Lesson 03.ppt PDF

Recap Lecture-2  Kleene Star Closure, Plus operation, recursive definition of languages, INTEGER, EVEN, factorial, PALINDROME, {anbn}, languages of strings (i) ending in a, (ii) beginning and ending in same letters, (iii) containing aa or bb (iv)containing exactly aa,...

Recap Lecture-2  Kleene Star Closure, Plus operation, recursive definition of languages, INTEGER, EVEN, factorial, PALINDROME, {anbn}, languages of strings (i) ending in a, (ii) beginning and ending in same letters, (iii) containing aa or bb (iv)containing exactly aa, 1  Regular Expression  As discussed earlier that a* generates Λ, a, aa, aaa, … and a+ generates a, aa, aaa, aaaa, …, so the language L1 = {Λ, a, aa, aaa, …} and L2 = {a, aa, aaa, aaaa, …} can simply be expressed by a* and a+, respectively. a* and a+ are called the regular expressions (RE) for L1 and L2 respectively. Note: a+, aa* and a*a generate L2. 2 Recursive definition of Regular Expression(RE) Step 1: Every letter of Σ including Λ is a regular expression. Step 2: If r1 and r2 are regular expressions then 1. (r1) 2. r1 r2 3. r1 + r2 and 4. r1* are also regular expressions. Step 3: Nothing else is a regular expression. 3 Defining Languages (continued)…  Method 3 (Regular Expressions)  Consider the language L={Λ, x, xx, xxx,…} of strings, defined over Σ = {x}. We can write this language as the Kleene star closure of alphabet Σ or L=Σ*={x}* this language can also be expressed by the regular expression x*.  Similarly the language L={x, xx, xxx,…}, defined over Σ = {x}, can be expressed by the regular expression x+. 4  Now consider another language L, consisting of all possible strings, defined over Σ = {a, b}. This language can also be expressed by the regular expression (a + b)*.  Now consider another language L, of strings having exactly double a, defined over Σ = {a, b}, then it’s regular expression may be b*aab* 5  Now consider another language L, of even length, defined over Σ = {a, b}, then it’s regular expression may be ((a+b)(a+b))*  Now consider another language L, of odd length, defined over Σ = {a, b}, then it’s regular expression may be (a+b)((a+b)(a+b))* or ((a+b)(a+b))*(a+b) 6  Remark  It may be noted that a language may be expressed by more than one regular expressions, while given a regular expression there exist a unique language generated by that regular expression. 7  Example:  Consider the language, defined over Σ={a , b} of words having at least one a, may be expressed by a regular expression (a+b)*a(a+b)*.  Consider the language, defined over Σ = {a, b} of words having at least one a and one b, may be expressed by a regular expression (a+b)*a(a+b)*b(a+b)*+ (a+b)*b(a+b)*a(a+b)*. 8  Consider the language, defined over Σ={a, b}, of words starting with double a and ending in double b then its regular expression may be aa(a+b)*bb  Consider the language, defined over Σ={a, b} of words starting with a and ending in b OR starting with b and ending in a, then its regular expression may be a(a+b)*b+b(a+b)*a 9 SummingUP Lecture 3 RE, Recursive definition of RE, defining languages by RE, { x}*, { x}+, {a+b}*, Language of strings having exactly one aa, Language of strings of even length, Language of strings of odd length, RE defines unique language (as Remark), Language of strings having at least one a, Language of strings havgin at least one a and one b, Language of strings starting with aa and ending in bb, Language of strings starting with and ending in different letters. 10