Deterministic One-Way Simulation of Two-Way Deterministic Finite Automata Over Small Alphabets

It is shown that a two-way deterministic finite automaton (2DFA) with [Formula: see text] states over an alphabet [Formula: see text] can be transformed to an equivalent one-way automaton (1DFA) with [Formula: see text] states, where [Formula: see text]. This reflects the fact that, by keeping the last processed symbol [Formula: see text] in memory, the simulating 1DFA can remember one of [Formula: see text] states in which the automaton moves by [Formula: see text] to the right, and a function that maps [Formula: see text] states moving to the left to [Formula: see text] states moving to the right; cf. ca. [Formula: see text] functions in the classical construction. A close lower bound of [Formula: see text] states is established using a 2-symbol alphabet, with witness languages defined by direction-determinate 2DFA. The same lower bound is also achieved with witness languages defined by sweeping 2DFA, at the expense of using a 5-symbol alphabet. In addition, the complexity of transforming a sweeping or a direction-determinate 2DFA to a 1DFA is shown to be exactly [Formula: see text].