Single Final State for NFA Fall 2004 COMP

Single Final State for NFA Fall 2004 COMP 335 1

Any NFA can be converted to an equivalent NFA with a single final state Fall 2004 COMP 335 2

Example NFA Equivalent NFA Fall 2004 COMP 335 3

NFA In General Equivalent NFA Single final state Fall 2004 COMP 335 4

Extreme Case NFA without final state Add a final state Without transitions Fall 2004 COMP 335 5

Properties of Regular Languages Fall 2004 COMP 335 6

For regular languages we will prove that: and Union: Concatenation: Are regular Languages Star: Reversal: Complement: Intersection: Fall 2004 COMP 335 7

We say: Regular languages are closed under Union: Concatenation: Star: Reversal: Complement: Intersection: Fall 2004 COMP 335 8

Regular language NFA Single final state Fall 2004 Single final state COMP 335 9

Example Fall 2004 COMP 335 10

Union NFA for Fall 2004 COMP 335 11

Example NFA for Fall 2004 COMP 335 12

Concatenation NFA for Fall 2004 COMP 335 13

Example NFA for Fall 2004 COMP 335 14

Star Operation NFA for Fall 2004 COMP 335 15

Example NFA for Fall 2004 COMP 335 16

Reverse NFA for 1. Reverse all transitions Fall 2004 2. Make initial state final state and vice versa COMP 335 17

Example Fall 2004 COMP 335 18

Complement 1. Take the DFA that accepts 2. Make final states non-final, and vice-versa Fall 2004 COMP 335 19

Example Fall 2004 COMP 335 20

Intersection De. Morgan’s Law: regular regular Fall 2004 COMP 335 21

Example regular Fall 2004 COMP 335 regular 22

Regular Expressions Fall 2004 COMP 335 23

Regular Expressions Regular expressions describe regular languages Example: describes the language Fall 2004 COMP 335 24

Recursive Definition Primitive regular expressions: Given regular expressions and Are regular expressions Fall 2004 COMP 335 25

Examples A regular expression: Not a regular expression: Fall 2004 COMP 335 26

Languages of Regular Expressions : language of regular expression Example: Fall 2004 COMP 335 27

Definition For primitive regular expressions : Fall 2004 COMP 335 28

Definition (continued) For regular expressions Fall 2004 COMP 335 and 29

Example Regular expression: Fall 2004 COMP 335 30

Example Regular expression Fall 2004 COMP 335 31

Example Regular expression Fall 2004 COMP 335 32

Example Regular expression = {all strings with at least two consecutive 0} Fall 2004 COMP 335 33

Example Regular expression = { all strings without two consecutive 0 } Fall 2004 COMP 335 34

Equivalent Regular Expressions Definition: Regular expressions and are equivalent if Fall 2004 COMP 335 35

Example = { all strings without two consecutive 0 } and are equivalent Reg. expressions Fall 2004 COMP 335 36

Regular Expressions and Regular Languages Fall 2004 COMP 335 37

Theorem Languages Generated by Regular Expressions Fall 2004 COMP 335 Regular Languages 38

Theorem - Part 1 Languages Generated by Regular Expressions 1. For any regular expression the language Fall 2004 Regular Languages is regular COMP 335 39

Theorem - Part 2 Languages Generated by Regular Expressions 2. For any regular language a regular expression Fall 2004 COMP 335 Regular Languages , there is with 40

Proof - Part 1 1. For any regular expression the language is regular Proof by induction on the size of Fall 2004 COMP 335 41

Induction Basis Primitive Regular Expressions: NFAs regular languages Fall 2004 COMP 335 42

Inductive Hypothesis Assume for regular expressions and that and are regular languages Fall 2004 COMP 335 43

Inductive Step We will prove: are regular Languages. Fall 2004 COMP 335 44

By definition of regular expressions: Fall 2004 COMP 335 45

By inductive hypothesis we know: and are regular languages We also know: Regular languages are closed under: Union Concatenation Star Fall 2004 COMP 335 46

Therefore: Are regular languages Fall 2004 COMP 335 47

And trivially: is a regular language Fall 2004 COMP 335 48

Proof – Part 2 2. For any regular language a regular expression there is with Proof by construction of regular expression Fall 2004 COMP 335 49

Since NFA is regular, take an that accepts it Single final state Fall 2004 COMP 335 50

From , construct an equivalent Generalized Transition Graph in which transition labels are regular expressions Example: Fall 2004 COMP 335 51

Another Example: Fall 2004 COMP 335 52

Reducing the states: Fall 2004 COMP 335 53

Resulting Regular Expression: Fall 2004 COMP 335 54

In General Removing states: Fall 2004 COMP 335 55

The final transition graph: The resulting regular expression: Fall 2004 COMP 335 56