D. Lehmann and S. Shelah. Reasoning with time and chance. Information and Control 53, pp. 165-198, 1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to prevent corrupt routers from correlating multiple paths originating from the same sender. 2 Markov Chain Model Checking We model the probabilistic behavior of a peer to peer communication system as a discrete time Markov chain (DTMC) which is a standard approach in probabilistic verification [LS82, HS84, Var85, HJ94]. Formally, a Markov chain can be defined as consisting in a finite set of states S, the initial state s 0 , the transition relation T : S S [0; 1] such that 8s 2 S P s 2S T (s; s ) 1, and a labeling function from states to a finite set of propositions L : S 2 AP . In our model, ....

D. Lehmann and S. Shelah. Reasoning with time and chance. Information and Control, 53(3):165--198, 1982.

....of [BE99] Below we assume a given Markov chain M = #. The following useful lemma states that if # 0 # n and # 0 is visited infinitely often, then almost surely # n is visited infinitely often. Lemma 2.5. If P (# i 1 , # i ) 0 for i = 1, n, then M (### 0 ### n ) 1. See e.g. [LS82] for a proof. Definition 2.6 (Attractors) A non empty set W a W of configurations is an attractor when ##W a ) 1 for all # W (2) The attractor is finite when W a is. Assume W a W is a finite attractor. We define GM (W a ) as the finite directed graph a , ## where the ....

D. Lehmann and S. Shelah. Reasoning with time and chance. Information and Control, 53:165--198, 1982.

....was uncovered by automated model checking and is reported for the first time in this paper. 2. Markov Chain Model Checking We model the probabilistic behavior of a peer to peer communication system as a discrete time Markov chain (DTMC) which is a standard approach in probabilistic verification [15, 13, 27, 12]. Formally, a Markov chain can be defined as consisting in a finite set of states S, the initial state s 0 , the transition relation T : S S [0; 1] such that 8s 2 S P s 2S T (s; s ) 1, and a labeling function from states to a finite set of propositions L : S 2 AP . In our model, ....

D. Lehmann and S. Shelah. Reasoning with time and chance. Information and Control, 53(3):165--198, 1982.

....logics. Predicate logics have known advantages for representing the veri cation problem due to their expressiveness and simplicity of formalizing of complicated properties. Logics with probabilities were considered before in the context of arti cial intelligence and veri cation. In early paper [LS82] there are modal operators that allow one to say with probability one or with probability greater that zero . Most work related to the veri cation used probalilistic expensions of temporal logics; in these extensions the probabilities can be used in very constraint ways. This work is surveyed, ....

....question is whether one can replace almost always decidable by decidable . Below some extensions of our results are described. A. Probabilities 0 and 1. Probabilities 0 and 1 play an important role in many questions related to speci cation and veri cation. Some probability logics, e.g. LS82] consider only probabilistic operators Prob =0 and Prob =1 . Theorem 4 can be strengthened as follows Theorem 7 Given a Finite Probabilistic Process M , a state s 0 of M and a parametrized completely closed formula in the class C with m parameters, one can compute for each parameter p i in ....

D. Lehmann and S. Shelah. Reasoning about time and chance. Information and Control, 53(3):165-198, 1982.

....which may or may not exhibit non determinism as well as probabilistic choice. Typically, the verification aims to establish qualitative properties, i.e. properties that are fulfilled by almost all executions, which amounts to showing that the property is satisfied with probability 1, see e.g. [2,3,19,21,32,33,43,49 51,58,59]. Although the above requirement of probability 1 is important in many cases, for some properties it is simply not the case that they are satisfied with probability 1, but instead with probability 1 # for some suitable # (an error) These quantitative properties, which are the focus of this ....

Lehmann D, Shelah S: Reasoning with Time and Chance, Information and Control 53: 165--198 (1982)

....to induce a natural probability distribution on this set of points. Let S fut denote the sample space assignment that assigns Pref i;c to p i at c, and let P fut denote the probability assignment induced by S fut . We remark that this is the probability assignment used in [HMT88] as well as [LS82]. In the probability space P fut i;c , any event that has already happened by the point c will have probability 1. Future events (that get decided further down the computation tree) still have nontrivial probabilities, which is why we have termed it a future probability assignment. Let us ....

D. Lehmann and S. Shelah. Reasoning about time and chance. Information and Control, 53:165--198, 1982.

....certain transitions non zero probability) our simple program succeeds in meeting its specification with probability one. There has been considerable work on this sort of verification i.e. testing if a given program and system will meet a given specification with probability 1 e.g. ACD91, HS83, LS83, HS84] Essentially they work with logics which reason about probabilistic systems themselves. In [LS83] a logic containing a straightforward logic of linear time has a new modality added to it. This new modality is read certainly and indicates that its argument happens with probability 1. ....

....probability one. There has been considerable work on this sort of verification i.e. testing if a given program and system will meet a given specification with probability 1 e.g. ACD91, HS83, LS83, HS84] Essentially they work with logics which reason about probabilistic systems themselves. In [LS83] a logic containing a straightforward logic of linear time has a new modality added to it. This new modality is read certainly and indicates that its argument happens with probability 1. Three different axiomatisations are also provided covering general, finite and bounded models. HS84] gives ....

