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 n states over an alphabet Sigma can be transformed to an equivalent one-way automaton (1DFA) with |Sigma| center dot phi(n) + 1 states, where phi(n) =max(k=1)(n-1)k(n-k+1) = n(n)- (nlnln n /ln n) +O( n/ln n). This reflects the fact that, by keeping the last processed symbol a in memory, the simulating 1DFA can remember one of k states in which the automaton moves by a to the right, and a function that maps n - k states moving to the left to k states moving to the right; cf. ca. n(n) functions in the classical construction. A close lower bound of phi(n) 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 phi(n).