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Hereditary history preserving bisimulation is a natural extension of bisimulation to the setting of so-called "true" concurrency. Somewhat surprisingly, this extension turns out to be undecidable, in general, for finite-state concurrent systems. In this paper, we show that for a substantial and useful class of finite-state concurrent systems - those whose semantics can be described in terms of Mazurkiewicz traces - hereditary history preserving is decidable.