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 non-standard 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.

Trade-off analysis meets probabilistic model checking

We survey the specification formalism of modal transition systems (MTS). We discuss various extensions of MTS, their relationships and modelling capabilities. The extensions include more involved modalities, quantitative aspects, or infinite state spaces. Further, we discuss problems arising in verification and analysis of these systems. We cover refinement checking, model checking and synthesis, standard logical and structural operations as used in specification theories as well as the respective tool support.

30 Years of Modal Transition Systems: Survey of Extensions and Analysis

This paper proposes a new theory of quantitative specifications. It generalizes the notions of step-wise refinement and compositional design operations from the Boolean to an arbitrary quantitative setting. Using a great number of examples, it is shown that this general approach permits to unify many interesting quantitative approaches to system design.

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General quantitative specification theories with modal transition systems

Weighted automata are non-deterministic automata where the transitions are equipped with weights. They can model quantitative aspects of systems like costs or energy consumption. The value of a run can be computed, for example, as the maximum, average, or discounted sum of transition weights. In multi-weighted automata, transitions carry several weights and can model, for example, the ratio between rewards and costs, or the efficiency of use of a primary resource under some upper bound constraint on a secondary resource. Here, we introduce a general model for multi-weighted automata as well as a multi-weighted MSO logic. In our main results, we show that this multi-weighted MSO logic and multi-weighted auto-mata are expressively equivalent both for finite and infinite words. The translation process is effective, leading to decidability results for our multi-weighted MSO logic.

Multi-weighted Automata and MSO Logic

Weighted automata are non-deterministic automata where the transitions are equipped with weights. They can model quantitative aspects of systems like costs or energy consumption. The value of a run can be computed, for example, as the maximum, limit average, or discounted sum of transition weights. In multi-weighted automata, transitions carry several weights and can model, for example, the ratio between rewards and costs, or the efficiency of use of a primary resource under some upper bound constraint on a secondary resource. Here, we introduce a general model for multi-weighted automata as well as a multi-weighted MSO logic. In our main results, we show that this multi-weighted MSO logic and multi-weighted automata are expressively equivalent both for finite and infinite words. The translation process is effective, leading to decidability results for our multi-weighted MSO logic.

Multiweighted modal automata provide a specification theory for multiweighted transition systems that have recently attracted interest in the context of energy games. We propose a simple fragment of CTL that is able to express properties about accumulated weights along maximal runs of multiweighted modal automata. Our logic is equipped with a game-based semantics and guarantees both soundness (formula satisfaction is propagated to the modal refinements) as well as completeness (formula non-satisfaction is propagated to at least one of its implementations). We augment our theory with a summary of decidability and complexity results of the generalized model checking problem, asking whether a specification -- abstracting the whole set of its implementations -- satisfies a given formula.

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Papers

A Logic for Accumulated-Weight Reasoning on Multiweighted Modal Automata