1. CSC312
Automata Theory
Lecture # 4
Languages-III

2. Languages of Regular Expressions
Example: Consider the alphabet Σ={a} The language of all words containing even number of a’s can be defined by the following RE (aa)* Example: Language of all words containing only odd no. of a’s can be defined by the following RE 1. (aaa)* 2. a(aa)*+ 3. a+(aa)* 4. a+a* 5. a+(aa)*a correct but inefficient due to repetition 6. (aa)*a or a(aa)*

3. Languages of Regular Expressions
Example: The language of all words having all possible combinations of a’s followed be one b can be described by the following RE.
1. a+b 2. a*+b 3. a*b 4. (+a+)b 5. a+b+b
Example: The language of all words in which all a’s (if any) comes before all the b’s (if any) can be defined the following RE
(ab)* 2. a*b*
3. a+b+ +  4. a+b* +  both are inefficient

4. Example:The language of all words of a’s & b’s that have atleast two letters, that begin & end with a’s & that have nothing but b’s inside (if any thing at all) can be defined by following RE.
Σ = {a, b}
(aba)* 2. ab*a ab+a 4. a+b*a+
all above are incorrect
5. ab*a

5. Example: Consider the alphabet Σ={a,b,c}
Example: Consider the alphabet Σ={a,b,c}. The language of all words that begins with either a or c, followed be any no. of b’s can be defined by following RE. (a+c)b* = ab* + cb* Example: The language of all words that ends with letter b can be defined be RE (a+b)*b

6. Note: All finite languages are regular whereas some infinite languages are regular & some are non-regular. In order to write RE for finite languages just include +’s between the words of finite languages. Consider the alphabet Σ={a,b}. The words of length 1 are two. Its RE will be a+b. Similarly the words of length 2 will be aa,ab,ba,bb. Its RE will be aa+ab+ba+bb or (a+b)(b+a). Similary the languages of all words whose length is 3 will be aaa,aab,aba,abb,bbb,bba,bab,baa. Thus we can conclude that all words of a’s and b’s of any length can be defined by (a+b)*.

7. Definition
For regular expressions and

8. Example
Regular expression:

9. Example
Regular expression

10. Example = { all strings with at least two consecutive 0 }
= { all strings without
two consecutive 0 }

11. Equivalent Regular Expressions
Definition:
Regular expressions and
are equivalent if

12. Examples (a+)* = (a*)+ = a* (a*+b)* = (a+b)*
(a+b)*ab(a+b)* = (a+b)*a(a+b)*b(a+b)*
(a+b)*= (a*b*)*
(a+b)*= (a*b*+b*a*)*
(aa+ab*)* = (aa+ab)*
a*a* = (aa)*

13. Regular Expressions and Regular Languages

14. Theorem
Languages
Generated by
Regular Expressions
Regular
Languages