Question

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 the following regular expressions.

(r1 + r2 + r1r2 + r2r1)*.

Answer 1

The simplest form of the regular expression (r1 + r2 + r1r2 + r2r1)* is simply (r1 + r2)*, as the Kleene star operation includes all concatenations and permutations of r1 and r2.

Step-by-step explanation:

To find the simplest form of the given regular expression (r1 + r2 + r1r2 + r2r1)*, we need to apply the principles of regular expression simplification.

Understanding the operations involved:

• '+' denotes the union of regular expressions.
• '*' is the Kleene star, which represents zero or more occurrences of the expression.

Considering the provided expression, any string that can be derived from r1 or r2, or any concatenation of r1 and r2 in any order, can be represented by the expression (r1 + r2)*. This is because the Kleene star already takes into account all possible concatenations and permutations including repeating sequences, making the explicit r1r2 and r2r1 portions redundant.

Therefore, the simplified regular expression equivalent to (r1 + r2 + r1r2 + r2r1)* is simply (r1 + r2)*.