Regular expressions and NFAs without Ε-transitions

• Authors: Georg Schnitger
• Venue: STACS'06 Proceedings of the 23rd Annual conference on Theoretical Aspects of Computer Science
• Year: 2006

We consider the problem of converting regular expressions of length n over an alphabet of size k into ε-free NFAs with as few transitions as possible. Whereas the previously best construction uses O(n ·min{k, log2n} ·log2n) transitions, we show that O( n · log22k · log2n ) transitions suffice. For small alphabets we further improve the upper bound to $O(n \cdot log_2 2k \cdot k^{L_k (n)+1})$, where Lk(n) = O(log$_{\rm 2}^{\rm *}$n). In particular, $n \cdot 2^{O({log}^*_2 n)}$ transitions and hence almost linear size suffice for the binary alphabet! Finally we show the lower bound Ω(n · log$_{\rm 2}^{\rm 2}$ 2k) and as a consequence the upper bound O(n · log$_{\rm 2}^{\rm 2}$n) of [7] for general alphabets is best possible. Thus the conversion problem is solved for large alphabets (k = nΩ(1)) and almost solved for small alphabets (k = O(1)). Classification. Automata and formal languages, descriptional complexity, nondeterministic automata, regular expressions.