Succinct Representations for (Non)Deterministic Finite Automata

Abstract

Deterministic finite automata are one of the simplest and most practical models of computation studied in automata theory. Their 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 \(\varSigma \) using \((\sigma -1) n\log n (1+o(1))\) bits, which can determine, given an input string x over \(\varSigma \), whether \(\mathcal {D}\) accepts x in optimal O(|x|) time. We also consider the case when there are \(N < \sigma n\) non-failure transitions, and obtain various time-space trade-offs in both the cases. When the input deterministic finite automaton \(\mathcal {A}\) is acyclic, not only we can improve the above space bound significantly to \((\sigma -1) (n-1)\log n+ O(n + \log ^2 \sigma )\) bits, we can also check if an input string x over \(\varSigma \) can be accepted by \(\mathcal {A}\) optimally in O(|x|) 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 \(\varSigma \) 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.