# Preimage problems for deterministic finite automata

## TL;DR

This paper investigates the computational complexity of various preimage-related problems in deterministic finite automata, including extending, totally extending, and resizing words, across different automaton classes and constraints.

## Contribution

It provides a comprehensive complexity analysis of three natural preimage problems in DFA, considering various automaton properties and restrictions.

## Key findings

- Complexity results vary across problem types and automaton classes.
- Some problems are computationally hard, others are tractable under certain conditions.
- The paper offers a complete complexity classification for all cases studied.

## Abstract

Given a subset of states $S$ of a deterministic finite automaton and a word $w$, the preimage is the subset of all states mapped to a state in $S$ by the action of $w$. We study three natural problems concerning words giving certain preimages. The first problem is whether, for a given subset, there exists a word \emph{extending} the subset (giving a larger preimage). The second problem is whether there exists a \emph{totally extending} word (giving the whole set of states as a preimage)---equivalently, whether there exists an \emph{avoiding} word for the complementary subset. The third problem is whether there exists a \emph{resizing} word. We also consider variants where the length of the word is upper bounded, where the size of the given subset is restricted, and where the automaton is strongly connected, synchronizing, or binary. We conclude with a summary of the complexities in all combinations of the cases.