Regular Languages A language is regular over  if it can be built from ;, {  }, and { a } for every a 2 , using operators union ( [ ), concatenation (. ) and Kleene star ( * ). Example : ( { a } [ { b } ) * { a } { a } ( { a } [ { b } ) * [ ( { a } [ { b } )* { b } { b } ( { a } [ { b } ) * [Section 3.1]

Regular Languages A language is regular over  if it can be built from ;, {  }, and { a } for every a 2 , using operators union ( [ ), concatenation (. ) and Kleene star ( * ). Example : ( { a } [ { b } ) * { a } { a } ( { a } [ { b } ) * [ ( { a } [ { b } )* { b } { b } ( { a } [ { b } )^* Simplify as : [Section 3.1]

A language is regular over  if it can be built from ;, {  }, and { a } for every a 2 , using operators union ( [ ), concatenation (. ) and Kleene star ( * ). Recursive definition of regular languages over  : [Section 3.1] Regular Languages

Recursive definition of regular expressions over  : 1) ;, {  }, and { a } for every a 2  are regular languages represented by regular expressions __, __, __, respectively. 2) If L 1, L 2 are regular languages represented by regular Expressions r 1 and r 2, then L 1 [ L 2, L 1 L 2, and L 1 * are regular languages represented by regular expressions _____, _____, _____, respectively. 3) No other expressions are regular expressions. [Section 3.1] Regular Expressions

Comments : - For convenience, we will also use r + and r i for i 2 N. - The priority of operators in decreasing order : Kleene’s star, concatenation, union. Parenthesize : a + b * c a + bc * [Section 3.1]

Regular Expressions Simplify the following regular expressions : - a * (a+  ) - a * a * - (a * b * ) * - (a+b) * ab(a+b) * + a * b * [Section 3.1]

Regular Expressions Give a regular expression for each of the following : - the language of all strings over {a,b} of even length, - the language of all strings over {a,b,c} in which all a’s precede all b’s, - the language of all strings over {a,b} with length greater than 3, - the language of all odd-length strings over {a,b} containing the string bb, - the language of all strings over {a,b} that do not contain the string aaa. Is the language { a k b k | k 2 N } regular ? [Section 3.1]

Finite Automata Our simplest model of computation : - has a tape with input, read once from left to right - has only a single memory register with the state [Sections 3.2, 3.3]

Finite Automata Example : the language of all strings over {a,b} with an even number of a’s. [Sections 3.2, 3.3]

Finite Automata More examples : - Language of strings over {a,b} with at least 3 a’s. - Language of strings over {a,b} containing substring aaba. - Language of strings over {a,b} not containing substring aaba. - Language of strings over {a,b} with even number of a’s and odd number of b’s. [Sections 3.2, 3.3]

Finite Automata A (deterministic) finite automaton (FA, in some books also abbreviated as DFA) is a 5-tuple (Q, ,q 0,A,  ) where - Q is a finite set of states -  is a finite alphabet (input symbols) - q o 2 Q is the initial state - A µ Q is a set of accepting states -  : Q £   Q is the transition function [Sections 3.2, 3.3]

Finite Automata Comment : We saw that we can represent a finite automaton by a transition diagram. Draw the transition diagram for ( {q a,q b }, {a,b}, q a, {q b },  ) where  is given by the following table (q a,a)  q a (q a,b)  q b (q b,a)  q a (q b,b)  q b Which strings does this FA accept ? [Sections 3.2, 3.3]

Finite Automata Defining the computation of an FA M=(Q, ,q 0,A,  ). Extended transition function  * : Q £  *  Q : 1) For every q 2 Q, let  * (q,  ) = 2) For every q 2 Q, y 2  *, and  2 , let  * (q,y  ) = We say that a string x 2  * is accepted by M if __________. A string which is not accepted by M is rejected by M. The language accepted by M, denoted by L(M) is the set of all strings accepted by M. [Sections 3.2, 3.3]

Finite Automata Kleene’s Thm: A language L over  is regular iff there exists a finite automaton that accepts L. Recall L = { a k b k | k ¸ 0 }. Is it regular ? [Section 3.4]

Operations on Finite Automata Let M 1 = (Q 1, ,q 1,A 1,  1 ) and M 2 = (Q 2, ,q 2,A 2,  2 ) be two FA’s. Union: We know that L(M 1 ) [ L(M 2 ) is regular. Construct an FA M such that L(M) = L(M 1 ) [ L(M 2 ). [Section 3.5]

Operations on Finite Automata Let M 1 = (Q 1, ,q 1,A 1,  1 ) and M 2 = (Q 2, ,q 2,A 2,  2 ) be two FA’s. Intersection: Is L(M 1 ) Å L(M 2 ) is regular ? Construct an FA M such that L(M) = L(M 1 ) Å L(M 2 ). [Section 3.5]

Operations on Finite Automata Let M 1 = (Q 1, ,q 1,A 1,  1 ) be an FA. Complement: Is L(M 1 )’ regular ? Construct an FA M such that L(M) = L(M 1 )’. [Section 3.5]

Operations on Finite Automata Let M 1 = (Q 1, ,q 1,A 1,  1 ) and M 2 = (Q 2, ,q 2,A 2,  2 ) be two FA’s. Difference: Construct an FA M such that L(M) = L(M 1 ) - L(M 2 ). [Section 3.5]

Closure Properties We just saw that the class of regular languages is closed under union, intersection, complement, and difference. Let L = { x 2 {a,b} * | x has the same number of a’s and b’s }. Is L regular ? [Section 3.5]