Recap: Kleene Closure

• $L^* = \{w \in \Sigma^* ~|~ \exists n \in \mathbb N. ~w \in L^n \}$
• think of L* as the set of strings you can make if you have a collection of stamps, one for each string in L, and you form every possible string that can be made from those stamps

Rethinking Regular languages

• several tools for showing language $L$ is regular
  • construct DFA for L
  • construct NFA for L
  • combine several simpler regular languages together via closure properties to form L
    • using this one today
• we’ll build up all regular languages

Regular expressions

• way of describing a language via a string representation
• conceptually, strings describing how to assemble a larger language out of smaller pieces

Atomic Regular Expressions

• three simple building blocks
• $\emptyset$ → regular expression that represents the empty language $\emptyset$
• For any $a \in \Sigma$, symbol $a$ is a regular expression for the language $\{a\}$
  • write out the letter a → describes the language $\{a\}$
• $\varepsilon$ → regular expression that represents the language $\{ \varepsilon \}$

Remember, $\{\varepsilon \} \neq \emptyset$, and $\{ \varepsilon \} \neq \varepsilon$.

Compound Regular Expressions (note: regex = regular expression)

• If $R_1$ and $R_2$ are regular expressions, $R_1R_2$ is a regular expression for the concatenation of the languages of $R_1$ and $R_2$
  • writing “hi” means concatenating $\{h\}$ and $\{i\}$ → $\{hi\}$
  • any string with just letters represents the set with just that string in it
• $R_1 \cup R_2$ is a regular expression for the union of $R_1$ and $R_2$
  • writing “hello $\cup$ goodbye” = $\{ hello, goodbye\}$
• If $R$ is a regular expression, $R^*$ is a regular expression for the Kleene closure of the language of $R$
  • e.g. $z^* = \{z\}^*=\{\varepsilon, z, zz, zzz, zzzz,\dots \}$
• parentheses in regular expressions → $(R)$ means the same thing as $R$