Abstract
In a recent paper, Dobronravov et al. (“On the length 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 \(10^{n/5}-1=\varOmega (1.584^n)\) 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 \(\varTheta (n)\). 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 \(7^{n/4}-1=\varOmega (1.626^n)\). Furthermore, the construction is adapted to use a fixed alphabet, resulting in shortest strings of length \(\varOmega (1.275^n)\). It is also shown that an n-state semi-direction-determinate 2DFA can be transformed to a one-way NFA with \(O(\frac{1}{\sqrt{n}} 3^n)\) states.
Research supported by Russian Science Foundation, project 18-11-00100.
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The authors are grateful to the anonymous reviewers for careful reading and for pertinent remarks.
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Krymski, S., Okhotin, A. (2020). Longer Shortest Strings in Two-Way Finite Automata. In: Jirásková, G., Pighizzini, G. (eds) Descriptional Complexity of Formal Systems. DCFS 2020. Lecture Notes in Computer Science(), vol 12442. Springer, Cham. https://doi.org/10.1007/978-3-030-62536-8_9
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