fall 2006costas busch - rpi1 regular expressions
Post on 19-Dec-2015
212 views
Embed Size (px)
TRANSCRIPT
- Slide 1
- Fall 2006Costas Busch - RPI1 Regular Expressions
- Slide 2
- Fall 2006Costas Busch - RPI2 Regular Expressions Regular expressions describe regular languages Example: describes the language
- Slide 3
- Fall 2006Costas Busch - RPI3 Recursive Definition Are regular expressions Primitive regular expressions: Given regular expressions and
- Slide 4
- Fall 2006Costas Busch - RPI4 Examples A regular expression: Not a regular expression:
- Slide 5
- Fall 2006Costas Busch - RPI5 Languages of Regular Expressions : language of regular expression Example
- Slide 6
- Fall 2006Costas Busch - RPI6 Definition For primitive regular expressions:
- Slide 7
- Fall 2006Costas Busch - RPI7 Definition (continued) For regular expressions and
- Slide 8
- Fall 2006Costas Busch - RPI8 Example Regular expression:
- Slide 9
- Fall 2006Costas Busch - RPI9 Example Regular expression
- Slide 10
- Fall 2006Costas Busch - RPI10 Example Regular expression
- Slide 11
- Fall 2006Costas Busch - RPI11 Example Regular expression = { all strings containing substring 00 }
- Slide 12
- Fall 2006Costas Busch - RPI12 Example Regular expression = { all strings without substring 00 }
- Slide 13
- Fall 2006Costas Busch - RPI13 Equivalent Regular Expressions Definition: Regular expressions and are equivalent if
- Slide 14
- Fall 2006Costas Busch - RPI14 Example = { all strings without substring 00 } and are equivalent regular expressions
- Slide 15
- Fall 2006Costas Busch - RPI15 Regular Expressions and Regular Languages
- Slide 16
- Fall 2006Costas Busch - RPI16 Theorem Languages Generated by Regular Expressions Regular Languages
- Slide 17
- Fall 2006Costas Busch - RPI17 Languages Generated by Regular Expressions Regular Languages Generated by Regular Expressions Regular Languages Proof:
- Slide 18
- 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
- Slide 19
- Fall 2006Costas Busch - RPI19 Induction Basis Primitive Regular Expressions: Corresponding NFAs regular languages
- Slide 20
- Fall 2006Costas Busch - RPI20 Inductive Hypothesis Suppose that for regular expressions and, and are regular languages
- Slide 21
- Fall 2006Costas Busch - RPI21 Inductive Step We will prove: Are regular Languages
- Slide 22
- Fall 2006Costas Busch - RPI22 By definition of regular expressions:
- Slide 23
- Fall 2006Costas Busch - RPI23 By inductive hypothesis we know: and are regular languages Regular languages are closed under: Union Concatenation Star We also know:
- Slide 24
- Fall 2006Costas Busch - RPI24 Therefore: Are regular languages is trivially a regular language (by induction hypothesis) End of Proof-Part 1
- Slide 25
- Fall 2006Costas Busch - RPI25 Using the regular closure of these operations, we can construct recursively the NFA that accepts Example:
- Slide 26
- 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
- Slide 27
- Fall 2006Costas Busch - RPI27 Since is regular, there is a NFA that accepts it Take it with a single final state
- Slide 28
- Fall 2006Costas Busch - RPI28 From construct the equivalent Generalized Transition Graph in which transition labels are regular expressions Example: Corresponding Generalized transition graph
- Slide 29
- Fall 2006Costas Busch - RPI29 Another Example: Transition labels are regular expressions
- Slide 30
- Fall 2006Costas Busch - RPI30 Reducing the states: Transition labels are regular expressions
- Slide 31
- Fall 2006Costas Busch - RPI31 Resulting Regular Expression:
- Slide 32
- Fall 2006Costas Busch - RPI32 In General Removing a state:
- Slide 33
- 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
- Slide 34
- Fall 2006Costas Busch - RPI34 Standard Representations of Regular Languages Regular Languages DFAs NFAs Regular Expressions
- Slide 35
- 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)