Longer Shortest Strings in Two-Way Finite Automata

In a recent paper, Dobronravov et al. ("On the length of of shortest strings accepted by two-way finite automata", DLT 2019) prove that the shortest string in a language recognized by an n state two-way finite automaton (2DFA) can be at least 7(n/5) - 1 symbols long, improved to 10(n/5) -1 = Omega(1.584(n)) in their latest contribution. The lower bound was obtained using "direction-determinate" 2DFA, which always remember their direction of motion at the last step, and used an alphabet of size Theta(n). In this paper, the method of Dobronravov et al. is extended to a new, more general class: the semi-direction-determinate 2DFA. This yields n-state 2DFA with shortest strings of length 7(n/4) -1 = Omega(1.626(n)). Furthermore, the construction is adapted to use a fixed alphabet, resulting in shortest strings of length Omega(1.275(n)). It is also shown that an n-state semi-direction-determinate 2DFA can be transformed to a one-way NFA with O(1/root n3(n)) states.