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Published byGenesis Whinery Modified over 2 years ago

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1 Context-Free Languages

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2 Regular Languages

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3 Context-Free Languages

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4 Pushdown Automata Context-Free Grammars stack automaton

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5 Context-Free Grammars

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6 Grammars Grammars express languages Example: the English language

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7

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8 A derivation of “the dog walks”:

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9 A derivation of “a cat runs”:

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10 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” }

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11 Notation VariableTerminal Production Rules

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12 Another Example Grammar: Derivation of sentence :

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13 Grammar: Derivation of sentence :

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14 Other derivations:

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15 Language of the grammar

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16 More Notation Grammar Set of variables Set of terminal symbols Start variable Set of Production rules

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17 Example Grammar :

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18 More Notation Sentential Form: A sentence that contains variables and terminals Example: Sentential Formssentence

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19 We write: Instead of:

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20 In general we write: If:

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21 By default:

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22 Example Grammar Derivations

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23 Grammar Example Derivations

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24 Another Grammar Example Grammar : Derivations:

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25 More Derivations

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26 Language of a Grammar For a grammar with start variable : String of terminals

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27 Example For grammar : Since:

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28 A Convenient Notation

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29 Example A context-free grammar : A derivation:

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30 A context-free grammar : Another derivation:

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31 (((( )))) Describes parentheses:

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32 A context-free grammar : A derivation: Example

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33 A context-free grammar : Another derivation:

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35 A context-free grammar : A derivation: Example

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36 A context-free grammar : A derivation:

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37 () ((( ))) (( )) Describes matched parentheses:

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38 Definition: Context-Free Grammars Grammar Productions of the form: String of variables and terminals VariablesTerminal symbols Start variable Variable

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39

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40 Definition: Context-Free Languages A language is context-free if and only if there is a context-free grammar with

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41 Derivation Order Leftmost derivation: Rightmost derivation:

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42 Leftmost derivation: Rightmost derivation:

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43 Derivation Trees

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48 Derivation Tree

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49 yield Derivation Tree

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50 Partial Derivation Trees Partial derivation tree

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51 Partial derivation tree

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52 Partial derivation tree sentential form yield

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53 Same derivation tree Sometimes, derivation order doesn’t matter Leftmost: Rightmost:

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54 Ambiguity

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55 leftmost derivation

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56 leftmost derivation

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57 Two derivation trees

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58 The grammar is ambiguous: stringhas two derivation trees

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59 stringhas two leftmost derivations The grammar is ambiguous:

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60 Definition: A context-free grammar is ambiguous if some string has: two or more derivation trees

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61 In other words: A context-free grammar is ambiguous if some string has: two or more leftmost derivations (or rightmost)

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62 Why do we care about ambiguity? take

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65 Correct result:

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66 We want to remove ambiguity Ambiguity is bad for programming languages

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67 We fix the ambiguous grammar: New non-ambiguous grammar:

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69 Unique derivation tree

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70 The grammar : is non-ambiguous: Every string has a unique derivation tree

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71 Another Ambiguous Grammar IF_STMTif EXPR then STMT if EXPR then STMT else STMT

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72 If expr1 then if expr2 then stmt1 else stmt2 IF_STMT expr1then elseifexpr2then STMT stmt1 if IF_STMT expr1thenelse ifexpr2then STMTstmt2 if stmt1 stmt2

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73 Inherent Ambiguity Some context free languages have only ambiguous grammars Example:

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74 The string has two derivation trees

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