  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 $\exists^{\log^{\omega}}\text{IFP}$ captures
the limited nondeterminism class $\beta\text{P}$. In order to study its
expressive power, we also design a new version of Ehrenfeucht-Fra\"iss\'e game
for this logic and show that our capturing result will not hold on the general
case, i.e. on all the finite structures.
