Costas Buch - RPI1 Simplifications of Context-Free Grammars.

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Simplifications of Context-Free Grammars

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Presentation transcript:

Costas Buch - RPI1 Simplifications of Context-Free Grammars

Costas Buch - RPI2 A Substitution Rule Substitute Equivalent grammar

Costas Buch - RPI3 A Substitution Rule Equivalent grammar Substitute

Costas Buch - RPI4 In general: Substitute equivalent grammar

Costas Buch - RPI5 Nullable Variables Nullable Variable:

Costas Buch - RPI6 Removing Nullable Variables Example Grammar: Nullable variable

Costas Buch - RPI7 Substitute Final Grammar

Costas Buch - RPI8 Unit-Productions Unit Production: (a single variable in both sides)

Costas Buch - RPI9 Removing Unit Productions Observation: Is removed immediately

Costas Buch - RPI10 Example Grammar:

Costas Buch - RPI11 Substitute

Costas Buch - RPI12 Remove

Costas Buch - RPI13 Substitute

Costas Buch - RPI14 Remove repeated productions Final grammar

Costas Buch - RPI15 Useless Productions Some derivations never terminate... Useless Production

Costas Buch - RPI16 Another grammar: Not reachable from S Useless Production

Costas Buch - RPI17 In general: if then variable is useful otherwise, variable is useless contains only terminals

Costas Buch - RPI18 A production is useless if any of its variables is useless Productions useless Variables useless

Costas Buch - RPI19 Removing Useless Productions Example Grammar:

Costas Buch - RPI20 First: find all variables that can produce strings with only terminals Round 1: Round 2:

Costas Buch - RPI21 Keep only the variables that produce terminal symbols: (the rest variables are useless) Remove useless productions

Costas Buch - RPI22 Second: Find all variables reachable from Use a Dependency Graph not reachable

Costas Buch - RPI23 Keep only the variables reachable from S Final Grammar (the rest variables are useless) Remove useless productions

Costas Buch - RPI24 Removing All Step 1: Remove Nullable Variables Step 2: Remove Unit-Productions Step 3: Remove Useless Variables

Costas Buch - RPI25 Normal Forms for Context-free Grammars

Costas Buch - RPI26 Chomsky Normal Form Each productions has form: variable or terminal

Costas Buch - RPI27 Examples: Not Chomsky Normal Form Chomsky Normal Form

Costas Buch - RPI28 Convertion to Chomsky Normal Form Example: Not Chomsky Normal Form

Costas Buch - RPI29 Introduce variables for terminals:

Costas Buch - RPI30 Introduce intermediate variable:

Costas Buch - RPI31 Introduce intermediate variable:

Costas Buch - RPI32 Final grammar in Chomsky Normal Form: Initial grammar

Costas Buch - RPI33 From any context-free grammar (which doesn’t produce ) not in Chomsky Normal Form we can obtain: An equivalent grammar in Chomsky Normal Form In general:

Costas Buch - RPI34 The Procedure First remove: Nullable variables Unit productions

Costas Buch - RPI35 Then, for every symbol : In productions: replace with Add production New variable:

Costas Buch - RPI36 Replace any production with New intermediate variables:

Costas Buch - RPI37 Theorem: For any context-free grammar (which doesn’t produce ) there is an equivalent grammar in Chomsky Normal Form

Costas Buch - RPI38 Observations Chomsky normal forms are good for parsing and proving theorems It is very easy to find the Chomsky normal form for any context-free grammar

Costas Buch - RPI39 Greinbach Normal Form All productions have form: symbolvariables

Costas Buch - RPI40 Examples: Greinbach Normal Form Not Greinbach Normal Form

Costas Buch - RPI41 Conversion to Greinbach Normal Form: Greinbach Normal Form

Costas Buch - RPI42 Theorem: For any context-free grammar (which doesn’t produce ) there is an equivalent grammar in Greinbach Normal Form

Costas Buch - RPI43 Observations Greinbach normal forms are very good for parsing It is hard to find the Greinbach normal form of any context-free grammar

Costas Buch - RPI44 The CYK Parser

Costas Buch - RPI45 The CYK Membership Algorithm Input: Grammar in Chomsky Normal Form String Output: find if

Costas Buch - RPI46 The Algorithm Grammar : String : Input example:

Costas Buch - RPI47

Costas Buch - RPI48

Costas Buch - RPI49

Costas Buch - RPI50

Costas Buch - RPI51 Therefore: Time Complexity: The CYK algorithm can be easily converted to a parser (bottom up parser) Observation: