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.
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