Abstract. We introduce a symbolic model checking procedure for Probabilistic Computation Tree Logic PCTL over labelled Markov chains as models. Model checking for probabilistic logics typically involves solving linear equation systems in order to ascertain the probability of a given formula holding in a state. Our algorithm is based on the idea of representing the matrices used in the linear equation systems by Multi-Terminal Binary Decision Diagrams (MTBDDs) introduced in Clarke et al [14]. Our procedure, based on the algorithm used by Hansson and Jonsson [24], uses BDDs to represent formulas and MTBDDs to represent Markov chains, and is efficient because it avoids explicit state space construction. A PCTL model checker is being implemented in Verus [9]. 1

We consider two-player games played for an infinite number of rounds, with ω-regular winning conditions. The games may be concurrent, in that the players choose their moves simultaneously and independently, and probabilistic, in that the moves determine a probability distribution for the successor state. We introduce quantitative game µ-calculus, and we show that the maximal probability of winning such games can be expressed as the fixpoint formulas in this calculus. We develop the arguments both for deterministic and for probabilistic concurrent games; as a special case, we solve probabilistic turn-based games with ω-regular winning conditions, which was also open. We also characterize the optimality, and the memory requirements, of the winning strategies. In particular, we show that while memoryless strategies suffice for winning games with safety and reachability conditions, Büchi conditions require the use of strategies with infinite memory. The existence of optimal strategies, as opposed to ε-optimal, is only guaranteed in games with safety winning conditions.

This paper investigates a probabilistic version of the concurrent constraint programming paradigm (CCP). The aim is to introduce the possibility to formulate so called "randomised algorithms" within the CCP framework. Differently from common approaches in (imperative) high-level programming languages, which rely on some kind of random() function, we introduce randomness in the very definition of the language by means of a probabilistic choice construct. This allows a program to make stochastic moves during its execution. We call the resulting language Probabilistic Concurrent Constraint Programming (PCCP). We present an operational semantics for PCCP by means of a probabilistic transition system such that the execution of a PCCP program may be seen as a stochastic process, i.e. as a random walk on the transition graph. The transition probabilities are given explicitly. This semantics captures a notion of observables which combines results of computations and the probability of those re...

Dijkstra's guarded-command language GCL contains explicit `demonic' nondeterminism, representing abstraction from (or ignorance of) which of two program fragments will be executed. We introduce probabilistic nondeterminism to the language, calling the result pGCL. Important is that both forms of nondeterminism are present --- both demonic and probabilistic: unlike earlier approaches, we do not deal only with one or the other. The programming logic of `weakest preconditions' for GCL becomes a logic of `greatest pre-expectations' for pGCL: we embed predicates (Boolean-valued expressions over state variables) into arithmetic by writing [P ], an expression that is 1 when P holds and 0 when it does not. Thus in a trivial sense [P ] is the probability that P is true, and such embedded predicates are the basis for the more elaborate arithmetic expressions that we call "expectations". pGCL is suitable for describing random algorithms, at least over discrete distributions. In our presentation o...

) 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...

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Abstract. Following earlier models, we lift standard deterministic and nondeterministic semantics of imperative programs to probabilistic semantics. This semantics allows for random external inputs of known or unknown probability and random number generators. We then propose a method of analysis of programs according to this semantics, in the general framework of abstract interpretation. This method lifts an “ordinary ” abstract lattice, for non-probabilistic programs, to one suitable for probabilistic programs. Our construction is highly generic. We discuss the influence of certain parameters on the precision of the analysis, basing ourselves on experimental results. 1

. 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...

Generalising Boolean-valued predicates to expectations --- functions from the state space into [0; 1] -- allows the definition of probabilistic temporal operators that treat explicit probabilities as well as demonic nondeterminism and divergence. The conventional operational interpretation of the temporal operators does not generalise so easily: although one may speak of "satisfying a predicate" in the standard case, it is not meaningful to "satisfy an expectation". That difficulty is avoided by giving the operational interpretation of the operators for the probabilistic case in terms of various kinds of gambling game.

We present a method for approximating the semantics of probabilistic programs to the purpose of constructing semantics-based analyses of such programs. The method resembles the one based on Galois connection as developed in the Cousot framework for abstract interpretation. The main difference between our approach and the standard theory of abstract interpretation is the choice of linear space structures instead of order-theoretic ones as semantical (concrete and abstract) domains. We show that our method generates "best approximations" according to an appropriate notion of precision defined in terms of a norm. Moreover, if re-casted in a order-theoretic setting these approximations are correct in the sense of classical abstract interpretation theory. We use Concurrent ...