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We investigate a subclass of well-structured transition systems (WSTS), the bounded--in the sense of Ginsburg and Spanier (Trans. AMS 1964) --complete deterministic ones, which we claim provide an adequate basis for the study of forward analyses as developed by Finkel and Goubault-Larrecq (ICALP 2009). Indeed, we prove that, unlike other conditions considered previously for the termination of forward analysis, boundedness is decidable. Boundedness turns out to be a valuable restriction for WSTS verification, as we show that it further allows to decide all ?-regular properties on the set of infinite traces of the system.