Fall 2006Costas Busch - RPI1 Regular Expressions

Fall 2006Costas Busch - RPI2 Regular Expressions Regular expressions describe regular languages Example: describes the language

Fall 2006Costas Busch - RPI3 Recursive Definition Are regular expressions Primitive regular expressions: Given regular expressions and

Fall 2006Costas Busch - RPI4 Examples A regular expression: Not a regular expression:

Fall 2006Costas Busch - RPI5 Languages of Regular Expressions : language of regular expression Example

Fall 2006Costas Busch - RPI6 Definition For primitive regular expressions:

Fall 2006Costas Busch - RPI7 Definition (continued) For regular expressions and

Fall 2006Costas Busch - RPI8 Example Regular expression:

Fall 2006Costas Busch - RPI9 Example Regular expression

Fall 2006Costas Busch - RPI10 Example Regular expression

Fall 2006Costas Busch - RPI11 Example Regular expression = { all strings containing substring 00 }

Fall 2006Costas Busch - RPI12 Example Regular expression = { all strings without substring 00 }

Fall 2006Costas Busch - RPI13 Equivalent Regular Expressions Definition: Regular expressions and are equivalent if

Fall 2006Costas Busch - RPI14 Example = { all strings without substring 00 } and are equivalent regular expressions

Fall 2006Costas Busch - RPI15 Regular Expressions and Regular Languages

Fall 2006Costas Busch - RPI16 Theorem Languages Generated by Regular Expressions Regular Languages

Fall 2006Costas Busch - RPI17 Languages Generated by Regular Expressions Regular Languages Generated by Regular Expressions Regular Languages Proof:

Fall 2006Costas Busch - RPI18 Proof - Part 1 For any regular expression the language is regular Languages Generated by Regular Expressions Regular Languages Proof by induction on the size of

Fall 2006Costas Busch - RPI19 Induction Basis Primitive Regular Expressions: Corresponding NFAs regular languages

Fall 2006Costas Busch - RPI20 Inductive Hypothesis Suppose that for regular expressions and, and are regular languages

Fall 2006Costas Busch - RPI21 Inductive Step We will prove: Are regular Languages

Fall 2006Costas Busch - RPI22 By definition of regular expressions:

Fall 2006Costas Busch - RPI23 By inductive hypothesis we know: and are regular languages Regular languages are closed under: Union Concatenation Star We also know:

Fall 2006Costas Busch - RPI24 Therefore: Are regular languages is trivially a regular language (by induction hypothesis) End of Proof-Part 1

Fall 2006Costas Busch - RPI25 Using the regular closure of these operations, we can construct recursively the NFA that accepts Example:

Fall 2006Costas Busch - RPI26 For any regular language there is a regular expression with Proof - Part 2 Languages Generated by Regular Expressions Regular Languages We will convert an NFA that accepts to a regular expression

Fall 2006Costas Busch - RPI27 Since is regular, there is a NFA that accepts it Take it with a single final state

Fall 2006Costas Busch - RPI28 From construct the equivalent Generalized Transition Graph in which transition labels are regular expressions Example: Corresponding Generalized transition graph

Fall 2006Costas Busch - RPI29 Another Example: Transition labels are regular expressions

Fall 2006Costas Busch - RPI30 Reducing the states: Transition labels are regular expressions

Fall 2006Costas Busch - RPI31 Resulting Regular Expression:

Fall 2006Costas Busch - RPI32 In General Removing a state:

Fall 2006Costas Busch - RPI33 The resulting regular expression: By repeating the process until two states are left, the resulting graph is Initial graphResulting graph End of Proof-Part 2

Fall 2006Costas Busch - RPI34 Standard Representations of Regular Languages Regular Languages DFAs NFAs Regular Expressions

Fall 2006Costas Busch - RPI35 When we say: We are given a Regular Language We mean:Language is in a standard representation (DFA, NFA, or Regular Expression)