## Conversion of regular expressions into realtime automata.(English)Zbl 1110.68063

Summary: We consider conversions of regular expressions into $$k$$-real-time finite state automata, i.e., automata in which the number of consecutive uses of $$\varepsilon$$-transitions, along any computation path, is bounded by a fixed constant $$k$$. For 2-real-time automata, i.e., for automata that cannot change the state, without reading an input symbol, more than two times in a row, we show that the conversion of a regular expression into such an automaton produces only $$O(n)$$ states, $$O(n\log n)$$ $$\varepsilon$$-transitions, and $$O(n)$$ alphabet-transitions. We also show how to easily transform these 2-real-time machines into 1-real-time automata, still with only $$O(n\log n)$$ edges. These results contrast with the known lower bound $$\Omega(n(\log n)^2/\log\log n)$$, holding for 0-real-time automata, i.e., for automata with no $$\varepsilon$$-transitions.

### MSC:

 68Q45 Formal languages and automata
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### References:

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