Example of Regular Expression

Ex. 6:
Find a regular expression corresponding to the language of all strings over the alphabet { a, b } that contain no more than one occurence of the string aa.

Solution:
If there is one substring aa in a string of the language, then that aa can be followed by any number of b. If an a comes after that aa, then that a must be preceded by b because otherwise there are two occurences of aa. Hence any string that follows aa is represented by ( b + ba )*. On the other hand if an a precedes the aa, then it must be followed by b. Hence a string preceding the aa can be represented by ( b + ab )*. Hence if a string of the language contains aa then it corresponds to the regular expression ( b + ab )*aa( b + ba )* .

If there is no aa but at least one a exists in a string of the language, then applying the same argument as for aa to a, ( b + ab )*a( b + ba )* is obtained as a regular expression corresponding to such strings.

If there may not be any a in a string of the language, then applying the same argument as for aa to , ( b + ab )*( b + ba )* is obtained as a regular expression corresponding to such strings.
Altogether ( b + ab )*( + a + aa )( b + ba )* is a regular expression for the language.

Examples of Regular Expressions and Regular Languages.

examples of Regular expressions and Regular Languages.
Ex.:
Let r1 and r2 be arbitrary regular expressions over some alphabet. Find a simple (the shortest and with the smallest nesting of * and +) regular expression which is equal to each of the following regular expressions.

(a) (r1 + r2 + r1r2 + r2r1)*
(b) (r1(r1 + r2)*)+

Solution:
One general strategy to approach this type of question is to try to see whether or not they are equal to simple regular expressions that are familiar to us such as a, a*, a+, (a + b)*, (a + b)+ etc.

(a) Since (r1 + r2)* represents all strings consisting of strings of r1 and/or r2 , r1r2 + r2r1 in the given regular expression is redundant, that is, they do not produce any strings that are not represented by (r1 + r2)*. Thus (r1 + r2 + r1r2 + r2r1)* is reduced to (r1 + r2)*.

(b) (r1(r1 + r2)*)+ means that all the strings represented by it must consist of one or more strings of (r1(r1 + r2)*). However, the strings of (r1(r1 + r2)*) start with a string of r1 followed by any number of strings taken arbitrarily from r1 and/or r2.
Thus anything that comes after the first r1 in (r1(r1 + r2)*)+ is represented by (r1 + r2)*. Hence (r1(r1 + r2)*) also represents the strings of (r1(r1 + r2)*)+, and conversely (r1(r1 + r2)*)+ represents the strings represented by (r1(r1 + r2)*). Hence (r1(r1 + r2)*)+ is reduced to (r1(r1 + r2)*).

In our next Posts, we will see some more good examples of Regular expressions and Regular Languages.