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PRISM: Probabilistic symbolic model checker
, 2002
"... Abstract. In this paper we describe PRISM, a tool being developed at the University of Birmingham for the analysis of probabilistic systems. PRISM supports two probabilistic models: continuoustime Markov chains and Markov decision processes. Analysis is performed through model checking such systems ..."
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Cited by 184 (15 self)
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Abstract. In this paper we describe PRISM, a tool being developed at the University of Birmingham for the analysis of probabilistic systems. PRISM supports two probabilistic models: continuoustime Markov chains and Markov decision processes. Analysis is performed through model checking such systems against specifications written in the probabilistic temporal logics PCTL and CSL. The tool features three model checking engines: one symbolic, using BDDs (binary decision diagrams) and MTBDDs (multiterminal BDDs); one based on sparse matrices; and one which combines both symbolic and sparse matrix methods. PRISM has been successfully used to analyse probabilistic termination, performance, dependability and quality of service properties for a range of systems, including randomized distributed algorithms, polling systems, workstation cluster and wireless cell communication. 1
Probabilistic Symbolic Model Checking with PRISM: A Hybrid Approach
 International Journal on Software Tools for Technology Transfer (STTT
, 2002
"... In this paper we introduce PRISM, a probabilistic model checker, and describe the ecient symbolic techniques we have developed during its implementation. PRISM is a tool for analysing probabilistic systems. It supports three models: discretetime Markov chains, continuoustime Markov chains and ..."
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Cited by 137 (26 self)
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In this paper we introduce PRISM, a probabilistic model checker, and describe the ecient symbolic techniques we have developed during its implementation. PRISM is a tool for analysing probabilistic systems. It supports three models: discretetime Markov chains, continuoustime Markov chains and Markov decision processes. Analysis is performed through model checking speci cations in the probabilistic temporal logics PCTL and CSL. Motivated by the success of model checkers such as SMV, which use BDDs (binary decision diagrams), we have developed an implementation of PCTL and CSL model checking based on MTBDDs (multiterminal BDDs) and BDDs. Existing work in this direction has been hindered by the generally poor performance of MTBDDbased numerical computation, which is often substantially slower than explicit methods using sparse matrices. We present a novel hybrid technique which combines aspects of symbolic and explicit approaches to overcome these performance problems. For typical examples, we achieve orders of magnitude speedup compared to MTBDDs and are able to almost match the speed of sparse matrices whilst maintaining considerable space savings.
Automatic verification of realtime systems with discrete probability distributions
 Theoretical Computer Science
, 1999
"... Abstract. We consider the timed automata model of [3], which allows the analysis of realtime systems expressed in terms of quantitative timing constraints. Traditional approaches to realtime system description express the model purely in terms of nondeterminism; however, we may wish to express the ..."
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Cited by 72 (27 self)
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Abstract. We consider the timed automata model of [3], which allows the analysis of realtime systems expressed in terms of quantitative timing constraints. Traditional approaches to realtime system description express the model purely in terms of nondeterminism; however, we may wish to express the likelihood of the system making certain transitions. In this paper, we present a model for realtime systems augmented with discrete probability distributions. Furthermore, using the algorithm of [5] with fairness, we develop a model checking method for such models against temporal logic properties which can refer both to timing properties and probabilities, such as, “with probability 0.6 or greater, the clock x remains below 5 until clock y exceeds 2”. 1
Model checking continuoustime Markov chains by transient analysis
, 2000
"... . The verification of continuoustime Markov chains (CTMCs) against continuous stochastic logic (CSL) [3, 6], a stochastic branchingtime temporal logic, is considered. CSL facilitates among others the specification of steadystate properties and the specification of probabilistic timing properties o ..."
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Cited by 69 (17 self)
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. The verification of continuoustime Markov chains (CTMCs) against continuous stochastic logic (CSL) [3, 6], a stochastic branchingtime temporal logic, is considered. CSL facilitates among others the specification of steadystate properties and the specification of probabilistic timing properties of the form P# #p(#1 U I #2 ), for state formulas #1 and #2 , comparison operator ##, probability p, and real interval I. The main result of this paper is that model checking probabilistic timing properties can be reduced to the problem of computing transient state probabilities for CTMCs. This allows us to verify such properties by using e#cient techniques for transient analysis of CTMCs such as uniformisation. A second result is that a variant of ordinary lumping equivalence (i.e., bisimulation), a wellknown notion for aggregating CTMCs, preserves the validity of all CSLformulas. In 12th Annual Symposium on Computer Aided Verification, CAV 2000, c # SpringerVerlag 2000 Chicago,...
Implementation of Symbolic Model Checking for Probabilistic Systems
, 2002
"... In this thesis, we present ecient implementation techniques for probabilistic model checking, a method which can be used to analyse probabilistic systems such as randomised distributed algorithms, faulttolerant processes and communication networks. A probabilistic model checker inputs a probabilist ..."
