Probabilistic Choice Operator Research Articles

Asynchronous Probabilistic Couplings in Higher-Order Separation Logic

Probabilistic couplings are the foundation for many probabilistic relational program logics and arise when relating random sampling statements across two programs. In relational program logics, this manifests as dedicated coupling rules that, e.g., say we may reason as if two sampling statements return the same value. However, this approach fundamentally requires aligning or "synchronizing" the sampling statements of the two programs which is not always possible. In this paper, we develop Clutch, a higher-order probabilistic relational separation logic that addresses this issue by supporting asynchronous probabilistic couplings. We use Clutch to develop a logical step-indexed logical relation to reason about contextual refinement and equivalence of higher-order programs written in a rich language with a probabilistic choice operator, higher-order local state, and impredicative polymorphism. Finally, we demonstrate our approach on a number of case studies. All the results that appear in the paper have been formalized in the Coq proof assistant using the Coquelicot library and the Iris separation logic framework.

Symbolic protocol verification with dice1

Symbolic protocol verification generally abstracts probabilities away, considering computations that succeed only with negligible probability, such as guessing random numbers or breaking an encryption scheme, as impossible. This abstraction, sometimes referred to as the perfect cryptography assumption, has shown very useful as it simplifies automation of the analysis. However, probabilities may also appear in the control flow where they are generally not negligible. In this paper we consider a framework for symbolic protocol analysis with a probabilistic choice operator: the probabilistic applied π-calculus. We define and explore the relationships between several behavioral equivalences. In particular we show the need for randomized schedulers and exhibit a counter-example to a result in a previous work that relied on non-randomized ones. As in other frameworks that mix both non-deterministic and probabilistic choices, schedulers may sometimes be unrealistically powerful. We therefore consider two subclasses of processes that avoid this problem. In particular, when considering purely non-deterministic protocols, as is done in classical symbolic verification, we show that a probabilistic adversary has – maybe surprisingly – a strictly superior distinguishing power for may testing, which, when the number of sessions is bounded, we show to coincide with purely possibilistic similarity.

A Gradual Probabilistic Lambda Calculus

Probabilistic programming languages have recently gained a lot of attention, in particular due to their applications in domains such as machine learning and differential privacy. To establish invariants of interest, many such languages include some form of static checking in the form of type systems. However, adopting such a type discipline can be cumbersome or overly conservative. Gradual typing addresses this problem by supporting a smooth transition between static and dynamic checking, and has been successfully applied for languages with different constructs and type abstractions. Nevertheless, its benefits have never been explored in the context of probabilistic languages. In this work, we present and formalize GPLC, a gradual source probabilistic lambda calculus. GPLC includes a binary probabilistic choice operator and allows programmers to gradually introduce/remove static type–and probability–annotations. The static semantics of GPLC heavily relies on the notion of probabilistic couplings, as required for defining several relations, such as consistency, precision, and consistent transitivity. The dynamic semantics of GPLC is given via elaboration to the target language TPLC, which features a distribution-based semantics interpreting programs as probability distributions over final values. Regarding the language metatheory, we establish that TPLC–and therefore also GPLC–is type safe and satisfies two of the so-called refined criteria for gradual languages, namely, that it is a conservative extension of a fully static variant and that it satisfies the gradual guarantee, behaving smoothly with respect to type precision.

DEGRADATION ANALYSIS OF PROBABILISTIC PARALLEL CHOICE SYSTEMS

Degradation analysis is used to analyze the useful lifetimes of systems, their failure rates, and various other system parameters like mean time to failure (MTTF), mean time between failures (MTBF), and the system failure rate (SFR). In many systems, certain possible parallel paths of execution that have greater chances of success are preferred over others. Thus we introduce here the concept of probabilistic parallel choice. We use binary and n-ary probabilistic choice operators in describing the selections of parallel paths. These binary and n-ary probabilistic choice operators are considered so as to represent the complete system (described as a series-parallel system) in terms of the probabilities of selection of parallel paths and their relevant parameters. Our approach allows us to derive new and generalized formulae for system parameters like MTTF, MTBF, and SFR. We use a generalized exponential distribution, allowing distinct installation times for individual components, and use this model to derive expressions for such system parameters.

Probabilistic Mobile Ambients

The calculus of Mobile Ambients has been introduced for expressing mobility and mobile computation. In this paper we present a probabilistic version of Mobile Ambients by augmenting the syntax of the original Ambient Calculus with a (guarded) probabilistic choice operator. To allow for the representation of both the probabilistic behaviour introduced through the new probabilistic choice operator and the nondeterminism present in the original Ambient Calculus we use probabilistic automata as the underpinning semantic model. The Ambient logic is a logic for Mobile Ambients that contains a novel treatment of both locations and hidden names. For specifying properties of Probabilistic Mobile Ambients, we extend this logic to specify probabilistic behaviour. In addition, to show the utility of our approach we present an example of a virus infecting a network.

