Translating regular expressions into small \(\epsilon\)-free nondeterministic finite automata. (English) Zbl 1014.68093

Summary: We prove that every regular expression of size \(n\) can be converted into an equivalent Nondeterministic \(\varepsilon\)-free Finite Automaton (NFA) with \(O(n\log n)^2)\) transitions in time \(O(n^2\log n)\). The best previously known conversions result in NFAs of worst-case size \(\Theta(n^2)\). We complement our result by proving an almost matching lower bound. We exhibit a sequence of regular expressions of size \(O(n)\) and show the number of transitions required in equivalent NFAs is \(\Omega(n\log n)\). This also proves there does not exist a linear-size conversion from regular expressions to NFAs.


68Q45 Formal languages and automata
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