We investigate zero-sum turn-based two-player stochastic games in which the objective of one player is to maximize the amount of rewards obtained during a play, while the other aims at minimizing it. We focus on games in which the minimizer plays in a fair way. We believe that these kinds of games enjoy interesting applications in software verification, where the maximizer plays the role of a system intending to maximize the number of “milestones” achieved, and the minimizer represents the behavior of some uncooperative but yet fair environment. Normally, to study total reward properties, games are requested to be stopping (i.e., they reach a terminal state with probability 1). We relax the property to request that the game is stopping only under a fair minimizing player. We prove that these games are determined, i.e., each state of the game has a value defined. Furthermore, we show that both players have memoryless and deterministic optimal strategies, and the game value can be computed by approximating the greatest-fixed point of a set of functional equations. We implemented our approach in a prototype tool, and evaluated it on an illustrating example and an Unmanned Aerial Vehicle case study.

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We present a formal characterization of fault-tolerant behaviors of computing systems via simulation relations. This formalization makes use of variations of standard simulation relations in order to compare the executions of a system that exhibits faults with executions where no faults occur; intuitively, the latter can be understood as a specification of the system and the former as a fault-tolerant implementation. By employing variations of standard simulation algorithms, our characterization enables us to algorithmically check fault-tolerance in polynomial time, i.e., to verify that a system behaves in an acceptable way even subject to the occurrence of faults. Furthermore, the use of simulation relations in this setting allows us to distinguish between the different levels of fault-tolerance exhibited by systems during their execution. We prove that each kind of simulation relation preserves a corresponding class of temporal properties expressed in CTL; more precisely, masking fault-tolerance preserves liveness and safety properties, nonmasking fault-tolerance preserves liveness properties, while failsafe fault-tolerance guarantees the preservation of safety properties. We illustrate the suitability of this formal framework through its application to standard examples of fault-tolerance.