Probabilistic input-driven pushdown automata

A probabilistic variant of input-driven pushdown automata (IDPDA), also known as visibly pushdown automata, is introduced. It is proved that these automata can be determinized: an n-state probabilistic IDPDA that accepts each string with probability at least $\lambda +\delta$ or at most $\lambda -\delta$, for some isolated cut-point $\lambda \in [0,1]$ with $\delta >0$, is transformed to a deterministic IDPDA recognizing the same language, with at most ${(1+\frac{1}{\delta })}^{n^2-n}$ states and at most $\left|{\Sigma }_{+1}\right|\cdot {(1+\frac{1}{\delta })}^{n^2-n}$ stack symbols, where ${\Sigma }_{+1}$ is the set of left bracket symbols in the alphabet. An asymptotically close lower bound is provided: for infinitely many n, there is a probabilistic IDPDA with $2n+3$ states and n stack symbols, and with $\delta =\frac{1}{270n}$, such that every equivalent deterministic IDPDA needs at least $7^{n^2/14}$ states and at least n stack symbols. A few special cases of automata with reduced determinization complexity are identified.