# Time complexity gap for two-way probabilistic finite-state automata

## Abstract

It is shown that if a two-way probabilistic finite-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 infinitely many inputs x, the expected running time of M on input x must exceed 2n(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 polynominal-time 2pfa to an equivalent two-way nondeterministic finite-state automaton or to an equivalent one-way deterministic finite-state automaton.