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"... Abstract We study the language equivalence problem for probabilistic pushdown automata (pPDA) and their subclasses. We show that the problem is interreducible with the multiplicity equivalence problem for context-free grammars, the decidability of which has been open for several decades. Interreduc ..."

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Abstract We study the language equivalence problem for probabilistic pushdown automata (pPDA) and their subclasses. We show that the problem is interreducible with the multiplicity equivalence problem for context-free grammars, the decidability of which has been open for several decades. Interreducibility also holds for pPDA with one control state. In contrast, for the case of a one-letter input alphabet we show that pPDA language equivalence (and hence multiplicity equivalence of context-free grammars) is in PSPACE and at least as hard as the polynomial identity testing problem.