Nondeterminism versus Determinism for Two-Way Finite Automata: Generalizations of Sipser’s Separation

Whether there exists an exponential gap between the size of a minimal deterministic two-way automaton and the size of a minimal nondeterministic two-way automaton for a specific regular language is a long standing open problem and surely one of the most challenging problems in automata theory. Twenty four years ago, Sipser [M. Sipser: Lower bounds on the size of sweeping automata. ACM STOC ’79, 360–364] showed an exponential gap between nondeterminism and determinism for the so-called sweeping automata which are automata whose head can reverse direction only at the endmarkers. Sweeping automata can be viewed as a special case of oblivious two-way automata with a number of reversals bounded by a constant.Our first result extends the result of Sipser to general oblivious two-way automata with an unbounded number of reversals. Using this extension we show our second result, namely an exponential gap between determinism and nondeterminism for two-way automata with the degree of non-obliviousness bounded by o(n) for inputs of length n. The degree of non-obliviousness of a two-way automaton is the number of distinct orders in which the tape cells are visited.