D. Lehmann and S. Shelah. Reasoning with time and chance. Information and Control, 53:165--198, 1983.

....serve as specification formalism for both qualitative and quantitative temporal properties. In the former case, a LTL specification just consists of a LTL formula f ; satisfaction of f in a state s means that f holds for almost all paths starting in s (i.e. with probability 1) Lehmann Shelah [LS82] present sound and complete axiomatizations for (a logic that subsumes) LTL interpreted over Markov chains of arbitrary size; thus, the framework of [LS82] can serve as a proof theoretic method for verifying qualitative properties for PLCSs. Quantitative properties can be expressed by a LTL ....

.... formula f ; satisfaction of f in a state s means that f holds for almost all paths starting in s (i.e. with probability 1) Lehmann Shelah [LS82] present sound and complete axiomatizations for (a logic that subsumes) LTL interpreted over Markov chains of arbitrary size; thus, the framework of [LS82] can serve as a proof theoretic method for verifying qualitative properties for PLCSs. Quantitative properties can be expressed by a LTL formula f and a lower bound probability p; satisfaction in a state s means that the probability for f is beyond the given lower bound p. 4 In [IN97] an ....

D. Lehmann and S. Shelah. Reasoning about time and chance. Information and Control, 53(3):165--198, 1982.

....suitable for expressing or reasoning about soft deadlines, since probabilities are not included. On the other hand, there are several examples in the litterature of modal logics that are extended with probabilities (but not time) e.g. PTL by Hart and Sharir [HS84] and TC by Lehman and Shelah [LS82]. However, these works only deal with properties that either hold with probability one or with a non zero probability. Probabilistic modal logics have been used in the verification of probabilistic algorithms. Mostly, the objective has been to verify that such algorithms satisfy certain properties ....

D. Lehmann and S. Shelah. Reasoning with time and chance. Information and Control, 53:165--198, 1982.

....as reliability and performance, require instead a probabilistic characterization of the system. The first applications of temporal logic to probabilistic systems consisted in studying which temporal logic properties are satisfied with probability 1 by systems modeled either as finite Markov chains [14, 18, 12, 1, 20] or as augmented Markov models exhibiting both nondeterministic and probabilistic behavior [22, 19, 5, 20] Subsequently, 10, 2] considered systems modeled by discrete Markov chains, and introduced the logics pCTL and pCTL , that can express quantitative bounds on the probability of system ....

D. Lehman and S. Shelah. Reasoning with time and chance. Information and Control, 53(3):165--198, 1982.

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D. Lehmann, S. Shelah (1982): Reasoning with Time and Chance. Information and Control, vol. 53 (3), pp. 165 - 198.

....a logic for reasoning about actions, probabilities, and time. His framework is purely logical and does not deal with legacy code. In addition, he studies a general logic, whereas our TP agents have a restricted syntactic form which makes them more suitable for computation. Lehmann and Shelah [19] were one of the rst to integrate time and probability. They developed a probabilistic temporal logic however, actions were not studied, and an algorithm by which an agent can decide what to do, given its operating principles, was not studied either. Dean and Kanazawa have also studied the ....

D. Lehmann and S. Shelah. Reasoning with time and chance. Information and Control, 53:165-198, 1982.

....testing satisfiablity and of model checking than CTL . It should also be mentioned that there are branching time logics with syntax like CTL or CTL , the formulas of which are interpreted over fair structures [6] Abrahamson structures (suffix and fusion closed) 2] and probabilistic structures [24]. These logics have also been shown to be decidable and to have the finite model property. 3 Temporal Logics on Partial Orders The aim of this section is to present the existing formal languages of temporal logic which are used to specify behaviours of concurrent systems represented by partial ....

D. Lehmann, S. Shelah (1982): Reasoning with Time and Chance. Information and Control, vol. 53 (3), pp. 165 - 198.

....from the product of the automaton and the process. Related work. Pioneering works on the verification of probabilistic systems have studied the satisfaction problem (i.e. satisfaction of the specification with probability one) for probabilistic systems modeled either as finite Markov chains [LS82,Pnu83,HS84] or as augmented Markov models comprising both nondeterministic and probabilistic behavior [Var85,PZ86,CY88] Specifications were given as temporal logic formulae or Buchi automata. Subsequently, HJ89,HJ94] have considered the probabilistic extension of CTL and a restricted class of ....

D. Lehman and S. Shelah. Reasoning with time and chance. Information and Control, 53(3):165--198, 1982.

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D. Lehmann and S. Shelah. Reasoning with time and chance. Information and Control 53, pp. 165-198, 1982.

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D. Lehmann and S. Shelah. Reasoning about time and chance. Information and Control, 53(3):165--198, 1982.

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D. Lehman and S. Shelah. Reasoning with time and chance. Information and Control, 53:165#198, 1982.

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D. Lehman and S. Shelah. Reasoning with time and chance. Information and Control, 53:165--198, 1982.

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D. Lehmann and S. Shelah. Reasoning with Time and Chance. Information and Control, 53:165--198, 1982.

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D. Lehmann and S. Shelah. Reasoning with time and chance. Information and Control, 53(3):165-198, 1982.

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D. Lehmann and S. Shelah. Reasoning about time and chance. Information and Control, 53(3):165-- 198, 1982.

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D. Lehmann and S. Shelah. Reasoning about time and chance. Information and Control, 53(3):165-- 198, 1982.

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D. Lehmann and S. Shelah. Reasoning with time and chance. Information and Control, 53(3):165--198, 1982.

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D. Lehman and S. Shelah. Reasoning with time and chance. Information and Control, 53(3):165--198, 1982.

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D. Lehmann and S. Shelah. Reasoning with time and chance. Information and Control 53(3) (1982) 165-198.

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