Hereditary first-order model checking is polynomial-time exactly for quantifier prefixes of the forms ∀*∃* and ∀*∃∀*, and coNP-complete otherwise for non-monadic signatures.
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Hereditary First-Order Logic: the tractable quantifier prefix classes
Hereditary first-order model checking is polynomial-time exactly for quantifier prefixes of the forms ∀*∃* and ∀*∃∀*, and coNP-complete otherwise for non-monadic signatures.