1.3 regular expressions

Formal Def of a Regular Expression

Say that R is a regular expression if R is: - $a$ for some $a$ in the alphabet $\Sigma$ - $\varepsilon$ - $\emptyset$ - ($R_1 \cup R_2$) - ($R_1 \circ R_2$) - ($R_1^*$)

($R_1 R_2$) are regualr expressions(inductive def)

some conclusions

• L(R): the language of R
• concatenating the empty set to any set yields the empty set, so $1 * \emptyset = \emptyset$
• $\emptyset * = \{\varepsilon\}$
• $R \cup \emptyset = R$
• $R \circ \varepsilon = R$

Equivalence with Finite Automata

• hint: regular language is one that is recognized by some finite automation
• a language is regular iff some regular expression describes it
• prove see textbook p67 p70
• GNFA: generalized nondeterministic finite automation, its transition arrows may have any regular expression as labels