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The classical decision problem, as it is understood today, is the quest for adelineation between the decidable and the undecidable parts of first-orderlogic based on elegant syntactic criteria. In this paper, we treat the conceptof separateness of variables and explore its applicability to the classicaldecision problem. Two disjoint sets of first-order variables are separated in agiven formula if variables from the two sets never co-occur in any atom of thatformula. This simple notion facilitates extending many well-known decidablefirst-order fragments significantly and in a way that preserves decidability.We will demonstrate that for several prefix fragments, several guardedfragments, the two-variable fragment, and for the fluted fragment. Altogether,we will investigate nine such extensions more closely. Interestingly, each ofthem contains the relational monadic first-order fragment without equality.Although the extensions exhibit the same expressive power as the respectiveoriginals, certain logical properties can be expressed much more succinctly. Inthree cases the succinctness gap cannot be bounded using any elementaryfunction.

Separateness of Variables -- A Novel Perspective on Decidable First-Order Fragments