A Logic that Captures $β$P on Ordered Structures

We extend the inflationary fixed-point logic, IFP, with a new kind of second-order quantifiers which have (poly-)logarithmic bounds. We prove that on ordered structures the new logic $\exists^{\log^ω}\text{IFP}$ captures the limited nondeterminism class $β\text{P}$. In order to study its expressive power, we also design a new version of Ehrenfeucht-Fraïssé game for this logic and show that our capturing result will not hold on the general case, i.e. on all the finite structures.

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