  We consider the termination/non-termination property of a class of loops.
Such loops are commonly used abstractions of real program pieces. Second-order
logic is a convenient language to express non-termination. Of course, such
property is generally undecidable. However, by restricting the language to
known decidable cases, we exhibit new classes of loops, the non-termination of
which is decidable. We present a bunch of examples.
