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Abstract. The paper addresses the problem of computing maximal conditional expected accumulated rewards until reaching a target state (briefly called maximal conditional expectations) in finite-state Markov decision processes where the condition is given as a reachability constraint. Conditional expectations of this type can, e.g., stand for the maximal expected termination time of probabilistic programs with non-determinism, under the condition that the program eventually terminates, or for the worst-case expected penalty to be paid, assuming that at least three deadlines are missed. The main results of the paper are (i) a polynomial-time algorithm to check the finiteness of maximal conditional expectations, (ii) PSPACE-completeness for the threshold problem in acyclic Markov decision processes where the task is to check whether the maximal conditional expectation exceeds a given threshold, (iii) a pseudo-polynomial-time algorithm for the threshold problem in the general (cyclic) case, and (iv) an exponential-time algorithm for computing the maximal conditional expectation and an optimal scheduler.

Many important performance and reliability measures can be formalized as the accumulated values of weight functions. In this paper, we introduce an extension of linear time logic including past (LTL) with new operators that impose constraints on the accumulated weight along path fragments. The fragments are characterized by regular conditions formalized by deterministic finite automata (monitor DFA). This new logic covers properties expressible by several recently proposed formalisms. We study the model-checking problem for weighted transition systems, Markov chains and Markov decision processes with rational weights. While the general problem is undecidable, we provide algorithms and sharp complexity bounds for several sublogics that arise by restricting the monitoring DFA.

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Probabilistic model checking is a well-established method for the automated quantitative system analysis. It has been used in various application areas such as coordination algorithms for distributed systems, communication and multimedia protocols, biological systems, resilient systems or security. In this paper, we report on the experiences we made in inter-disciplinary research projects where we contribute with formal methods for the analysis of hardware and software systems. Many performance measures that have been identified as highly relevant by the respective domain experts refer to multiple objectives and require a good balance between two or more cost or reward functions, such as energy and utility. The formalization of these performance measures requires several concepts like quantiles, conditional probabilities and expectations and ratios of cost or reward functions that are not supported by state-ofthe-art probabilistic model checkers. We report on our current work in this direction, including applications in the field of software product line verification.