Succinct Representations for (Non)Deterministic Finite Automata

Deterministic finite automata are one of the simplest and most practical models computation studied in theory. Their 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 \(\varSigma \) \((\sigma -1) n\log (1+o(1))\) bits, can determine, given an input string x \), whether accepts O(|x|) time. We consider case when there \(N < \sigma n\) non-failure transitions, obtain various time-space trade-offs both cases. When {A}\) acyclic, not only improve above space bound significantly (n-1)\log n+ O(n + \log ^2 )\) check if be accepted by optimally exhibit {N}\) n^2+n\) space, such that x, decide efficiently \(O(n^2|x|)\) Finally, provide time efficient algorithms performing several standard operations as union, intersection complement languages automata.