# Succinct Representation for (Non)Deterministic Finite Automata

## TL;DR

This paper introduces space-efficient data structures for representing deterministic and non-deterministic finite automata, enabling fast string acceptance queries and standard automata operations with optimal or near-optimal space and time complexity.

## Contribution

The authors develop succinct representations for both deterministic and non-deterministic finite automata, achieving optimal space and query time for acyclic automata and efficient algorithms for automata operations.

## Key findings

- Succinct data structure for deterministic finite automata with optimal space and query time.
- Improved space bounds for acyclic deterministic automata with optimal acceptance checking time.
- Succinct representation of non-deterministic finite automata enabling efficient acceptance decision.

## Abstract

Deterministic finite automata are one of the simplest and most practical models of computation studied in automata theory. Their conceptual extension is the non-deterministic finite automata which also have plenty of applications. In this article, we study these models through the lens of succinct data structures where our ultimate goal is to encode these mathematical objects using information-theoretically optimal number of bits along with supporting queries on them efficiently. Towards this goal, we first design a succinct data structure for representing any deterministic finite automaton $\mathcal{D}$ having $n$ states over a $\sigma$-letter alphabet $\Sigma$ using $(\sigma-1) n\log n + O(n \log \sigma)$ bits of space, which can determine, given an input string $x$ over $\Sigma$, whether $\mathcal{D}$ accepts $x$ in $O(|x| \log \sigma)$ time, using constant words of working space. When the input deterministic finite automaton is acyclic, not only we can improve the above space-bound significantly to $(\sigma -1) (n-1)\log n+ 3n + O(\log^2 \sigma) + o(n)$ bits, we also obtain optimal query time for string acceptance checking. More specifically, using our succinct representation, we can check if a given input string $x$ can be accepted by the acyclic deterministic finite automaton using time proportional to the length of $x$, hence, the optimal query time. We also exhibit a succinct data structure for representing a non-deterministic finite automaton $\mathcal{N}$ having $n$ states over a $\sigma$-letter alphabet $\Sigma$ using $\sigma n^2+n$ bits of space, such that given an input string $x$, we can decide whether $\mathcal{N}$ accepts $x$ efficiently in $O(n^2|x|)$ time. Finally, we also provide time and space-efficient algorithms for performing several standard operations such as union, intersection, and complement on the languages accepted by deterministic finite automata.