This document contains information about converting regular expressions to finite automata. It 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. It then provides steps for converting a regular expression to an NFA-Λ and converting an NFA-Λ to a finite automaton. The document concludes by recommending reviewing a textbook chapter on these topics.

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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