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Under the assumption that the Polynomial-Time Hierarchy does not collapse
we show for a regular language :
the unbalanced polynomial-time leaf language class determined by
equals iff is existentially but not
quantifierfree definable in FO[<, min, max, +1, −1].
Furthermore, no such
class lies properly between NP and co-1-NP or NP⊕co-NP.
The proofs rely on a result of Pin and Weil
characterizing
the automata of existentially first-order definable languages.
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