We consider pushdown systems that store, instead of a single word, a Mazurkiewicz trace on its stack. These systems are special cases of valence automata over graph monoids and subsume multi-stack systems. We identify a class of such systems that allow to decide the first-order theory of their configuration graph with reachability. This result complements results by D'Osualdo, Meyer, and Zetzsche (namely the decidability for arbitrary pushdown systems under a severe restriction on the dependence alphabet).
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Dietrich Kuske (2026) studied this question.
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