Monads and Quantitative Equational Theories for Nondeterminism and Probability

The monad of convex sets of probability distributions is a well-known tool for modelling the combination of nondeterministic and probabilistic computational effects. In this work we lift this monad from the category of sets to the category of extended metric spaces, by means of the Hausdorff and Kantorovich metric liftings. Our main result is the presentation of this lifted monad in terms of the quantitative equational theory of convex semilattices, using the framework of quantitative algebras recently introduced by Mardare, Panangaden and Plotkin

Up-To Techniques for Generalized Bisimulation Metrics

Bisimulation metrics allow us to compute distances between the behaviors of probabilistic systems. In this paper we present enhancements of the proof method based on bisimulation metrics, by extending the theory of up-to techniques to (pre)metrics on discrete probabilistic concurrent processes. Up-to techniques have proved to be a powerful proof method for showing that two systems are bisimilar, since they make it possible to build (and thereby check) smaller relations in bisimulation proofs. We define soundness conditions for up-to techniques on metrics, and study compatibility properties that allow us to safely compose up-to techniques with each other. As an example, we derive the soundness of the up-to-bisimilarity-metric-and-context technique. The study is carried out for a generalized version of the bisimulation metrics, in which the Kantorovich lifting is parametrized with respect to a distance function. The standard bisimulation metrics, as well as metrics aimed at capturing multiplicative properties such as differential privacy, are specific instances of this general definition

The Theory of Traces for Systems with Nondeterminism, Probability, and Termination

This paper studies trace-based equivalences for systems combining nondeterministic and probabilistic choices. We show how trace semantics for such processes can be recovered by instantiating a coalgebraic construction known as the generalised powerset construction. We characterise and compare the resulting semantics to known definitions of trace equivalences appearing in the literature. Most of our results are based on the exciting interplay between monads and their presentations via algebraic theories.Comment: This paper is an extended version of a LICS 2019 paper "The Theory of Traces for Systems with Nondeterminism and Probability". It contains all the proofs, additional explanations, material, and example

Universal Quantitative Algebra for Fuzzy Relations and Generalised Metric Spaces

We present a generalisation of the theory of quantitative algebras of Mardare, Panangaden and Plotkin where (i) the carriers of quantitative algebras are not restricted to be metric spaces and can be arbitrary fuzzy relations or generalised metric spaces, and (ii) the interpretations of the algebraic operations are not required to be nonexpansive. Our main results include: a novel sound and complete proof system, the proof that free quantitative algebras always exist, the proof of strict monadicity of the induced Free-Forgetful adjunction, the result that all monads (on fuzzy relations) that lift finitary monads (on sets) admit a quantitative equational presentation.Comment: Appendix remove

Behavioral Equivalences for Higher-Order Languages with Probabilities

Higher-order languages, whose paradigmatic example is the lambda-calculus, are languages with powerful operators that are capable of manipulating and exchanging programs themselves. This thesis studies behavioral equivalences for programs with higher-order and probabilistic features. Behavioral equivalence is formalized as a contextual, or testing, equivalence, and two main lines of research are pursued in the thesis. The first part of the thesis focuses on contextual equivalence as a way of investigating the expressiveness of different languages. The discriminating powers offered by higher-order concurrent languages (Higher-Order pi-calculi) are compared with those offered by higher-order sequential languages (à la lambda-calculus) and by first-order concurrent languages (à la CCS). The comparison is carried out by examining the contextual equivalences induced by the languages on two classes of first-order processes, namely nondeterministic and probabilistic processes. As a result, the spectrum of the discriminating powers of several varieties of higher-order and first-order languages is obtained, both in a nondeterministic and in a probabilistic setting. The second part of the thesis is devoted to proof techniques for contextual equivalence in probabilistic lambda-calculi. Bisimulation-based proof techniques are studied, with particular focus on deriving bisimulations that are fully abstract for contextual equivalence (i.e., coincide with it). As a first result, full abstraction of applicative bisimilarity and similarity are proved for a call-by-value probabilistic lambda-calculus with a parallel disjunction operator. Applicative bisimulations are however known not to scale to richer languages. Hence, more robust notions of bisimulations for probabilistic calculi are considered, in the form of environmental bisimulations. Environmental bisimulations are defined for pure call-by-name and call-by-value probabilistic lambda-calculi, and for a (call-by-value) probabilistic lambda-calculus extended with references (i.e., a store). In each case, full abstraction results are derived

On Applicative Similarity, Sequentiality, and Full Abstraction

International audienceWe study how applicative bisimilarity behaves when instantiated on a call-by-value probabilistic λ-calculus, endowed with Plotkin's parallel disjunction operator. We prove that congruence and coincidence with the corresponding context relation hold for both bisimilarity and similarity, the latter known to be impossible in sequential languages

