The maximum length of shortest accepted strings for direction-determinate two-way finite automata

TL;DR

This paper determines the maximum length of the shortest accepted string for direction-determinate two-way finite automata, establishing an exact formula, and constructs automata with exponentially long shortest accepted strings.

Contribution

It provides an exact formula for the maximum shortest accepted string length in direction-determinate automata and constructs automata with exponentially long shortest strings.

Findings

Maximum length for direction-determinate automata is inom{n}{loor(n/2)} - 1

Constructed automata with shortest accepted strings of length rac{3}{4} imes 2^n - 1

Exact maximum length established for all n 

Abstract

It is shown that, for every $n \geqslant 2$, the maximum length of the shortest string accepted by an $n$-state direction-determinate two-way finite automaton is exactly $\binom{n}{\lfloor\frac{n}{2}\rfloor}-1$ (direction-determinate automata are those that always remember in the current state whether the last move was to the left or to the right). For two-way finite automata of the general form, a family of $n$-state automata with shortest accepted strings of length $\frac{3}{4} \cdot 2^n - 1$ is constructed.