## 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.

### MSC:

 68Q45 Formal languages and automata
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### References:

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