The Theory of Traces for Systems with Nondeterminism and Probability

International audienceThis paper studies trace-based equivalences for systems combining nondeterministic and probabilistic choices. We show how trace semantics for such processes can be recovered by instantiating a coalgebraic construction known as the generalised powerset construction. We characterise and compare the resulting semantics to known definitions of trace equivalences appearing in the literature. Most of our results are based on the exciting interplay between monads and their presentations via algebraic theories

Équivalences comportementales pour les langages d’ordre supérieur avec probabilité

Higher-order languages, whose paradigmatic example is the λ-calculus, are languages with powerful operators that are capable of manipulating and exchanging programs themselves. This thesis studies behavioral equivalences for programs with higher-order and probabilistic features. Behavioral equivalence is formalized as a contextual, or testing, equivalence, and two main lines of research are pursued in the thesis.The first part of the thesis focuses on contextual equivalence as a way of investigating the expressiveness of different languages. The discriminating powers offered by higher- order concurrent languages (Higher-Order π-calculi) are compared with those offered by higher-order sequential languages (à la λ-calculus) and by first-order concurrent languages (à la CCS). The comparison is carried out by examining the contextual equivalences induced by the languages on two classes of first-order processes, namely nondeterministic and probabilistic processes. As a result, the spectrum of the discriminating powers of several varieties of higher-order and first-order languages is obtained, both in a nondeterministic and in a probabilistic setting.The second part of the thesis is devoted to proof techniques for contextual equivalence in probabilistic λ-calculi. Bisimulation-based proof techniques are studied, with particular focus on deriving bisimulations that are fully abstract for contextual equivalence (i.e., coincide with it). As a first result, full abstraction of applicative bisimilarity and similarity are proved for a call-by-value probabilistic λ-calculus with a parallel disjunction operator. Applicative bisimulations are however known not to scale to richer languages. Hence, more robust notions of bisimulations for probabilistic calculi are considered, in the form of environmental bisimulations. Environmental bisimulations are defined for pure call- by-name and call-by-value probabilistic λ-calculi, and for a (call-by-value) probabilistic λ-calculus extended with references (i.e., a store). In each case, full abstraction results are derived.Les langages d'ordre supérieur, dont l'exemple paradigmatique est le λ-calcul, sont des langages avec des opérateurs capables de manipuler et d'échanger des programmes. Cette thèse étudie les équivalences comportementales pour des programmes ayant des caractéristiques probabilistes et d'ordre supérieur. L'équivalence comportementale est formalisée comme un contexte et deux axes de recherche sont poursuivis dans la thèse.La première partie de la thèse se concentre sur l'équivalence contextuelle comme moyen d'étudier l'expressivité des différentes langages. Les pouvoirs discriminants offerts par les langages concurrents d'ordre supérieur (comme le π-calcul d'ordre supérieur) sont comparés à ceux offerts par les langages séquentiels d'ordre supérieur (comme le λ-calcul) et par les langages concurrents de premier ordre (comme le CCS). La comparaison est réalisée en examinant les équivalences contextuelles induites par les langages sur deux classes de processus de premier ordre: les processus non déterministes, et les processus probabilistes. En conséquence, on obtient le spectre des pouvoirs discriminants de plusieurs variétés de langages de premier ordre et d'ordre supérieur, dans un cadre non déterministe et dans un cadre probabiliste.La deuxième partie de la thèse est consacrée aux techniques de preuve pour l'équivalence contextuelle dans les λ-calculs probabilistes. Les techniques de preuve basées sur la bisimulation sont étudiées, avec un accent particulier sur la dérivation de bisimulations qui sont complètement abstraites pour l'équivalence contextuelle (c'est-à-dire qui coïncident avec elle). Comme premier résultat, l'abstraction complète de la bisimilarité applicative et de la similarité est prouvée pour un λ-calcul probabiliste en appel par valeur avec un opérateur de disjonction parallèle. Les bisimulations applicatives sont cependant connues pour ne pas être applicables à des langages plus riches. Par conséquent, des notions plus robustes de bisimulation pour les calculs probabilistes sont considérées, sous la forme de bisimulations environnementales. Les bisimulations environnementales sont définies pour des λ-calculs probabilistes en appel par nom et en appel par valeur, et pour un λ-calcul probabiliste (en appel par valeur) avec références. Dans chaque cas, des résultats d'abstraction complets sont prouvés

Beyond Nonexpansive Operations in Quantitative Algebraic Reasoning

International audienc