# Detecting Unrealizable Specifications of Distributed Systems

Bernd Finkbeiner and Leander Tentrup

Writing formal specifications for distributed systems is difficult. Even simple consistency requirements often turn out to be unrealizable because of the complicated information flow in the distributed system: not every information is available in every component, and information transmitted from other components may arrive with a delay or not at all, especially in the presence of faults. The problem of checking the distributed realizability of a temporal specification is, in general, undecidable. Semi-algorithms for synthesis, such as bounded synthesis, are only useful in the positive case, where they construct an implementation for a realizable specification, but not in the negative case: if the specification is unrealizable, the search for the implementation never terminates. In this paper, we introduce *counterexamples to distributed realizability* and present a method for the detection of such counterexamples for specifications given in linear-time temporal logic (LTL). A counterexample consists of a set of paths, each representing a different sequence of inputs from the environment, such that, no matter how the components are implemented, the specification is violated on *at least one* of these paths. We present a method for finding such counterexamples both for the classic distributed realizability problem and for the distributed realizability problem with faulty nodes. Our method considers, incrementally, larger and larger sets of paths until a counterexample is found. While counterexamples for full LTL may consist of infinitely many paths, we give a semantic characterization such that the required number of paths can be bounded. For this fragment, we thus obtain a decision procedure. Experimental results, obtained with a QBF-based prototype implementation, show that our method finds simple errors very quickly, and even problems with high combinatorial complexity, like the Byzantine Generals’ Problem, are tractable.

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