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We consider linear cost-register automata (equivalent to weighted automata)over the semiring of nonnegative rationals, which generalise probabilisticautomata. The two problems of boundedness and zero isolation ask whether thereis a sequence of words that converge to infinity and to zero, respectively. Inthe general model both problems are undecidable so we focus on the copylesslinear restriction. There, we show that the boundedness problem is decidable.  As for the zero isolation problem we need to further restrict the class. Weobtain a model, where zero isolation becomes equivalent to universalcoverability of orthant vector addition systems (OVAS), a new model in the VASfamily interesting on its own. In standard VAS runs are considered only in thepositive orthant, while in OVAS every orthant has its own set of vectors thatcan be applied in that orthant. Assuming Schanuel's conjecture is true, weprove decidability of universal coverability for three-dimensional OVAS, whichimplies decidability of zero isolation in a model with at most threeindependent registers.

The boundedness and zero isolation problems for weighted automata over nonnegative rationals