Mathematical Preliminaries and Notation

Set theory

Basics of set theory

What is a FSM/A

• States
• • Non-final state
• • Final state

Acceptors / Transducers

Finite Automata: Deterministic Finite Accepters, Non-Deterministic Finite Accepters

Equivalence of Deterministic and Non-Deterministic Finite Accepters

Reduction of the number of states in Finite Automata

Regular Expressions, Connection between Regular Expressions and Regular Languages Regular Grammars

---

Regular Expressions (Regex)

Regular expressions (regex) are sequences of characters that define search patterns. They are commonly used for string searching, matching, and manipulation.

---

Basic Regex Symbols

Symbol	Meaning
.	Matches any character except a newline
*	Matches 0 or more of the preceding element
+	Matches 1 or more of the preceding element
?	Matches 0 or 1 of the preceding element
\d	Matches any digit (equivalent to [0-9])
\w	Matches any word character (alphanumeric)
\s	Matches any whitespace character
\b	Word boundary
^	Start of a string
$	End of a string
[]	Character class
()	Capturing group

---

Examples of Common Regex Patterns

1. Match a valid email address:

[a-zA-Z0-9._%+-]+@[a-zA-Z0-9.-]+\.[a-zA-Z]{2,}

This pattern matches standard email formats like user@example.com.

2. Match a phone number:

\(\d{3}\)\s?\d{3}-\d{4}

Matches phone numbers in the format (123) 456-7890.

3. Match a date in format DD/MM/YYYY:

\b\d{2}/\d{2}/\d{4}\b

Matches dates such as 06/10/2024.

---

Advanced Patterns

• Non-capturing group: Use (?: ...) for a group without capturing its match.
  • Example: (?:foo|bar) matches foo or bar without storing the result.
• Lookaheads and Lookbehinds:
  • Positive look ahead: (?=...) matches a group that is followed by a certain pattern.
  • Negative look ahead: (?!...) matches a group that is NOT followed by a certain pattern.
  • Positive look behind: (?<=...) matches a group that is preceded by a certain pattern.
  • Negative look behind: (?<!...) matches a group that is NOT preceded by a certain pattern.