A Time Complexity Gap for Two-Way Probabilistic Finite-State Automata

It is shown that if a 2-way probabilistic nite state automaton (2pfa) M recognizes a nonregular language L with error probability bounded below 1/2, then there is a positive constant b (depending on M) such that, for innnitely many inputs x, the expected running time of M on input x must exceed 2 n b where n is the length of x. This complements a result of Freivalds showing that 2pfa's can recognize certain nonregular languages in exponential expected time. It also establishes a time complexity gap for 2pfa's, since any regular language can be recognized by some 2pfa in linear time. Other results give roughly exponential upper and lower bounds on the worst-case increase in the number of states when converting a polynomial-time 2pfa to an equivalent 2-way nondeterministic nite-state automaton or to an equivalent 1-way deterministic nite-state automaton.