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44
Compositional design methodology with constraint Markov chains
 in: International Conference on Quantitative Evaluation of Systems, QEST, IEEE Computer Society
"... Notions of specification, implementation, satisfaction, and refinement, together with operators supporting stepwise design, constitute a specification theory. We construct such a theory for Markov Chains (MCs) employing a new abstraction of a Constraint MC. Constraint MCs permit rich constraints on ..."
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Cited by 13 (7 self)
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Notions of specification, implementation, satisfaction, and refinement, together with operators supporting stepwise design, constitute a specification theory. We construct such a theory for Markov Chains (MCs) employing a new abstraction of a Constraint MC. Constraint MCs permit rich constraints on probability distributions and thus generalize prior abstractions such as Interval MCs. Linear (polynomial) constraints suffice for closure under conjunction (respectively parallel composition). This is the first specification theory for MCs with such closure properties. We discuss its relation to simpler operators for known languages such as probabilistic process algebra. Despite the generality, all operators and relations are computable. I.
PROBMELA: a modeling language for communicating probabilistic processes
, 2004
"... Building automated tools to address the analysis of reactive probabilistic systems requires a simple, but expressive input language with a formal semantics based on a probabilistic operational model that can serve as starting point for verification algorithms. We introduce a higher level description ..."
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Cited by 13 (4 self)
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Building automated tools to address the analysis of reactive probabilistic systems requires a simple, but expressive input language with a formal semantics based on a probabilistic operational model that can serve as starting point for verification algorithms. We introduce a higher level description language for probabilistic parallel programs with shared variables, message passing via synchronous and (perfect or lossy) fifo channels and atomic regions and provide a structured operational semantics. Applied to finitestate systems, the semantics can serve as basis for the algorithmic generation of a Markov decision process that models the stepwise behavior of the given system.
On Automated Verification of Probabilistic Programs
"... Abstract. We introduce a simple procedural probabilistic programming language which is suitable for coding a wide variety of randomised algorithms and protocols. This language is interpreted over finite datatypes and has a decidable equivalence problem. We have implemented an automated equivalence c ..."
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Cited by 13 (6 self)
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Abstract. We introduce a simple procedural probabilistic programming language which is suitable for coding a wide variety of randomised algorithms and protocols. This language is interpreted over finite datatypes and has a decidable equivalence problem. We have implemented an automated equivalence checker, which we call apex, for this language, based on game semantics. We illustrate our approach with three nontrivial case studies: (i) Herman’s selfstabilisation algorithm; (ii) an analysis of the average shape of binary search trees obtained by certain sequences of random insertions and deletions; and (iii) the problem of anonymity in the Dining Cryptographers protocol. In particular, we record an exponential speedup in the latter over stateoftheart competing approaches. 1
Game relations and metrics
 In LICS’07
, 2007
"... We consider twoplayer games played over finite state spaces for an infinite number of rounds. At each state, the players simultaneously choose moves; the moves determine a successor state. It is often advantageous for players to choose probability distributions over moves, rather than single moves. ..."
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Cited by 13 (3 self)
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We consider twoplayer games played over finite state spaces for an infinite number of rounds. At each state, the players simultaneously choose moves; the moves determine a successor state. It is often advantageous for players to choose probability distributions over moves, rather than single moves. Given a goal (e.g., “reach a target state”), the question of winning is thus a probabilistic one: “what is the maximal probability of winning from a given state?”. On these game structures, two fundamental notions are those of equivalences and metrics. Given a set of winning conditions, two states are equivalent if the players can win the same games with the same probability from both states. Metrics provide a bound on the difference in the probabilities of winning across states, capturing a quantitative notion of state “similarity”. We introduce equivalences and metrics for twoplayer game structures, and we show that they characterize the difference in probability of winning games whose goals are expressed in the quantitative µcalculus. The quantitative µcalculus can express a large set of goals, including reachability, safety, and ωregular properties. Thus, we claim that our relations and metrics provide the canonical extensions to games, of the classical notion of bisimulation for transition systems. We develop our results both for equivalences and metrics, which generalize bisimulation, and for asymmetrical versions, which generalize simulation.
Testing Semantics for Probabilistic LOTOS
, 1995
"... In this paper we present a probabilistic extension of LOTOS which is upward compatible with LOTOS. We present testing semantics for the reactive and generative models described in [vGSST90]. While there is a certain lose of the meaning of probabilities in the reactive model, testing with probabilist ..."
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Cited by 12 (8 self)
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In this paper we present a probabilistic extension of LOTOS which is upward compatible with LOTOS. We present testing semantics for the reactive and generative models described in [vGSST90]. While there is a certain lose of the meaning of probabilities in the reactive model, testing with probabilistic tests proves to be too strong, because it does not relate behavior expressions which we expect to be equivalent. This is why we introduce the limited generative model, where tests are not allowed to have explicit probabilities. We give a fully abstract characterization for the reactive model, while we give alternative characterizations (based on a set of essential tests) for the generative and limited generative models. We also present some algebraic laws for each of the models, including some laws which establish the difference between the three models.
