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Is there any systematic (algorithmic) method to convert a regular expression like

$(a^*)(b)(b^*)(a^*)c$

to an automaton (let's say a Buchi automaton)?

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    $\begingroup$ Regular expressions connote languages of finite words, whereas Büchi automata accept infinite words. On the other hand, regular expressions are classically equivalent in power to DFAs, and there is a very simple algorithm that converts a regular expression to an NFA. You can find it in any decent textbook. $\endgroup$ – Yuval Filmus Jan 17 '17 at 18:13
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There are (at least) three algorithms that convert a regular expression to an equivalent finite-state automaton:

Büchi automata enter the picture if you are working with $\omega$-regular expressions. See e.g. this description of the algorithm for constructing an equivalent Büchi automaton from an $\omega$-regular expression.

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