On Existentially First-Order Definable Languages and Their Relation to NP

Under the assumption that the Polynomial-Time Hierarchy does not collapse we show that a regular language L determines NP as an unbalanced polynomial-time leaf language if and only if L is existentially but not quantifierfree definable in FO[<, min, max, −1, +1]. The proof relies on the result of Pin & Weil [PVV97] characterizing the automata of existentially first-order definable languages.

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