The fundamentals of probabilistic model checking for Markovian models and temporal properties have been studied extensively in the past 20 years. Research on methods for computing conditional probabilities for temporal properties under temporal conditions is, however, comparably rare. For computing conditional probabilities or expected values under ω-regular conditions in Markov chains, we introduce a new transformation of Markov chains that incorporates the effect of the condition into the model. For Markov decision processes, we show that the task to compute maximal reachability probabilities under reachability conditions is solvable in polynomial time, while it was conjectured to be computationally hard. Using adaptions of known automata-based methods, our algorithm can be generalized for computing the maximal conditional probabilities for ω-regular events under ω-regular conditions. The feasibility of our algorithms is studied in two benchmark examples.

The concept of features provides an elegant way to specify families of systems. Given a base system, features encapsulate additional functionalities that can be activated or deactivated to enhance or restrict the base system’s behaviors. Features can also facilitate the analysis of families of systems by exploiting commonalities of the family members and performing an all-in-one analysis, where all systems of the family are analyzed at once on a single family model instead of one-by-one. Most prominent, the concept of features has been successfully applied to describe and analyze (software) product lines. We present the tool ProFeat that supports the feature-oriented engineering process for stochastic systems by probabilistic model checking. To describe families of stochastic systems, ProFeat extends models for the prominent probabilistic model checker Prism by feature-oriented concepts, including support for probabilistic product lines with dynamic feature switches, multi-features and feature attributes. ProFeat provides a compact symbolic representation of the analysis results for each family member obtained by Prism to support, e.g., model repair or refinement during feature-oriented development. By means of several case studies we show how ProFeat eases family-based quantitative analysis and compare one-by-one and all-in-one analysis approaches.

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Unambiguous automata are nondeterministic automata in which every word has at most one accepting run. In this paper we give a polynomial-time algorithm for model checking discrete-time Markov chains against ω-regular specifications represented as unambiguous automata. We furthermore show that the complexity of this model checking problem lies in NC: the subclass of P comprising those problems solvable in poly-logarithmic parallel time. These complexity bounds match the known bounds for model checking Markov chains against specifications given as deterministic automata, notwithstanding the fact that unambiguous automata can be exponentially more succinct than deterministic automata. We report on an implementation of our procedure, including an experiment in which the implementation is used to model check LTL formulas on Markov chains.Over infinite words, not only are UBA as expressive as non-deterministic Büchi automata [1], they can also be exponentially more succinct than deterministic automata. For example, for a fixed k ∈ N the language "eventually b occurs and a appears k steps before the first b" over the alphabet {a, b, c} is recognized by an UBA with k+1 states (shown on the left-hand side of Figure 1). On the other hand, a deterministic automaton for this language requires at least 2 k states, regardless of the acceptance condition, as it needs to store the positions of the a's among the last k input symbols. Languages of this type arise in a number of contexts, e.g., absence of unsolicited response in a communication protocol-if a message is received, then it has been sent in the recent past.The exponential succinctness of UBA relative to deterministic automata is also manifested in translations of linear temporal logic (LTL) to automata. The non-deterministic Büchi automata that are obtained from LTL formulas by applying the classical closure algorithm of [38,37] are unambiguous. The generated automata moreover enjoy the separation property:different states have disjoint languages. Thus, while the generation of deterministic ω-automata from LTL formulas incurs a double-exponential blow-up in the worst case, the translation of LTL formulas into separated UBA incurs only a single exponential blow-up. This fact has been observed by several authors, see e.g. [15,33], and adapted for LTL with step parameters [39,10].In the context of probabilistic model checking, UBA provide an elegant alternative to deterministic automata for computing probabilities of ω-regular properties on finite-state Markov chains. A polynomial-time model checking procedure for UBA that represent safety properties was given [4], while [15] gives a polynomial-time algorithm for separated UBA. However, separation is a rather strong restriction, and non-separated UBA (and even DBA) can be exponentially more succinct than separated UBA, see [8]. Furthermore, algorithms for the generation of (possibly non-separated) UBA from LTL formulas that are more compact than the separated UBA generated by the classical closure algorithm have been real...

Probabilistic model checking (PMC) is a well-established and powerful method for the automated quantitative analysis of parallel distributed systems. Classical PMC-approaches focus on computing probabilities and expectations in Markovian models annotated with numerical values for costs and utility, such as energy and performance. Usually, the utility gained and the costs invested are dependent and a trade-off analysis is of utter interest.In this paper, we provide an overview on various kinds of nonstandard multi-objective formalisms that enable to specify and reason about the trade-off between costs and utility. In particular, we present the concepts of quantiles, conditional probabilities and expectations as well as objectives on the ratio between accumulated costs and utility. Such multi-objective properties have drawn very few attention in the context of PMC and hence, there is hardly any tool support in state-of-the-art model checkers. Furthermore, we broaden our results towards combined quantile queries, computing conditional probabilities those conditions are expressed as formulas in probabilistic computation tree logic, and the computation of ratios which can be expected on the long-run.

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