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Cited by 50 (18 self)
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In this thesis, we present ecient implementation techniques for probabilistic model checking, a method which can be used to analyse probabilistic systems such as randomised distributed algorithms, faulttolerant processes and communication networks. A probabilistic model checker inputs a probabilistic model and a speci cation, such as \the message will be delivered with probability 1", \the probability of shutdown occurring is at most 0.02" or \the probability of a leader being elected within 5 rounds is at least 0.98", and can automatically verify if the speci cation is true in the model.
Symbolic Model Checking of Concurrent Probabilistic Systems Using MTBDDs
, 2000
"... Symbolic model checking for purely probabilistic processes using MTBDDs [12] was introduced in [4] and further developed in [20, 3]. In this paper we consider models for concurrent probabilistic systems similar to those of [28, 7, 5] and the concurrent Markov chains of [35, 13], which extend the ..."
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Cited by 35 (16 self)
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Symbolic model checking for purely probabilistic processes using MTBDDs [12] was introduced in [4] and further developed in [20, 3]. In this paper we consider models for concurrent probabilistic systems similar to those of [28, 7, 5] and the concurrent Markov chains of [35, 13], which extend the purely probabilistic processes through the addition of nondeterministic choice. As a specification formalism we use the probabilistic branchingtime temporal logic PBTL of [5, 7], which allows us to express properties such as "under any scheduling of nondeterministic choices, the probability of OE holding until / is true is at least 0.78". In [5, 7] it is shown that the verification of "until" properties can be reduced to a linear programming problem and solved with the help of e.g. the simplex algorithm, but no symbolic model checking is considered. Based on the algorithms of [5, 7], we derive symbolic model checking procedure for PBTL over concurrent probabilistic systems using MTBDDs, and extend it with fairness constraints. We furthermore implement an experimental model checker using the Colorado University Decision Diagram (CUDD) package [32]. Our key contribution is an implementation of the simplex algorithm in terms of MTBDDs.
Symbolic model checking for probabilistic processes using MTBDDs and the Kronecker representation
 In Tools and Algorithms for the Analysis and Construction of Systems, LNCS 1785
, 2000
"... Abstract. This paper reports on experimental results with symbolic model checking of probabilistic processes based on MultiTerminal Binary Decision Diagrams (MTBDDs). We consider concurrent probabilistic systems as models; these allow nondeterministic choice between probability distributions and ar ..."
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Cited by 24 (2 self)
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Abstract. This paper reports on experimental results with symbolic model checking of probabilistic processes based on MultiTerminal Binary Decision Diagrams (MTBDDs). We consider concurrent probabilistic systems as models; these allow nondeterministic choice between probability distributions and are particularly well suited to modelling distributed systems with probabilistic behaviour, e.g. randomized consensus algorithms and probabilistic failures. As a specification formalism we use the probabilistic branchingtime temporal logic PBTL which allows one to express properties such as “under any scheduling of nondeterministic choices, the probability of φ holding until ψ is true is at least 0.78/at most 0.04 ”. We adapt the Kronecker representation of (Plateau 1985), which yields a very compact MTBDD encoding of the system. We implement an experimental model checker using the CUDD package and demonstrate that model construction and reachabilitybased model checking is possible in a matter of seconds for certain classes of systems consisting of up to 10 30 states. 1
Symbolic Representations and Analysis of Large Probabilistic Systems
 In Validation of Stochastic Systems
, 2004
"... Abstract. This paper describes symbolic techniques for the construction, representation and analysis of large, probabilistic systems. Symbolic approaches derive their efficiency by exploiting highlevel structure and regularity in the models to which they are applied, increasing the size of the stat ..."
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Cited by 15 (2 self)
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Abstract. This paper describes symbolic techniques for the construction, representation and analysis of large, probabilistic systems. Symbolic approaches derive their efficiency by exploiting highlevel structure and regularity in the models to which they are applied, increasing the size of the state spaces which can be tackled. In general, this is done by using data structures which provide compact storage but which are still efficient to manipulate, usually based on binary decision diagrams (BDDs) or their extensions. In this paper we focus on BDDs, multivalued decision diagrams (MDDs), multiterminal binary decision diagrams (MTBDDs) and matrix diagrams. 1
Symbolic Representations and Analysis of Large State Spaces
 In this Proceedings
, 2003
"... This paper describes symbolic techniques for the construction, representation and analysis of large, probabilistic systems. Symbolic approaches derive their eciency by exploiting highlevel structure and regularity in the models to which they are applied, increasing the size of the state spaces ..."
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Cited by 3 (1 self)
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This paper describes symbolic techniques for the construction, representation and analysis of large, probabilistic systems. Symbolic approaches derive their eciency by exploiting highlevel structure and regularity in the models to which they are applied, increasing the size of the state spaces which can be tackled. In general, this is done by using data structures which provide compact storage but which are still e cient to manipulate, usually based on binary decision diagrams (BDDs) or their extensions. In this paper we focus on BDDs, multivalued decision diagrams (MDDs), multiterminal binary decision diagrams (MTBDDs) and matrix diagrams.