Will $L = \{a^* b^*\}$ be classified as a regular language?

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Will $L = \{a^* b^*\}$ be classified as a regular language?

I am confused because I know that $L = \{a^n b^n\}$ is not regular. What difference does the kleene star make?

Asked By : user6268553

Answered By : Yuval Filmus

A language is regular, by definition, if it is accepted by some DFA. (This is at least one common definition.) Can you think of a DFA accepting the language?

A well-known result (that is proved in many textbooks) states that the language of a regular expression is regular. Since $a^* b^*$ is a regular expression, its language must be regular (if you believe this result).

Finally, to answer your question (what difference does the Kleene star make): in the language $\{a^n b^n : n \geq 0\}$, we need to count the number of $a$s and $b$s; in the language $a^*b^*$ we don't.

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Question Source : http://cs.stackexchange.com/questions/56745