Regular Expressions 2018 Dr Sandip Mal 1 Regular

Regular Expressions 2018 Dr. Sandip Mal 1

Regular Expressions Regular expressions describe regular languages Example: describes the language 2018 Dr. Sandip Mal 2

Recursive Definition Primitive regular expressions: Given regular expressions and Are regular expressions 2018 Dr. Sandip Mal 3

Examples A regular expression: Not a regular expression: 2018 Dr. Sandip Mal 4

Languages of Regular Expressions : language of regular expression Example 2018 Dr. Sandip Mal 5

Definition For primitive regular expressions: 2018 Dr. Sandip Mal 6

Definition (continued) For regular expressions 2018 Dr. Sandip Mal and 7

Example Regular expression: 2018 Dr. Sandip Mal 8

Example Regular expression 2018 Dr. Sandip Mal 9

Example Regular expression 2018 Dr. Sandip Mal 10

Example Regular expression = { all strings containing substring 00 } 2018 Dr. Sandip Mal 11

Example Regular expression = { all strings without substring 00 } 2018 Dr. Sandip Mal 12

Equivalent Regular Expressions Definition: Regular expressions and are equivalent if 2018 Dr. Sandip Mal 13

Example = { all strings without substring 00 } and are equivalent regular expressions 2018 Dr. Sandip Mal 14

Regular Expressions and Regular Languages 2018 Dr. Sandip Mal 15

Theorem Languages Generated by Regular Expressions 2018 Dr. Sandip Mal Regular Languages 16

Proof: Languages Generated by Regular Expressions Regular Languages 2018 Dr. Sandip Mal 17

Proof - Part 1 Languages Generated by Regular Expressions Regular Languages For any regular expression the language is regular Proof by induction on the size of 2018 Dr. Sandip Mal 18

Induction Basis Primitive Regular Expressions: Corresponding NFAs regular languages 2018 Dr. Sandip Mal 19

Inductive Hypothesis Suppose that for regular expressions and , and are regular languages 2018 Dr. Sandip Mal 20

Inductive Step We will prove: Are regular Languages 2018 Dr. Sandip Mal 21

By definition of regular expressions: 2018 Dr. Sandip Mal 22

By inductive hypothesis we know: and are regular languages We also know: Regular languages are closed under: Union Concatenation Star 2018 Dr. Sandip Mal 23

Therefore: Are regular languages is trivially a regular language (by induction hypothesis) 2018 Dr. Sandip Mal End of Proof-Part 1 24

Using the regular closure of these operations, we can construct recursively the NFA that accepts Example: 2018 Dr. Sandip Mal 25

Proof - Part 2 Languages Generated by Regular Expressions Regular Languages For any regular language there is a regular expression with We will convert an NFA that accepts to a regular expression 2018 Dr. Sandip Mal 26

Since NFA is regular, there is a that accepts it Take it with a single final state 2018 Dr. Sandip Mal 27

From construct the equivalent Generalized Transition Graph in which transition labels are regular expressions Example: 2018 Corresponding Generalized transition graph Dr. Sandip Mal 28

Another Example: Transition labels are regular expressions 2018 Dr. Sandip Mal 29

Reducing the states: Transition labels are regular expressions 2018 Dr. Sandip Mal 30

Resulting Regular Expression: 2018 Dr. Sandip Mal 31

In General Removing a state: 2018 Dr. Sandip Mal 32

By repeating the process until two states are left, the resulting graph is Initial graph Resulting graph The resulting regular expression: 2018 Dr. Sandip Mal End of Proof-Part 2 33

Standard Representations of Regular Languages DFAs NFAs 2018 Dr. Sandip Mal Regular Expressions 34

When we say: We mean: We are given a Regular Language is in a standard representation (DFA, NFA, or Regular Expression) 2018 Dr. Sandip Mal 35