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...e Computation Tree Logic (CTL) [20]. In the case of CTMCs, we employ the temporal logic Continuous Stochastic Logic (CSL) developed originally by Aziz et al. [4, 5] and since extended by Baier et al. =-=[10]-=-, also based on CTL. Algorithms for stochastic model checking were originally introduced in [62, 23, 29, 5, 10], derive from conventional model checking, numerical linear algebra and standard techniqu...

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...incide. The validation of biochemical networks by means of T-invariants is demonstrated in [HK04]. Model checking has been very popular for the verification of technical systems since the eighties. A good starting point for qualitative model checking (CTL, LTL) is [CGP01]. For biochemical networks, qualitative model checking (Boolean semantics) has been introduced in [EKL+02], [CF03], and analytical stochastic model checking in [CVGO06], [HKN+06]. CSL, the stochastic counterpart to CTL, has been originally introduced in [AAB00], and extended in Petri Nets for Systems and Synthetic Biology 259 [CBK03]. PRISM [PNK06] provides an efficient numerical implementation for CSL model checking, also exploiting symbolic representations. Approximative model checking of CSL using discrete event simulation of probabilistic models has been proposed in [YS02] and implemented in the tool Ymer [YKNP06]. The simulative stochastic model checker MC2 has been inspired by the idea of approximative LTL checking of deterministic simulation runs, proposed in [APUM03], [CCRFS06], [FR07]. The Biocham approach [CCRFS06], as it stands now, restricts itself to a branching time Boolean semantics for qualitative models, ...

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...specified real-valued time bound, provided that all paths to these states obey certain properties, is at least/at most some probability value. Model-checking algorithms for CSL have been presented in =-=[8]-=-, and prototypical software implementations are available: for instance, one based on sparse matrices [38] and a symbolic tool based on multi-terminal binary decision diagrams [46]. Apart from present...

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...her the model satisfies the property with at least probability θ. Existing algorithms for solving the PMC problem fall into one of two categories. The first category comprises numerical methods (e.g. =-=[2,3,12,16,31]-=-) which can compute the probability with which the property holds with high precision. Numerical methods are generally only suitable for small systems (≈ 106 to 107 states). In a Biological System, th...

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...v chains and Markov decision processes) can be found on the PRISM website [16]. Model checking algorithms. The basic algorithms required for model checking are those proposed for PCTL [9] and for CSL =-=[5]-=-. Algorithms for the reward operators, which are based on standard techniques for Markov reward models, can be found in [13]. The key ingredients of these algorithms can be classified into two groups:...

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... or whether the ignition succeeds within 1ms with probability at least 0.99. In fact, several temporal logics have been developed in order to express these and other types of probabilistic properties =-=[3, 11, 1]-=-. The Probabilistic Model Checking (PMC) problem is to decide whether a stochastic model satisfies a temporal logic property with a probability greater than or equal to a certain threshold. More forma...

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...ov chains (DTMCs and CTMCs, for short), and variants thereof with costs. The advantages of probabilistic bisimulation [34] in this setting are manifold. It preserves the validity of PCTL [20] and CSL =-=[2, 6]-=- formulas, variants of CTL for the discrete- and continuous-time probabilistic setting, respectively. It implies ordinary lumpability of Markov chains [10], an aggregation technique for Markov chains ...