Simplifications of ContextFree Grammars 1 A Substitution Rule

Simplifications of Context-Free Grammars 1

A Substitution Rule equivalent grammar substitute B 2

In general: substitute B equivalent grammar 3

Useless Productions 4

never terminates. . . 5

Useless Production 6

Another grammar: 7

Not reachable from S Useless Production 8

In general: If Then Otherwise is useful is useless 9

Removing Useless Productions Example Grammar 10

First: find all variables that produce strings with only terminals Round 1: Round 2: 11

Keep only the variables that produce terminal symbols 12

Second: Find all variables reachable from S Dependency Graph not reachable 13

Keep only the variables reachable from S Final Grammar 14

Nullable Variables Nullable Variable: 15

Removing Nullable Variables Example Grammar: Nullable variable 16

Substitute Final Grammar 17

Unit-Productions Unit Production: 18

Removing Unit Productions Observation: Is removed immediately 19

Example Grammar: 20

substitute 21

22

substitute 23

Remove repeated productions Final grammar 24

Removing All 1. Remove Nullable Variables 2. Remove Unit-Productions 3. Remove Useless Variables 25

Normal Forms for Context-free Grammars 26

Chomsky Normal Form All productions have form: OR variable terminal 27

Examples: Chomsky Normal Form NOT Chomsky Normal Form 28

Convertion to Chomsky Normal Form Example: NOT Chomsky Normal Form 29

Introduce variables for terminals: 30

Introduce intermediate variables: 31

Final grammar in Chomsky Normal Form: Initial grammar 32

In general: From any context-free grammar Not in Chomsky Normal Form We can obtain: An equivalent grammar in Chomsky Normal Form 33

The Procedure: Remove nullable variables unit productions 34

For every symbol : Add new variable Add production In productions, replace with 35

Replace any production: with: New variables: 36

Theorem: For any context-free grammar there is an equivalent grammar in Chomsky Normal Form 37

Greinbach Normal Form All rules have form: symbol variables 38

Examples: Greinbach Normal Form NOT Greinbach Normal Form 39

Conversion to Greinbach Normal Form 40

Theorem: For any context-free grammar there is an equivalent grammar in Greinbach Normal Form 41

An Application of Chomsky Normal Forms 42

The CYK Membership Algorithm Inputs: • Grammar in Chomsky Normal Form • String w Output: find if 43

Algorithm: Input Examples Grammar String : : 44

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Therefore: Time Complexity: It can be easily converted to a parser 49