Costas Busch - RPI1 Grammars

Costas Busch - RPI2 Grammars Grammars express languages Example: the English language

Costas Busch - RPI3

4 A derivation of “the dog walks”:

Costas Busch - RPI5 A derivation of “a cat runs”:

Costas Busch - RPI6 Language of the grammar: L = { “a cat runs”, “a cat walks”, “the cat runs”, “the cat walks”, “a dog runs”, “a dog walks”, “the dog runs”, “the dog walks” }

Costas Busch - RPI7 Notation VariableTerminal Production Rules

Costas Busch - RPI8 Another Example Grammar: Derivation of sentence :

Costas Busch - RPI9 Grammar: Derivation of sentence :

Costas Busch - RPI10 Other derivations:

Costas Busch - RPI11 Language of the grammar

Costas Busch - RPI12 More Notation Grammar Set of variables Set of terminal symbols Start variable Set of Production rules

Costas Busch - RPI13 Example Grammar :

Costas Busch - RPI14 More Notation Sentential Form: A sentence that contains variables and terminals Example: Sentential Formssentence

Costas Busch - RPI15 We write: Instead of:

Costas Busch - RPI16 In general we write: If:

Costas Busch - RPI17 By default:

Costas Busch - RPI18 Example Grammar Derivations

Costas Busch - RPI19 Grammar Example Derivations

Costas Busch - RPI20 Another Grammar Example Grammar : Derivations:

Costas Busch - RPI21 More Derivations

Costas Busch - RPI22 Language of a Grammar For a grammar with start variable : String of terminals

Costas Busch - RPI23 Example For grammar : Since:

Costas Busch - RPI24 A Convenient Notation

Costas Busch - RPI25 Linear Grammars

Costas Busch - RPI26 Linear Grammars Grammars with at most one variable at the right side of a production Examples:

Costas Busch - RPI27 A Non-Linear Grammar Grammar : Number of in string

Costas Busch - RPI28 Another Linear Grammar Grammar :

Costas Busch - RPI29 Right-Linear Grammars All productions have form: Example: or string of terminals

Costas Busch - RPI30 Left-Linear Grammars All productions have form: Example: or string of terminals

Costas Busch - RPI31 Regular Grammars

Costas Busch - RPI32 Regular Grammars A regular grammar is any right-linear or left-linear grammar Examples:

Costas Busch - RPI33 Observation Regular grammars generate regular languages Examples:

Costas Busch - RPI34 Regular Grammars Generate Regular Languages

Costas Busch - RPI35 Theorem Languages Generated by Regular Grammars Regular Languages

Costas Busch - RPI36 Theorem - Part 1 Languages Generated by Regular Grammars Regular Languages Any regular grammar generates a regular language

Costas Busch - RPI37 Theorem - Part 2 Languages Generated by Regular Grammars Regular Languages Any regular language is generated by a regular grammar

Costas Busch - RPI38 Proof – Part 1 Languages Generated by Regular Grammars Regular Languages The language generated by any regular grammar is regular

Costas Busch - RPI39 The case of Right-Linear Grammars Let be a right-linear grammar We will prove: is regular Proof idea: We will construct NFA with

Costas Busch - RPI40 Grammar is right-linear Example:

Costas Busch - RPI41 Construct NFA such that every state is a grammar variable: special final state

Costas Busch - RPI42 Add edges for each production:

Costas Busch - RPI43

Costas Busch - RPI44

Costas Busch - RPI45

Costas Busch - RPI46

Costas Busch - RPI47

Costas Busch - RPI48 Grammar NFA

Costas Busch - RPI49 In General A right-linear grammar has variables: and productions: or

Costas Busch - RPI50 We construct the NFA such that: each variable corresponds to a node: special final state

Costas Busch - RPI51 For each production: we add transitions and intermediate nodes ………

Costas Busch - RPI52 For each production: we add transitions and intermediate nodes ………

Costas Busch - RPI53 Resulting NFA looks like this: It holds that:

Costas Busch - RPI54 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

Costas Busch - RPI55 Since is left-linear grammar the productions look like:

Costas Busch - RPI56 Construct right-linear grammar Left linear Right linear

Costas Busch - RPI57 Construct right-linear grammar Left linear Right linear

Costas Busch - RPI58 It is easy to see that: Since is right-linear, we have: Regular Language Regular Language Regular Language

Costas Busch - RPI59 Proof - Part 2 Languages Generated by Regular Grammars Regular Languages Any regular language is generated by some regular grammar

Costas Busch - RPI60 Proof idea: Let be the NFA with. Construct from a regular grammar such that Any regular language is generated by some regular grammar

Costas Busch - RPI61 Since is regular there is an NFA such that Example:

Costas Busch - RPI62 Convert to a right-linear grammar

Costas Busch - RPI63

Costas Busch - RPI64

Costas Busch - RPI65

Costas Busch - RPI66 In General For any transition: Add production: variableterminalvariable

Costas Busch - RPI67 For any final state: Add production:

Costas Busch - RPI68 Since is right-linear grammar is also a regular grammar with