Consider the logic whose $\tau$-sentences are the sentences in $IFP(\tau \cup \{<\})$, and the satisfaction relation is given by $\mathfrak{A} \models^* \phi$ if $(\mathfrak{A}, <)  \models \phi$ for every ordering $<$ of $\mathfrak{A}$. Why does this logic not capture PTIME? I suppose that it is because a machine cannot quickly determine whether whether all linear orderings satisfy the sentence, but is there a formal proof of this?
Why does IFP< not capture PTIME?
cc.complexity theory;lo.logic