We observe that equivalence is not a robust concept in the presence of numerical information - such as probabilities - in the model. We develop a metric analogue of weak bisimulation in the spirit of our earlier work on metric analogues for strong bisimulation. We give a fixed point characterization of the metric. This makes available coinductive reasoning principles and allows us to prove metric analogues of the usual algebraic laws for process combinators. We also show that quantitative properties of interest are continuous with respect to the metric, which says that if two processes are close in the metric then observable quantitative properties of interest are indeed close. As an important example of this we show that nearby processes have nearby channel capacities - a quantitative measure of their propensity to leak information.

The notion of process equivalence of probabilistic processes is sensitive to the exact probabilities of transitions. Thus, a slight change in the transition probabilities will result in two equivalent processes being deemed no longer equivalent. This instability is due to the quantitative nature of probabilistic processes. In a situation where the process behaviour has a quantitative aspect there should be a more robust approach to process equivalence. This paper studies a metric between labelled Markov processes. This metric has the property that processes are at zero distance if and only if they are bisimilar. The metric is inspired by earlier work on logics for characterizing bisimulation and is related, in spirit, to the Hutchinson metric.

The notion of process equivalence of probabilistic processes is sensitive to the exact probabilities of transitions. Thus, a slight change in the transition probabilities will result in two equivalent processes being deemed no longer equivalent. This instability is due to the quantitative nature of probabilistic processes. In a situation where the process behaviour has a quantitative aspect there should be a more robust approach to process equivalence. This paper studies a metric between labelled Markov processes. This metric has the property that processes are at zero distance if and only if they are bisimilar. The metric is inspired by earlier work on logics for characterizing bisimulation and is related, in spirit, to the Kantorovich metric.

) Vineet Gupta Radha Jagadeesan Prakash Panangaden y vgupta@mail.arc.nasa.gov radha@cs.luc.edu prakash@cs.mcgill.ca Caelum Research Corporation Dept. of Math. and Computer Sciences School of Computer Science NASA Ames Research Center Loyola University--Lake Shore Campus McGill University Moffett Field CA 94035, USA Chicago IL 60626, USA Montreal, Quebec, Canada Abstract This paper describes a stochastic concurrent constraint language for the description and programming of concurrent probabilistic systems. The language can be viewed both as a calculus for describing and reasoning about stochastic processes and as an executable language for simulating stochastic processes. In this language programs encode probability distributions over (potentially infinite) sets of objects. We illustrate the subtleties that arise from the interaction of constraints, random choice and recursion. We describe operational semantics of these programs (programs are run by sampling random choices), deno...

. Lossy channel systems (LCSs) are models for communicating systems where the subprocesses are linked via unbounded FIFO channels which might lose messages. Link protocols, such as the Alternating Bit Protocol and HDLC can be modelled with these systems. The decidability of several verification problems of LCSs has been investigated by Abdulla & Jonsson [AJ93,AJ94], e.g. they have shown that the reachability problem for LCSs is decidable while LTL model checking is not. In this paper, we consider probabilistic LCSs (which are LCSs where the transitions are augmented with appropriate probabilities) as introduced by [IN97] and show that the question of whether or not a linear time property holds with probability 1 is decidable. More precisely, we show how LTL nX model checking for (certain types of) probabilistic LCSs can be reduced to a reachability problem in a (non-probabilistic) LCS where the latter can be solved with the methods of [AJ93]. 1 1 Introduction Traditiona...

. We extend cc to allow the specification of a discrete probability distribution for random variables. We demonstrate the expressiveness of pcc by synthesizing combinators for default reasoning. We extend pcc uniformly over time, to get a synchronous reactive probabilistic programming language, Timed pcc. We describe operational and denotational models for pcc (and Timed pcc). The key feature of the denotational model(s) is that parallel composition is essentially set intersection. We show that the denotational model of pcc (resp. Timed pcc) is conservative over cc (resp. tcc). We also show that the denotational models are fully abstract for an operational semantics that records probability information. 1 Introduction Concurrent constraint programming(CCP, [Sar93]) is an approach to computation which uses constraints for the compositional specification of concurrent systems. It replaces the traditional notion of a store as a valuation of variables with the notion of a store as a cons...

Abstract There are many applications where the precise time at which an event will occur (or has occurred) is uncertain. Temporal probabilistic logic programs (TPLPs) allow a programmer to express knowledge about such events. In this paper, we develop a model theory, fixpoint theory, and proof theory for TPLPs, and show that the fixpoint theory may be used to enumerate consequences of a TPLP in a sound and complete manner. Likewise the proof theory provides a sound and complete inference system. Last, but not least, we provide complexity results for TPLPs, showing in particular, that reasonable classes of TPLPs have polynomial data complexity. 1 Introduction There are a vast number of applications where uncertainty and time are indelibly intertwined. For example, the US Postal Service (USPS) as well as most commercial shippers have detailed statistics on how long shipments take to reach their destinations. Likewise, we are working on a Viennese historical land deed application where the precise time at which certain properties passed from one owner to another is also highly uncertain. Historical radio carbon dating methods are yet another source of uncertainty, providing approximate information about when a piece was created. Logical reasoning in situations involving temporal uncertainty is definitely important. For example, an individual querying the USPS express mail tracking system may want to know when he can expect his package to be delivered today-- he may then choose to stay home during the period when the probability of delivery seems very high, and leave a note authorizing the delivery official to leave the package by the door at other times.

In this paper, we consider mixed systems containing both stochastic and non-stochastic 1 components. To compose such systems, we introduce a general combinator which allows the specification of an arbitrary mixed system in terms of elementary components of only two types. Thus, systems are obtained hierarchically, by composing subsystems, where each subsystem can be viewed as an "increment" in the decomposition of the full system. The resulting mixed stochastic system specifications are generally not "executable", since they do not necessarily permit the incremental simulation of the system variables. Such a simulation requires compiling the dependency relations existing between the system variables. Another issue involves finding the most likely internal states of a stochastic system from a set of observations. We provide a small set of primitives for transforming mixed systems, which allows the solution of the two problems of incremental simulation and estimation of stocha...

when using the new models. Note that the endogenous approach to probabilistic logic is also useful and, actually, widely used. By endogenous approach we mean that we tinker with the classical models in order to make them suitable for a specific type of probabilistic reasoning. For instance, if we want a logic for reasoning about probabilistic transition systems (probabilistic automata) we can modify the Kripke models of dynamic logic by labelling the transition pairs (pairs of the accessibility relation) with probabilities [12, 15]. As another example of the endogenous approach, consider the probabilization of first-order logic obtained by enriching Later on, the relationship to epistemic logic was made clear in [3, 2]. the domain of individuals with a probability distribution [1], having in mind notions like almost everywhere. Returning to quantum logic, the exogenous approach seems promising for several reasons: (i) it can be applied to any given logic ; (ii) it settles once

We reinterpret the modal ¯-calculus [16] to act over expectations rather than predicates, where expectations generalise predicates by taking states into the interval [0; 1] rather than the two-point set f0; 1g; our interest is in the idioms of the ¯-calculus that correspond to operators of temporal logic, in particular in establishing reasoning principles for them. Our model includes both probabilistic and demonic choice, and the interpretation exploits the characterisation [20] of probabilistic healthiness for expectation transformers to determine the algebraic properties of probabilistic next, and thence those of the other probabilistic temporal operators. The results confirm that many nonprobabilistic temporal axioms have close probabilistic analogues, and that they can indeed be used for quantitative reasoning about probabilistic behaviour. Often the proofs are just generalisations of the original nonprobabilistic versions. In the ¯-calculus more generally the new interpretation ...