Most decidability results concerning well-structured transition systems apply to the finitely branching variant. Yet some models (inserting automata, ω-Petri nets, …) are naturally infinitely branching. Here we develop tools to handle infinitely branching WSTS by exploiting the crucial property that in the (ideal) completion of a well-quasi-ordered set, downward-closed sets are finite unions of ideals. Then, using these tools, we derive decidability results and we delineate the undecidability frontier in the case of the termination, the control-state maintainability and the coverability problems.Coverability and boundedness under new effectivity conditions are shown decidable.
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