1. Automata

1.5 The Central Concept of Automata Theory

1.5.1 Alphabet

• Alphabet is the finite, non-empty Set of the symbol
• Using the $\sum$ symbol stand for the Alphabet

1.5.2 String

String, sometime called word is a finite sequence of symbols, which choose from the Alphabet.

• Empty String

If the string has no symbol, it is empty String, use the $\epsilon$ stand for it.

• String’s lenth

The number of symbol in the the string called the lenth of the string, use the $|w|$ stand for the lenth of string $w$

• The power of Alphabet

Use the exponent of the Alphabet like $\sum^k$ stand for the Set of string, which lenth is $k$, in $\sum$
Notice that $\sum^0 = \{\epsilon\}$

The set of all strings in an Alphabet, use $\sum^*$
If remove the $\epsilon$ in the normal Alphabet, the rest is called $\sum^+$
$\sum^+ = \sum^0\bigcup\sum^1\bigcup\sum^2\bigcup\dots$

$\sum^*=\sum^+\bigcup\{\epsilon\}$