Lecture on Regular Expressions, Languages, and Transition Functions

Overview

• Discussion on regular expressions (regex)
• Focus on union and concatenation in regex
• Explanation with Cartesian products

Key Points

Regular Expressions (Regex)

• Definitions and specifications for union and concatenation
• Regular expressions are denoted for languages L and M.

Union of Languages

• Represented as L + R
• Example: If L and M are two languages, their union (L ∪ M) combines all elements of both languages.

Concatenation

• Defined as the linking of two string sequences
• Represented as L.R or L followed by M

Cartesian Products

• Use in defining state transitions
• Key components in automaton
  • States (Q)
  • Alphabet (Σ)
  • Transition function (Δ)
  • Start state (Q0)
  • Final states (F)

Transition Functions

Definitions and Examples

• Illustrates how transition functions work using Cartesian products

• Transition function (Δ) for the union state involves combining the individual transitions from the component languages.

• Example demonstration:

  • Δ1: Transition function for first DFA
  • Δ2: Transition function for second DFA
  • Combined Transition: Δ(Q0, a) = (Δ1(Q0, a), Δ2(Q1, a))

• Start state and final states are also determined through Cartesian products.

Summary

• Regular expressions represent languages and can be combined through union and concatenation.
• Transition functions are key to understanding state changes in automata and can be defined using Cartesian products.