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| {{expert-subject|date=February 2012}}
| | Irwin Butts is what my wife enjoys to call me though I don't truly like becoming known as like that. Since she was eighteen she's been operating as a meter reader but she's usually needed her own business. To do aerobics is a thing that I'm totally addicted to. North Dakota is our beginning location.<br><br>Feel free to visit my page :: [http://www.streaming.iwarrior.net/user/WMcneil home std test] |
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| '''Probabilistic Computation Tree Logic''' (PCTL) is an extension of [[computation tree logic]] (CTL) which allows for probabilistic quantification of described properties. It has been defined in the paper by Hansson and Jonsson.<ref>http://citeseer.ist.psu.edu/hansson94logic.html</ref> | |
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| PCTL is a useful [[logic]] for stating soft deadline properties, e.g. "after a request for a service, there is at least a 98% probability that the service will be carried out within 2 seconds". Akin CTL suitability for model-checking PCTL extension is widely used as a property specification language for probabilistic model checkers.
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| == PCTL syntax ==
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| One of the possible syntax of PCTL is defined as follows:
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| <center> | |
| <math> | |
| \phi ::= p | \neg p | \phi \lor \phi | \phi \land \phi | \mathcal{P}_{\sim\lambda}(\phi \mathcal{U} \phi) |
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| \mathcal{P}_{\sim\lambda}(\square\phi)
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| </math>
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| </center>
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| Therein, <math>\sim \in \{ <, \leq, \geq, > \}</math> is comparison operator and <math>\lambda</math> is a probability threshold.
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| <br>
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| Formulas of PCTL are interpreted over discrete [[Markov chains]]. An interpretation structure
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| is a quadruple <math>K = \langle S, s^i, \mathcal{T}, L \rangle</math>, where
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| *<math>S</math> is a finite set of states,
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| *<math>s^i \in S</math> is an initial state,
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| *<math>\mathcal{T}</math> is a transition probability function, <math>\mathcal{T} : S \times S \to [0,1] </math>, such that for all <math>s \in S</math> we have <math>\sum_{s'\in S} \mathcal{T}(s,s')=1</math>, and
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| *<math>L</math> is a labeling function, <math>L:S\to2^A</math>, assigning atomic propositions to states.
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| <br>
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| A path <math>\sigma</math> from a state <math>s_0</math> is an infinite sequence of states
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| <math>s_0 \to s_1 \to \dots \to s_n \to \dots </math>. The n-th state of the path is denoted as <math>\sigma[n]</math>
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| and the prefix of <math>\sigma</math> of length <math>n</math> is denoted as <math>\sigma\uparrow n</math>.
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| == Probability measure ==
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| A probability measure <math>\mu_m</math> of the set of path with the common prefix of length <math>n</math> is equal to the product of transitions probabilitites along the prefix of the path:
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| <center><math>
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| \mu_m(\{\sigma \in X : \sigma\uparrow n = s_0 \to \dots \to s_n \}) = \mathcal{T}(s_0,s_1) \times\dots\times\mathcal{T}(s_{n-1},s_n)
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| </math></center>
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| For <math>n = 0</math> the probability measure is equal to <math>\mu_m(\{\sigma \in X : \sigma\uparrow 0 = s_0 \}) = 1</math>.
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| == Satisfaction relations ==
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| Satisfaction relations <math>s \models_K f</math>, <math>\sigma \models_K f</math> are inductively defined as follows:
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| * <math>s \models_K a</math> if and only if <math>a \in L(s)</math>,
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| * <math>s \models_K \neg f</math> if and only if not <math>s \models_K f</math>,
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| * <math>s \models_K f_1 \lor f_2</math> if and only if <math>s \models_K f_1</math> or <math>s \models_K f_2</math>,
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| * <math>s \models_K f_1 \land f_2</math> if and only if <math>s \models_K f_1</math> and <math>s \models_K f_2</math>,
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| * <math>s \models_K \mathcal{P}_{\sim\lambda}(f_1 \mathcal{U} f_2)</math> if and only if <math>\mu_m(\{\sigma : \sigma[0] = s \land (\exists i)\sigma[i] \models_K f_2 \land (\forall 0 \leq j < i) \sigma[j] \models_K f_1\}) \sim \lambda</math>, and
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| * <math>s \models_K \mathcal{P}_{\sim\lambda}(\square f)</math> if and only if <math>\mu_m(\{\sigma : \sigma[0] = s \land (\forall i \geq 0)\sigma[i] \models_K f\}) \sim \lambda</math>.
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| ==References==
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| <references />
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| [[Category:Temporal logic]]
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Irwin Butts is what my wife enjoys to call me though I don't truly like becoming known as like that. Since she was eighteen she's been operating as a meter reader but she's usually needed her own business. To do aerobics is a thing that I'm totally addicted to. North Dakota is our beginning location.
Feel free to visit my page :: home std test