This document contains information about converting regular expressions to finite automata. It begins with questions about the relationships between regular expressions, finite automata, and languages. It then discusses Kleene's Theorem, which states that any language that can be defined by a regular expression can also be defined by a finite automaton, and vice versa. The document provides steps for converting a regular expression to an NFA-Λ and then converting the NFA-Λ to a finite automaton. It concludes with a reminder to review these concepts in the textbook.

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2. Converting Regular ExpressionsConverting Regular Expressions
into Finite Automatainto Finite Automata
CSE2303 Formal Methods I
Lecture 5

3. OverviewOverview
• Questions
• Kleene’s Theorem
• Consequences
• Convert Regular Expressions to NFA-Λ
• Convert NFA-Λ to FA

4. QuestionsQuestions
• Can every language which is represented by
a regular expression be described by a
finite automaton?
• Can every language which is described by a
finite automaton be represented by a
regular expression?
• Can every language be represented by a
regular expression or a finite automaton?

5. Kleene’s TheoremKleene’s Theorem
Any language which can be defined by
– Regular Expressions
– Finite Automaton
– Nondeterministic Finite Automaton (NFA)
– NFA-Λ
– Transition Graphs
– Generalised Transistion Graphs
can be defined by any of the other methods

6. The Complement of RegularThe Complement of Regular
Language is a Regular LanguageLanguage is a Regular Language
Outline of Proof:
– Suppose we have a Regular Language.
– Therefore we have a regular expression that
defines the language.
– So, by Kleene’s Theorem, there is a FA that
defines this language.
– We can convert this FA into one that defines the
complement the language.
– So, by Kleene’s Theorem, there is a regular
expression that defines the complement.

7. Kleene’s TheoremKleene’s Theorem
Regular Expression
Finite Automaton
NFA-ΛGTG
TG NFA

8. How to convert aHow to convert a
Regular ExpressionRegular Expression
into ainto a
NFA-NFA-ΛΛ

9. Converting Regular ExpressionConverting Regular Expression
to NFA-to NFA-ΛΛ
Start with the graph.
- +
Regular Expression

10. Apply the following rules until all edges are
labeled with a letter or Λ.
1. Delete any edge labeled with φ.
2. Transform any edge like
RS
R
into
S

11. 3. Transform any edge like:
into
R + S
R
S

12. 4. Transform any edge like:
into
R*
R
Λ Λ

13. 1-
a
b
Λ
2 3+
4
5
Λ
Λ
Λ
6 7
Λ
Λ
a
a
(a* + aa*b)*(a* + aa*b)*

14. How to convert aHow to convert a
NFA-NFA-ΛΛ
into ainto a
FAFA

15. a
bb
-
a,b a,ba
3
2
41 +
a b
Start {1} {1,2} {1,3}
{1,2} {1,2,4} {1,3}
{1,3} {1,2} {1,3,4}
Final {1,2,4} {1,2,4} {1,3,4}
Final {1,3,4} {1,2,4} {1,3,4}

16. 1-
a b
Λ
2 3 4 5+
Λ Λ Λ
a b
Start/Final {1,2,3,4,5} {2,3,4,5} {4,5}
Final {2,3,4,5} {2,3,4,5} {4,5}
Final {4,5} φ {4,5}
φ φ φ

17. 1-
a
b
Λ
2 3+
4
5
Λ
Λ
Λ
6 7
Λ
Λ
a
a
a b
Start /Final{1,2,3,4} {5,4,6,7,2,3} φ
Final {2,3,4,5,6,7} {5,4,6,7,2,3} {2,3,4}
Final {2,3,4} {5,4,6,7,2,3} φ
φ φ φ

18. RevisionRevision
• Understand Kleene’s Theorem
• Be able to convert Regular Expressions into NFA-Λ
• Be able to convert NFA-Λ into a Finite Automaton
• Read
– Text Book Chapter 8