The logical characterization via SO∃ and FO(LFP) is order-sensitive: without a built-in linear order, FO(LFP) fails to capture P, undermining the biconditional's claim to equivalence with the full complexity-theoretic question.

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Biconditional(in formal logic): A logical statement that says two things are true if and only if each other is true; it's a two-way relationship (like saying 'you can vote if and only if you're 18').
Complexity-theoretic(in computer science): Related to the study of how difficult or resource-intensive it is to solve a computational problem (like how much time or memory a computer program needs).
Equivalence(what classical and intuitionistic logic disagree about): Two statements are equivalent when they mean exactly the same thing and always have the same truth value.
FO(LFP)(Descriptive complexity; extends expressive power of first-order logic): The extension of first-order logic with new relation symbols LFP_{ψ(R,x-vec)} for each formula ψ(R,x-vec) in which the relation variable appears only positively, with atomic formulas LFP_{ψ(R,x-vec)}(t-vec) interpreted as holding iff t-vec is in the least fixed point of the monotone operator induced by ψ.

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