This paper studies the decidability and computational complexity of checking probabilistic simulation pre-order between probabilistic pushdown automata (pPDA) and (probabilistic)finite-state systems.

We show that checking classical and combined probabilistic similarity are EXPTIME-complete in both directions and become polynomial if both the number of control states of the pPDA and the size of the finite-state system are fixed. These results show that checking probabilistic similarity is as hard as checking similarity

in the standard, i.e., non-probabilistic setting.