One-Way Bounded-Error Probabilistic Pushdown Automata and Kolmogorov Complexity - (Preliminary Report).

TLDR: It is proved that a well-known language—the set of palindromes—cannot be recognized by any bounded-error ppda; in other words, this language stays outside of BPCFL.

Abstract: One-way probabilistic pushdown automata (or ppda’s) are a simple model of randomized computation with last-in first-out memory device known as stacks and, when error probabilities are bounded away from 1 / 2, ppda’s can characterize a family of bounded-error probabilistic context-free languages (BPCFL). We resolve a fundamental question raised by Hromkovic and Schnitger [Inf. Comput. 208 (2010) 982–995] concerning the limitation of the language recognition power of bounded-error ppda’s. More specifically, we prove that a well-known language—the set of palindromes—cannot be recognized by any bounded-error ppda; in other words, this language stays outside of BPCFL. Furthermore, we show that, with bounded-error probability, no ppda can determine whether the center bit of input string is 1 (one). For those impossibility results, we utilize a complexity measure of algorithmic information known as Kolmogorov complexity. In our proofs, we first transform ppda’s into an ideal shape and then lead to a key lemma by employing a Kolmogorov complexity argument.