SUMMARYMany fatal accidents involving safety-critical reactive systems have occurred in unexpected situations, which were not considered during the design and test phases of system development. To prevent such accidents, reactive systems should be designed to respond appropriately to any request from an environment at any time. Verifying this property during the specification phase reduces the development costs of safety-critical reactive systems. This property of a specification is commonly known as realizability. The complexity of the realizability problem is 2EXPTIME-complete. We have introduced the concept of strong satisfiability, which is a necessary condition for realizability. Many practical unrealizable specifications are also strongly unsatisfiable. In this paper, we show that the complexity of the strong satisfiability problem is EXPSPACE-complete. This means that strong satisfiability offers the advantage of lower complexity for analysis, compared to realizability. Moreover, we show that the strong satisfiability problem remains EXPSPACE-complete even when only formulae with a temporal depth of at most 2 are allowed.
Linear temporal logic (LTL) synthesis is a formal method for automatically composing a reactive system that realizes a given behavioral specification described in LTL if the specification is realizable. Even if the whole specification is unrealizable, it is preferable to synthesize a best-effort reactive system. That is, a system that maximally realizes its partial specifications. Therefore, we categorized specifications into must specifications (which should never be violated) and desirable specifications (the violation of which may be unavoidable). In this paper, we propose a method for synthesizing a reactive system that realizes all must specifications and strongly endeavors to satisfy each desirable specification. The general form of the desirable specifications without assumptions is Gϕ, which means "ϕ B Takashi always holds". In our approach, the best effort to satisfy Gϕ is to maximize the number of steps satisfying ϕ in the interaction. To quantitatively evaluate the number of steps, we used a mean-payoff objective based on LTL formulae. Our method applies the Safraless approach to construct safety games from given must and desirable specifications, where the must specification can be written in full LTL and may include assumptions. It then transforms the safety games constructed from the desirable specifications into mean-payoff games and finally composes a reactive system as an optimal strategy on a synchronized product of the games.
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