_:ne3fee7930e814185bcb6af35698c6dd6xb7
<http://schema.org/description>
"The classical decision problem, as it is understood today, is the quest for a\ndelineation between the decidable and the undecidable parts of first-order\nlogic based on elegant syntactic criteria. In this paper, we treat the concept\nof separateness of variables and explore its applicability to the classical\ndecision problem. Two disjoint sets of first-order variables are separated in a\ngiven formula if variables from the two sets never co-occur in any atom of that\nformula. This simple notion facilitates extending many well-known decidable\nfirst-order fragments significantly and in a way that preserves decidability.\nWe will demonstrate that for several prefix fragments, several guarded\nfragments, the two-variable fragment, and for the fluted fragment. Altogether,\nwe will investigate nine such extensions more closely. Interestingly, each of\nthem contains the relational monadic first-order fragment without equality.\nAlthough the extensions exhibit the same expressive power as the respective\noriginals, certain logical properties can be expressed much more succinctly. In\nthree cases the succinctness gap cannot be bounded using any elementary\nfunction."
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