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Dynamic Logic
 Handbook of Philosophical Logic
, 1984
"... ed to be true under the valuation u iff there exists an a 2 N such that the formula x = y is true under the valuation u[x=a], where u[x=a] agrees with u everywhere except x, on which it takes the value a. This definition involves a metalogical operation that produces u[x=a] from u for all possibl ..."
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Cited by 826 (7 self)
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ed to be true under the valuation u iff there exists an a 2 N such that the formula x = y is true under the valuation u[x=a], where u[x=a] agrees with u everywhere except x, on which it takes the value a. This definition involves a metalogical operation that produces u[x=a] from u for all possible values a 2 N. This operation becomes explicit in DL in the form of the program x := ?, called a nondeterministic or wildcard assignment. This is a rather unconventional program, since it is not effective; however, it is quite useful as a descriptive tool. A more conventional way to obtain a square root of y, if it exists, would be the program x := 0 ; while x < y do x := x + 1: (1) In DL, such programs are firstclass objects on a par with formulas, complete with a collection of operators for forming compound programs inductively from a basis of primitive programs. To discuss the effect of the execution of a program on the truth of a formula ', DL uses a modal construct <>', which
A Logic for Reasoning about Time and Reliability
 Formal Aspects of Computing
, 1994
"... We present a logic for stating properties such as, "after a request for service there is at least a 98% probability that the service will be carried out within 2 seconds". The logic extends the temporal logic CTL by Emerson, Clarke and Sistla with time and probabilities. Formulas are inter ..."
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Cited by 256 (1 self)
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We present a logic for stating properties such as, "after a request for service there is at least a 98% probability that the service will be carried out within 2 seconds". The logic extends the temporal logic CTL by Emerson, Clarke and Sistla with time and probabilities. Formulas are interpreted over discrete time Markov chains. We give algorithms for checking that a given Markov chain satisfies a formula in the logic. The algorithms require a polynomial number of arithmetic operations, in size of both the formula and This research report is a revised and extended version of a paper that has appeared under the title "A Framework for Reasoning about Time and Reliability" in the Proceeding of the 10 th IEEE Realtime Systems Symposium, Santa Monica CA, December 1989. This work was partially supported by the Swedish Board for Technical Development (STU) as part of Esprit BRA Project SPEC, and by the Swedish Telecommunication Administration. the Markov chain. A simple example is inc...
PROBABILISTIC PREDICATE TRANSFORMERS
, 1995
"... Predicate transformers facilitate reasoning about imperative programs, including those exhibiting demonic nondeterministic choice. Probabilistic predicate transformers extend that facility to programs containing probabilistic choice, so that one can in principle determine not only whether a program ..."
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Cited by 109 (32 self)
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Predicate transformers facilitate reasoning about imperative programs, including those exhibiting demonic nondeterministic choice. Probabilistic predicate transformers extend that facility to programs containing probabilistic choice, so that one can in principle determine not only whether a program is guaranteed to establish a certain result, but also its probability of doing so. We bring together independent work of Claire Jones and Jifeng He, showing how their constructions can be made to correspond � from that link between a predicatebased and a relationbased view of probabilistic execution we are able to propose `probabilistic healthiness conditions', generalising those of Dijkstra for ordinary predicate transformers. The associated calculus seems suitable for exploring further the rigorous derivation of imperative probabilistic programs.
Symbolic model checking for probabilistic processes
 IN PROCEEDINGS OF ICALP '97
, 1997
"... 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 stat ..."
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Cited by 84 (29 self)
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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 MultiTerminal 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].
On probabilistic model checking
, 1996
"... Abstract. This tutorial presents an overview of model checking for both discrete and continuoustime Markov chains (DTMCs and CTMCs). Model checking algorithms are given for verifying DTMCs and CTMCs against specifications written in probabilistic extensions of temporal logic, including quantitative ..."
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Cited by 64 (10 self)
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Abstract. This tutorial presents an overview of model checking for both discrete and continuoustime Markov chains (DTMCs and CTMCs). Model checking algorithms are given for verifying DTMCs and CTMCs against specifications written in probabilistic extensions of temporal logic, including quantitative properties with rewards. Example properties include the probability that a fault occurs and the expected number of faults in a given time period. We also describe the practical application of stochastic model checking with the probabilistic model checker PRISM by outlining the main features supported by PRISM and three realworld case studies: a probabilistic security protocol, dynamic power management and a biological pathway. 1
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 55 (20 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.
