretation of the Identity Type

We explore a quantitative interpretation of 2-dimensional intuitionistic type theory (ITT) in which the identity type is interpreted as a "type of differences". We show that a fragment of ITT, that we call difference type theory (dTT), yields a general logical framework to talk about quantitative properties of programs like approximate equivalence and metric preservation. To demonstrate this fact, we show that dTT can be used to capture compositional reasoning in presence of errors, since any program can be associated with a "derivative" relating errors in input with errors in output. Moreover, after relating the semantics of dTT to the standard weak factorization systems semantics of ITT, we describe the interpretation of dTT in some quantitative models developed for approximate program transformations, incremental computing, program differentiation and differential privacy.

cs.LO↗

arXiv2021

On Counting Propositional Logic

We study counting propositional logic as an extension of propositional logic with counting quantifiers. We prove that the complexity of the underlying decision problem perfectly matches the appropriate level of Wagner's counting hierarchy, but also that the resulting logic admits a satisfactory proof-theoretical treatment. From the latter, a type system for a probabilistic lambda-calculus is derived in the spirit of the Curry-Howard correspondence, showing the potential of counting propositional logic as a useful tool in several fields of theoretical computer science.

cs.LO↗

arXiv2021

What's Decidable about (Atomic) Polymorphism

Due to the undecidability of most type-related properties of System F like type inhabitation or type checking, restricted polymorphic systems have been widely investigated (the most well-known being ML-polymorphism). In this paper we investigate System Fat, or atomic System F, a very weak predicative fragment of System F whose typable terms coincide with the simply typable ones. We show that the type-checking problem for Fat is decidable and we propose an algorithm which sheds some new light on the source of undecidability in full System F. Moreover, we investigate free theorems and contextual equivalence in this fragment, and we show that the latter, unlike in the simply typed lambda-calculus, is undecidable.

cs.LO↗

arXiv2021

On Generalized Metric Spaces for the Simply Typed Lambda-Calculus (Extended Version)

Generalized metrics, arising from Lawvere's view of metric spaces as enriched categories, have been widely applied in denotational semantics as a way to measure to which extent two programs behave in a similar, although non equivalent, way. However, the application of generalized metrics to higher-order languages like the simply typed lambda calculus has so far proved unsatisfactory. In this paper we investigate a new approach to the construction of cartesian closed categories of generalized metric spaces. Our starting point is a quantitative semantics based on a generalization of usual logical relations. Within this setting, we show that several families of generalized metrics provide ways to extend the Euclidean metric to all higher-order types.

cs.LO↗