A separation between P and NP would therefore manifest as a class of structures expressible in SO∃ but not in FO(LFP)

The logic \(\textsf{SO}(\texttt{LFP})\) and \(\textsf{SO}(\texttt{TC})\) are defined analogously by adding these operators to \(\textsf{SO}\) and allowing them to apply to formulas containing second-order variables. e. models \(\mathcal{A}\) for structures interpreting \(\leq\) as a linear order on \(A\)). Immerman (1999, p. 3 as “increas[ing] our intuition that polynomial time is a class whose fundamental nature goes beyond the machine models with which it is usually defined”. e. \(\textbf{P} \