The complexity properties of probabilistic automata with isolated cut point

A probabilistic automaton (PA) which accepts a language with e-isolated cut point 1 2 corresponds to a PA which computes with ( 1 2-e) bounded error probability. Let P(L, e) be the minimal number of states of a PA necessary for accepting a language L with e-isolated cut point 1 2. It is shown that there are languages Lk, 1 < k < ∞ and an infinite sequence of numbers 0 < e1 < e2 < ... < 1 2 such that for all i ≥1, P(Lk,ei) P(Lk, ei+1) → 0 when k→∞. It is also shown that the probabilistic recognition of the language Wk is more effective than that of the Lk. © 1988