fall 2003costas busch - rpi1 linear grammars grammars with at most one variable at the right side of...
Post on 22-Dec-2015
215 views
Embed Size (px)
TRANSCRIPT
- Slide 1
- Fall 2003Costas Busch - RPI1 Linear Grammars Grammars with at most one variable at the right side of a production Examples:
- Slide 2
- Fall 2003Costas Busch - RPI2 A Non-Linear Grammar Grammar : Number of in string
- Slide 3
- Fall 2003Costas Busch - RPI3 Another Linear Grammar Grammar :
- Slide 4
- Fall 2003Costas Busch - RPI4 Right-Linear Grammars All productions have form: Example: or string of terminals
- Slide 5
- Fall 2003Costas Busch - RPI5 Left-Linear Grammars All productions have form: Example: or string of terminals
- Slide 6
- Fall 2003Costas Busch - RPI6 Regular Grammars
- Slide 7
- Fall 2003Costas Busch - RPI7 Regular Grammars A regular grammar is any right-linear or left-linear grammar Examples:
- Slide 8
- Fall 2003Costas Busch - RPI8 Observation Regular grammars generate regular languages Examples:
- Slide 9
- Fall 2003Costas Busch - RPI9 Regular Grammars Generate Regular Languages
- Slide 10
- Fall 2003Costas Busch - RPI10 Theorem Languages Generated by Regular Grammars Regular Languages
- Slide 11
- Fall 2003Costas Busch - RPI11 Theorem - Part 1 Languages Generated by Regular Grammars Regular Languages Any regular grammar generates a regular language
- Slide 12
- Fall 2003Costas Busch - RPI12 Theorem - Part 2 Languages Generated by Regular Grammars Regular Languages Any regular language is generated by a regular grammar
- Slide 13
- Fall 2003Costas Busch - RPI13 Proof Part 1 Languages Generated by Regular Grammars Regular Languages The language generated by any regular grammar is regular
- Slide 14
- Fall 2003Costas Busch - RPI14 The case of Right-Linear Grammars Let be a right-linear grammar We will prove: is regular Proof idea: We will construct NFA with
- Slide 15
- Fall 2003Costas Busch - RPI15 Grammar is right-linear Example:
- Slide 16
- Fall 2003Costas Busch - RPI16 Construct NFA such that every state is a grammar variable: special final state
- Slide 17
- Fall 2003Costas Busch - RPI17 Add edges for each production:
- Slide 18
- Fall 2003Costas Busch - RPI18
- Slide 19
- Fall 2003Costas Busch - RPI19
- Slide 20
- Fall 2003Costas Busch - RPI20
- Slide 21
- Fall 2003Costas Busch - RPI21
- Slide 22
- Fall 2003Costas Busch - RPI22
- Slide 23
- Fall 2003Costas Busch - RPI23 Grammar NFA
- Slide 24
- Fall 2003Costas Busch - RPI24 In General A right-linear grammar has variables: and productions: or
- Slide 25
- Fall 2003Costas Busch - RPI25 We construct the NFA such that: each variable corresponds to a node: special final state
- Slide 26
- Fall 2003Costas Busch - RPI26 For each production: we add transitions and intermediate nodes
- Slide 27
- Fall 2003Costas Busch - RPI27 For each production: we add transitions and intermediate nodes
- Slide 28
- Fall 2003Costas Busch - RPI28 Resulting NFA looks like this: It holds that:
- Slide 29
- Fall 2003Costas Busch - RPI29 The case of Left-Linear Grammars Let be a left-linear grammar We will prove: is regular Proof idea: We will construct a right-linear grammar with
- Slide 30
- Fall 2003Costas Busch - RPI30 Since is left-linear grammar the productions look like:
- Slide 31
- Fall 2003Costas Busch - RPI31 Construct right-linear grammar Left linear Right linear
- Slide 32
- Fall 2003Costas Busch - RPI32 Construct right-linear grammar Left linear Right linear
- Slide 33
- Fall 2003Costas Busch - RPI33 It is easy to see that: Since is right-linear, we have: Regular Language Regular Language Regular Language
- Slide 34
- Fall 2003Costas Busch - RPI34 Proof - Part 2 Languages Generated by Regular Grammars Regular Languages Any regular language is generated by some regular grammar
- Slide 35
- Fall 2003Costas Busch - RPI35 Proof idea: Let be the NFA with. Construct from a regular grammar such that Any regular language is generated by some regular grammar
- Slide 36
- Fall 2003Costas Busch - RPI36 Since is regular there is an NFA such that Example:
- Slide 37
- Fall 2003Costas Busch - RPI37 Convert to a right-linear grammar
- Slide 38
- Fall 2003Costas Busch - RPI38
- Slide 39
- Fall 2003Costas Busch - RPI39
- Slide 40
- Fall 2003Costas Busch - RPI40
- Slide 41
- Fall 2003Costas Busch - RPI41 In General For any transition: Add production: variableterminalvariable
- Slide 42
- Fall 2003Costas Busch - RPI42 For any final state: Add production:
- Slide 43
- Fall 2003Costas Busch - RPI43 Since is right-linear grammar is also a regular grammar with