Discrete random variables over domains

In this paper we initiate the study of discrete random variables over domains. Our work is inspired by that of Daniele Varacca, who devised indexed valuations as models of probabilistic computation within domain theory. Our approach relies on new results about commutative monoids defined on domains that also allow actions of the non-negative reals. Using our approach, we define two such families of real domain monoids, one of which allows us to recapture Varacca’s construction of the Plotkin indexed valuations over a domain. Each of these families leads to the construction of a family of discrete random variables over domains, the second of which forms the object level of a continuous endofunctor on the categories RB (domains that are retracts of bifinite domains), and on FS (domains where the identity map is the directed supremum of deflations finitely separated from the identity). The significance of this last result lies in the fact that there is no known category of continuous domains that is closed under the probabilistic power domain, which forms the standard approach to modelling probabilistic choice over domains. The fact that RB and FS are Cartesian closed and also are closed under a power domain of discrete random variables means we can now model, e.g. the untyped lambda calculus extended with a probabilistic choice operator, implemented via random variables.

A translation of [formula omitted] into a class of timed-probabilistic Petri nets

TPAL p is an algebraic language for the description of concurrent systems with capabilities to express timed and probabilistic behaviours, as well as urgent interactions. In this paper we present the main features of the language, its operational semantics, and a translation of TPAL p terms into a particular class of timed-probabilistic Petri nets. The language includes a probabilistic choice operator, a timed prefix operator, and an urgent prefix operator, as well as some other operators that we may find in classical process algebras. An important feature of the language is that urgency is considered at any instant by executing as many urgent actions as possible, with the goal of complying in a great extent with the urgent actions indicated in the user specifications.

Domain equations for probabilistic processes

In this paper we consider Milner's calculus CCS enriched by a probabilistic choice operator. The calculus is given operational semantics based on probabilistic transition systems. We define operational notions of preorder and equivalence as probabilistic extensions of the simulation preorder and the bisimulation equivalence respectively. We extend existing category-theoretic techniques for solving domain equations to the probabilistic case and give two denotational semantics for the calculus. The first, ‘smooth’, semantic model arises as a solution of a domain equation involving the probabilistic powerdomain and solved in the category CONT⊥ of continuous domains. The second model also involves an appropriately restricted probabilistic powerdomain, but is constructed in the category CUM of complete ultra-metric spaces, and hence is necessarily ‘discrete’. We show that the domain-theoretic semantics is fully abstract with respect to the simulation preorder, and that the metric semantics is fully abstract with respect to bisimulation.

Domain Equations for Probabilistic Processes (Extended Abstract)

In this paper we consider Milner's calculus CCS enriched by a probabilistic choice operator. The calculus is given operational semantics based on probabilistic transition systems. We define operational notions of preorder and equivalence as probabilistic extensions of the simulation preorder and the bisimulation equivalence respectively. We extend existing category-theoretic techniques for solving domain equations to the probabilistic case and give two denotational semantics for the calculus. The first, “smooth”, semantic model arises as a solution of a domain equation involving the probabilistic powerdomain and solved in the category CONT⊥ of continuous domains. The second model also involves appropriately restricted probabilistic powerdomain, but is constructed in the category CUM of complete ultra-metric spaces, and hence is necessarily “discrete”. We show that the domain-theoretic semantics is fully abstract with respect to the simulation preorder, and that the metric semantics is fully abstract with respect to bisimulation.

Probabilistic and prioritized models of timed CSP

In this paper we present two languages that are refinements of timed CSP (Davies and Schneider, this volume): a probabilistic language, and a fully deterministic language with a notion of priority. In the first part of the paper we describe the deterministic language and its semantic model. The syntax is based upon that of timed CSP except some of the operators are refined so as to remove all nondeterminism; this produces prioritized operators. The semantics for our language represents a process as the set of possible behaviours for the process, where a behaviour models the priorities for different actions. A number of algebraic laws for our language are given and the model is illustrated with two examples. In the second part of the paper, we extend the language by adding a probabilistic choice operator. We produce a semantic model for our language which gives the probabilities of different behaviours occurring, as well as modelling the relative priorities for events within a behaviour. The model is illustrated with an example of a communications protocol transmitting messages over an unreliable medium.