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A path of M is of the form π = s0 a0 s1 a1 s2 · · · where ai ∈ δ(si ) and P(si , ai , si+1 ) > 0 for each i ≥ 0. Advances in Quantitative Veriﬁcation for Ubiquitous Computing 45 To reason formally about the behaviour of MDPs, we use the notion of strategies (also called adversaries or policies), which resolve all the nondeterministic choices in a model. Formally, a strategy is a function σ that maps a ﬁnite path ending in s to an action in δ(s) based on the history of choices made so far. Under a particular strategy, the behaviour of an MDP is fully probabilistic and we can deﬁne a probability space over the possible paths through the model using standard construction .
In: Berry, D. ) REFSQ 2011. LNCS, vol. 6606, pp. 151–165. Springer, Heidelberg (2011) 28. com/products/simulink/ 29. : Diﬀerential automata and their discrete simulations. Non-Linear Analysis 11(6), 665–683 (1987) 30. : A formal description of hybrid systems. A. ) HS 1995. LNCS, vol. 1066, pp. 511–530. Springer, Heidelberg (1996) Advances in Quantitative Verification for Ubiquitous Computing Marta Kwiatkowska Department of Computer Science, University of Oxford, Oxford, OX1 3QD Abstract. Ubiquitous computing, where computers ‘disappear’ and instead sensor-enabled and software-controlled devices assist us in everyday tasks, has become an established trend.
Examples of PCTL properties are: – Pmax=? [ F lost ] - “the maximum probability, over all possible strategies, of the protocol losing a message”; – Pmin=? [ F≤10 deliver ] - “the minimum probability, over all possible strategies, of the protocol delivering a message within 10 time steps”; – P≥1 [ near supplies U exit ] - “under all possible strategies, with probability 1, the robot always remains near supplies until exiting”. Model checking PCTL over MDPs requires a combination of graph-based algorithms and numerical solution techniques.