Succinct Representation for (Non)Deterministic Finite Automata

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