1. Language Recognition (12.4)
  Longin Jan Latecki
Temple University
Based on slides by Costas Busch from the course
and …

2. Three Equivalent Representations
Regular
expressions
Each
can
describe
the others
Regular
languages
Finite
automata
Kleene’s Theorem:
For every regular expression, there is a deterministic finite-state automaton that defines the same language, and vice versa.

3. EXAMPLE 1
Consider the language { ambn | m, n  N}, which is represented by the regular expression a*b*.
A regular grammar for this language can be written as follows:
S  | aS | B
B  b | bB.

4. Regular Expression
Regular Grammar a*
S   | aS
(a+b)*
S   | aS | bS
a* + b*
S   | A | B
A  a | aA
B  b | bB
a*b
S  b | aS
ba*
S  bA
A   | aA
(ab)*
S   | abS

5. NFAs  Regular grammars Thus, the language recognized by FSA is a regular language
Every NFA can be converted into a corresponding regular grammar and vice versa.
Each symbol A of the grammar is associated with a non-terminal node of the NFA sA, in particular, start symbol
S is associated with the start state sS.
Every transition is associated with a grammar production:
T(sA,a) = sB  A  aB.
Every production B   is associated with final state sB.

6. Equivalent FSA and regular grammar, Ex. 4, p. 772.
G=(V,T,S,P)
V={S, A, B, 0, 1} with
S=s0, A=s1, and B=s2,
T={0,1}, and productions are
S  0A | 1B | 1 | λ
A  0A | 1B | 1
B  0A | 1B | 1 | λ

7. Kleene’s Theorem Languages Generated by Languages Recognized by FSA
Regular Expressions
Languages Recognized by FSA

8. We will show:
Languages
Generated by
Regular Expressions
Languages Recognized by FSA
Languages
Generated by
Regular Expressions
Languages Recognized by FSA

9. For any regular expression
Proof - Part 1
Languages
Generated by
Regular Expressions
Languages Recognized by FSA
For any regular expression
the language is recognized by FSA (= is a regular language)
Proof by induction on the size of

10. Induction Basis
Primitive Regular Expressions:
NFAs
regular
languages

11. Inductive Hypothesis Assume for regular expressions and that
and are regular languages

12. Inductive Step
We will prove:
Are regular
Languages

13. By definition of regular expressions:

14. By inductive hypothesis we know:
and are regular languages
Regular languages are closed under:
Union
Concatenation
Star
We need to show:
This fact is illustrated in Fig. 2 on p. 769.

15. Therefore:
Are regular
languages
And trivially:
is a regular language

16. Proof - Part 2 Languages Generated by Languages Recognized by FSA
Regular Expressions
Languages Recognized by FSA
For any regular language there is
a regular expression with
Proof by construction of regular expression

17. Since is regular take the
NFA that accepts it
Single final state

18. From construct the equivalent
Generalized Transition Graph
in which transition labels are regular expressions
Example:

19. Another Example:

20. Reducing the states:

21. Resulting Regular Expression:

22. In General
Removing states:

23. The final transition graph:
The resulting regular expression:

24. Three Equivalent Representations
Regular
expressions
Each
can
describe
the others
Regular
languages
Finite
automata