Model Checking for Probability and Time: From Theory to Practice
 In Proc. Logic in Computer Science
, 2003
"... Probability features increasingly often in software and hardware systems: it is used in distributed coordination and routing problems, to model faulttolerance and performance, and to provide adaptive resource management strategies. Probabilistic model checking is an automatic procedure for establi ..."
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Cited by 48 (1 self)
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Probability features increasingly often in software and hardware systems: it is used in distributed coordination and routing problems, to model faulttolerance and performance, and to provide adaptive resource management strategies. Probabilistic model checking is an automatic procedure for establishing if a desired property holds in a probabilistic model, aimed at verifying probabilistic specifications such as "leader election is eventually resolved with probability 1", "the chance of shutdown occurring is at most 0.01%", and "the probability that a message will be delivered within 30ms is at least 0.75". A probabilistic model checker calculates the probability of a given temporal logic property being satisfied, as opposed to validity. In contrast to conventional model checkers, which rely on reachability analysis of the underlying transition system graph, probabilistic model checking additionally involves numerical solutions of linear equations and linear programming problems. This paper reports our experience with implementing PRISM (www.cs.bham.ac.uk/dxp/ prism/), a Probabilistic Symbolic Model Checker, demonstrates its usefulness in analysing realworld probabilistic protocols, and outlines future challenges for this research direction.
Concurrent Reachability Games
, 2008
"... We consider concurrent twoplayer games with reachability objectives. In such games, at each round, player 1 and player 2 independently and simultaneously choose moves, and the two choices determine the next state of the game. The objective of player 1 is to reach a set of target states; the objecti ..."
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Cited by 44 (18 self)
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We consider concurrent twoplayer games with reachability objectives. In such games, at each round, player 1 and player 2 independently and simultaneously choose moves, and the two choices determine the next state of the game. The objective of player 1 is to reach a set of target states; the objective of player 2 is to prevent this. These are zerosum games, and the reachability objective is one of the most basic objectives: determining the set of states from which player 1 can win the game is a fundamental problem in control theory and system verification. There are three types of winning states, according to the degree of certainty with which player 1 can reach the target. From type1 states, player 1 has a deterministic strategy to always reach the target. From type2 states, player 1 has a randomized strategy to reach the target with probability 1. From type3 states, player 1 has for every real ε> 0 a randomized strategy to reach the target with probability greater than 1 − ε. We show that for finite state spaces, all three sets of winning states can be computed in polynomial time: type1 states in linear time, and type2 and type3 states in quadratic time. The algorithms to compute the three sets of winning states also enable the construction of the winning and spoiling strategies.
Recursive Markov decision processes and recursive stochastic games
 In Proc. of 32nd Int. Coll. on Automata, Languages, and Programming (ICALP’05
, 2005
"... Abstract. We introduce Recursive Markov Decision Processes (RMDPs) and Recursive Simple Stochastic Games (RSSGs), and study the decidability and complexity of algorithms for their analysis and verification. These models extend Recursive Markov Chains (RMCs), introduced in [EY05a,EY05b] as a natural ..."
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Cited by 35 (9 self)
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Abstract. We introduce Recursive Markov Decision Processes (RMDPs) and Recursive Simple Stochastic Games (RSSGs), and study the decidability and complexity of algorithms for their analysis and verification. These models extend Recursive Markov Chains (RMCs), introduced in [EY05a,EY05b] as a natural model for verification of probabilistic procedural programs and related systems involving both recursion and probabilistic behavior. RMCs define a class of denumerable Markov chains with a rich theory generalizing that of stochastic contextfree grammars and multitype branching processes, and they are also intimately related to probabilistic pushdown systems. RMDPs & RSSGs extend RMCs with one controller or two adversarial players, respectively. Such extensions are useful for modeling nondeterministic and concurrent behavior, as well as modeling a system’s interactions with an environment. We provide a number of upper and lower bounds for deciding, given an RMDP (or RSSG) A and probability p, whether player 1 has a strategy to force termination at a desired exit with probability at least p. We also address “qualitative ” termination questions, where p = 1, and model checking questions. 1
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.