Regular Expressions - Automata Theory

What are Regular Expressions?

A Regular Expression is mathematically defined as follows −

• ε is a Regular Expression which express the language including an empty string. (L (ε) = {ε})
• φ is a Regular Expression expressed with an empty language. (L (φ) = { })
• x is a Regular Expression where L = {x}
• If X is a Regular Expression expressed in the language L(X)and Y is a Regular Expression denoting the language L(Y), then
• X + Y is a Regular Expression corresponding to the language L(X) ∪ L(Y) where L(X+Y) = L(X) ∪ L(Y).
• X . Y is a Regular Expression corresponding to the language L(X) . L(Y) where L(X.Y) = L(X) . L(Y)
• R* is a Regular Expression corresponding to the language L(R*)where L(R*) = (L(R))*
• If we apply any of the rules several times from 1 to 5, they are Regular Expressions.

Some RE Examples

Regular Expressions Regular Set
(0 + 10*) L = { 0, 1, 10, 100, 1000, 10000, … }
(0*10*) L = {1, 01, 10, 010, 0010, …}
(0 + ε)(1 + ε) L = {ε, 0, 1, 01}
(a+b)* Set of strings of a’s and b’s of any length including the null string. So L = { ε, a, b, aa , ab , bb , ba, aaa…….}
(a+b)*abb Set of strings of a’s and b’s ending with the string abb. So L = {abb, aabb, babb, aaabb, ababb, …………..}
(11)* Set consisting of even number of 1’s including empty string, So L= {ε, 11, 1111, 111111, ……….}
(aa)*(bb)*b Set of strings consisting of even number of a’s followed by odd number of b’s , so L = {b, aab, aabbb, aabbbbb, aaaab, aaaabbb, …………..}
(aa + ab + ba + bb)* String of a’s and b’s of even length can be obtained by concatenating any combination of the strings aa, ab, ba and bb including null, so L = {aa, ab, ba, bb, aaab, aaba, …………..}

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