Jones and Plotkin give a general construction for forming a probabilistic powerdomain over any directed-complete partial order [Jon90, JP89]. We apply their technique to the failures/divergences semantic model for Communicating Sequential Processes [Hoa85]. The resulting probabilistic model supports a new binary operator, probabilistic choice, and retains all operators of CSP including its two existing forms of choice. An advantage of using the general construction is that it is easy to see which CSP identities remain true in the probabilistic model. A surprising consequence however is that probabilistic choice distributes through all other operators; such algebraic mobility means that the syntactic position of the choice operator gives little information about when the choice actually must occur. That in turn leads to some interesting interaction between probability and nondeterminism. A simple communications protocol is used to illustrate the probabilistic algebra, and several sugg...

Published results show that various models may be obtained by combining parallel composition with probability and with or without non-determinism. In this paper we treat this problem in the setting of process algebra in the form of ACP. First, probabilities are introduced by an operator for the internal probabilistic choice. In this way we obtain the Basic Process Algebra with probabilistic choice prBPA.After- wards, prBPA is extended with parallel composition to ACP # .Wegive the axiom system for ACP # and a complete operational semantics that preserves the interleaving model for the dynamic concurrent processes. Considering the PAR protocol, a communication protocol that can be used in the case of unreliable channels, we investigate the applicability of ACP # . Using in addition only the priority operator and the preabstraction operator we obtain a recursive specification of the behaviour of the protocol that can be viewed as a Markov chain. 1

this paper we will define this operator within the framework of BPA " . This is the concurrency theory BPA (Basic Process Algebra) extended with the empty process

Probability, be it inherent or explicitly introduced, has become an important issue in the verification of programs. In this paper we study a formalism which allows reasoning about programs which can act probabilistically. To describe probabilistic programs, a basic programming language with an operator for probabilistic choice is introduced and a denotational semantics is given for this language. To specify properties of probabilistic programs, standard first order logic predicates are insufficient, so a notion of probabilistic predicates is introduced. A Hoare-style proof system to check properties of probabilistic programs is given. The proof system for a sublanguage is shown to be sound and complete; the properties that can be derived are exactly the valid properties. Finally some typical examples illustrate the use of the probabilistic predicates and the proof system.

For a process language with both nondeterministic and probabilistic choice, and a form of failure a transition system is given from which, in a modular way, various operational models corresponding to various interpretations of nondeterminism and probability can be obtained. The effect of failure of one component for the system as a whole is treated differently in each interpretation. The same approach is followed for an extension of the language with a parallel operator. The adopted concurrency model is of a distributed nature and assumes that progress is guaranteed if nonfailing components exist. To this end the notion of a take-over of a failing component is incorporated in the transition system. It is shown that the modular way in which the transition system can yield different semantical models applies to this setting as well.

In this report a comparative semantics is given for a language L p containing probabilistic and non-deterministic choice. The effects of interpreting these operators as local or global are investigated. For three of the possible combinations an operational model and a denotational model are given and compared. First models for local probabilistic choice and local non-deterministic choice are given using a generative approach. By adjusting these models slightly models for global probability and local non-determinism are obtained. Finally models for local probability and global non-determinism are presented using a stratified approach. For use with the denotational models a construction of a complete ultra-metric space of finite multisets is given. 1 Introduction The goal of this paper is to construct comparative semantics for a language combining non-determinism and probabilistic choice. The main interest is the interplay between these two concepts. Since many of the interesting proper...

In the paper we present an ACP-like process algebra which can be used to model both probabilistic and time behaviour of parallel systems. This process algebra is obtained by extension of untimed probabilistic process algebra with constructors that allow the explicit specification of timing aspects. In this paper we concentrate on giving axioms and deduction rules for these constructors. We give two probabilistic process algebras with discrete time. The first one only manipulates with processes that may be initialized within the current time slice or may delay a finite and fixed number of time slices. Later, we add processes whose execution can be postponed for an arbitrary number of time slices.

Process algebras are standard formalisms for compositionally describing systems by the dependencies of their observable synchronous communication. In concurrent systems, parallel composition introduces resolvable nondeterminism, i.e., nondeterminism that will be resolved in later design phases or by the operating system. Sometimes it is also important to express inherent nondeterminism for equal (communication) labels. Here, we give operational and axiomatic semantics to a process algebra having a parallel operator interpreted as concurrent and having a choice operator interpreted as inherent, not only w.r.t. different, but also w.r.t. equal next-step actions. In order to handle the different kinds of nondeterminism, the operational semantics uses µ-automata as underlying semantical model. Soundness and completeness of our axiom system w.r.t. the operational semantics is shown.

The ø-angelic choice is an operator that captures the behaviour of the external choice of CSP in a branching time setting. The idea of the ø-angelic choice is to delay any choice until an observable action happens. In this way, this new operator avoids preemption introduced by internal actions (ø actions). It is studied in theories with abstraction, more precisely, branching bisimulation and ø-bisimulation. In addition, an ilustrative example of application is given. Keywords: theory of concurrency, semantics of reactive systems, process algebra, nondeterminism, specification and verification methodologies 1 Introduction Process algebras [BW90] aim to deal with Semantic of Reactive Systems [Pnu85]. These systems, whose classical examples are industrial control systems, drivers of computer networks, components of operating systems or protocols of comunication, are mainly characterized by the fact that they can be described as maintaining an interaction with their environment. It is no...

The ø-angelic choice is an operator that captures the behaviour of the external choice of CSP in a branching time setting. The idea of the ø-angelic choice is to delay any choice until an observable action happens. In this way, this new operator avoids preemption introduced by internal actions (ø actions). It is studied in theories with abstraction, more precisely, branching bisimulation and ø-bisimulation and failure semantics. In addition, an ilustrative example of application is given. Keywords: theory of concurrency, semantics of reactive systems, process algebra, nondeterminism, specification and verification methodologies 1 Introduction The aim of this article is to introduce a new choice operator for process algebras [BW90] in order to deal with preemptive contexts. This operator is called ø-angelic choice and denoted by 2. The ø-angelic choice is strongly based in the external choice of CSP [BHR84, Hoa85] which was presented in a failure model. I showed in [D'A94] that extern...