We investigate fundamental properties of three-valued semantics for abstract dialectical frameworks (ADFs). In particular, we deal with realizability, i.e., the question whether there exists an ADF that has a given set of interpretations as its semantics. We provide necessary and sufficient conditions that hold for a set of three-valued interpretations whenever there is an ADF realizing it under admissible, complete, grounded, or preferred semantics. Moreover, we discuss how to construct such an ADF in case of realizability. Our results lay the ground for studying the expressiveness of ADFs under three-valued semantics. As a first application we study implications of our results on the existence of certain join operators on ADFs.
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