We study the relationship between least and
inflationary fixed-point logic.
By results of Gurevich and Shelah from 1986, it has been known
that on finite structures both logics have the same expressive
power. On infinite structures however, the question whether there is
a formula in IFP not equivalent to any LFP-formula was still open.
In this paper, we settle the question by showing that both logics
are equally expressive on arbitrary structures.
The proof will also establish the strictness of the
nesting-depth hierarchy for IFP on some infinite
structures. Finally, we show that the alternation hierarchy for
IFP collapses to the first level on all structures, i.e. the
complement of an inflationary fixed-point is an inflationary
fixed-point itself.