Single Final State for NFA Fall 2004 COMP 335

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

Example NFA Equivalent NFA Fall 2004 COMP 335

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

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

Properties of Regular Languages Fall 2004 COMP 335

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

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

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

Example Fall 2004 COMP 335

Union NFA for Fall 2004 COMP 335

Example NFA for Fall 2004 COMP 335

Concatenation NFA for Fall 2004 COMP 335

Example NFA for Fall 2004 COMP 335

Star Operation NFA for Fall 2004 COMP 335

Example NFA for Fall 2004 COMP 335

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

Example Fall 2004 COMP 335

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

Example Fall 2004 COMP 335

Intersection DeMorgan’s Law: regular Fall 2004 COMP 335

Example regular regular regular Fall 2004 COMP 335

Regular Expressions Fall 2004 COMP 335

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

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

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

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

Definition For primitive regular expressions : Fall 2004 COMP 335

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

Example Regular expression: Fall 2004 COMP 335

Example Regular expression Fall 2004 COMP 335

Example Regular expression Fall 2004 COMP 335

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

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

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

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

Regular Expressions and Regular Languages Fall 2004 COMP 335

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

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

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

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

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

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

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

By definition of regular expressions: Fall 2004 COMP 335

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

Therefore: Are regular languages Fall 2004 COMP 335

And trivially: is a regular language Fall 2004 COMP 335

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

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

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

Another Example: Fall 2004 COMP 335

Reducing the states: Fall 2004 COMP 335

Resulting Regular Expression: Fall 2004 COMP 335

In General Removing states: Fall 2004 COMP 335

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