Logical, Metric, and Algorithmic Characterisations of Probabilistic Bisimulation
, 2011
"... Many behavioural equivalences or preorders for probabilistic processes involve a lifting operation that turns a relation on states into a relation on distributions of states. We show that several existing proposals for lifting relations can be reconciled to be different presentations of essentially ..."
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Cited by 11 (5 self)
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Many behavioural equivalences or preorders for probabilistic processes involve a lifting operation that turns a relation on states into a relation on distributions of states. We show that several existing proposals for lifting relations can be reconciled to be different presentations of essentially the same lifting operation. More interestingly, this lifting operation nicely corresponds to the Kantorovich metric, a fundamental concept used in mathematics to lift a metric on states to a metric on distributions of states, besides the fact the lifting operation is related to the maximum flow problem in optimisation theory. The lifting operation yields a neat notion of probabilistic bisimulation, for which we provide logical, metric, and algorithmic characterisations. Specifically, we extend the HennessyMilner logic and the modal mucalculus with a new modality, resulting in an adequate and an expressive logic for probabilistic bisimilarity, respectively. The correspondence of the lifting operation and the Kantorovich metric leads to a natural characterisation of bisimulations as pseudometrics which are postfixed points of a monotone function. We also present an “on the fly ” algorithm to check if two states in a finitary system are related by probabilistic bisimilarity, exploiting the close relationship
Markovian Processes go Algebra
, 1994
"... We propose a calculus MPA for reasoning about random behaviour through time. In contrast to classical calculi each atomic action is supposed to happen after a delay that is characterized by a certain exponentially distributed random variable. The operational semantics of the calculus defines markovi ..."
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Cited by 8 (4 self)
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We propose a calculus MPA for reasoning about random behaviour through time. In contrast to classical calculi each atomic action is supposed to happen after a delay that is characterized by a certain exponentially distributed random variable. The operational semantics of the calculus defines markovian labelled transition systems as a combination of classical actionoriented transition systems and markovian processes, especially continuous time markov chains. This model allows to calculate performance measures (e.g. response times), as well as purely functional statements (e.g. occurences of deadlocks). In order to reflect different behavioural aspects we define a hierarchy of bisimulation equivalences and show that they are all congruences. Finally we present syntactic laws characterizing markovian bisimulation equivalence, our central notion of equivalence, and show that these laws form a sound and complete axiomatization for finite processes. 1 Introduction In recent years reasoning...
Deriving syntax and axioms for quantitative regular behaviours
, 2009
"... We present a systematic way to generate (1) languages of (generalised) regular expressions, and (2) sound and complete axiomatizations thereof, for a wide variety of quantitative systems. Our quantitative systems include weighted versions of automata and transition systems, in which transitions ar ..."
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Cited by 8 (3 self)
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We present a systematic way to generate (1) languages of (generalised) regular expressions, and (2) sound and complete axiomatizations thereof, for a wide variety of quantitative systems. Our quantitative systems include weighted versions of automata and transition systems, in which transitions are assigned a value in a monoid that represents cost, duration, probability, etc. Such systems are represented as coalgebras and (1) and (2) above are derived in a modular fashion from the underlying (functor) type of these coalgebras. In previous work, we applied a similar approach to a class of systems (without weights) that generalizes both the results of Kleene (on rational languages and DFA’s) and Milner (on regular behaviours and finite LTS’s), and includes many other systems such as Mealy and Moore machines. In the present paper, we extend this framework to deal with quantitative systems. As a consequence, our results now include languages and axiomatizations, both existing and new ones, for many different kinds of probabilistic systems.
A Uniform Framework for Modeling Nondeterministic, Probabilistic, Stochastic, or Mixed Processes and their Behavioral Equivalences
, 2013
"... Labeled transition systems are typically used as behavioral models of concurrent processes. Their labeled transitions define a onestep statetostate reachability relation. This model can be generalized by modifying the transition relation to associate a state reachability distribution with any pai ..."
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Cited by 8 (4 self)
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Labeled transition systems are typically used as behavioral models of concurrent processes. Their labeled transitions define a onestep statetostate reachability relation. This model can be generalized by modifying the transition relation to associate a state reachability distribution with any pair consisting of a source state and a transition label. The state reachability distribution is a function mapping each possible target state to a value that expresses the degree of onestep reachability of that state. Values are taken from a preordered set equipped with a minimum that denotes unreachability. By selecting suitable preordered sets, the resulting model, called ULTraS from Uniform Labeled Transition System, can be specialized to capture wellknown models of fully nondeterministic processes (LTS), fully probabilistic processes (ADTMC), fully stochastic processes (ACTMC), and nondeterministic and probabilistic (MDP) or nondeterministic and stochastic (CTMDP) processes. This uniform treatment of different behavioral models extends to behavioral equivalences. They can be defined on ULTraS by relying on appropriate measure functions that express the degree of reachability of a set of states when performing multistep computations. It is shown that the specializations of bisimulation, trace, and testing equivalences for the different classes of ULTraS coincide with the behavioral equivalences defined in the literature over traditional models except when nondeterminism and probability/stochasticity coexist; then new equivalences pop up.