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**Simplifications of Context-Free Grammars**

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A Substitution Rule Equivalent grammar Substitute

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A Substitution Rule Substitute Equivalent grammar

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In general: Substitute equivalent grammar

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Nullable Variables Nullable Variable:

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**Removing Nullable Variables**

Example Grammar: Nullable variable

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Final Grammar Substitute

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

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**Removing Unit Productions**

Observation: Is removed immediately

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Example Grammar:

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Substitute

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Remove

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Substitute

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**Remove repeated productions**

Final grammar

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**Useless Productions Useless Production**

Some derivations never terminate...

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Another grammar: Useless Production Not reachable from S

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**then variable is useful**

In general: contains only terminals if then variable is useful otherwise, variable is useless

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**A production is useless**

if any of its variables is useless Productions useless Variables useless

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**Removing Useless Productions**

Example Grammar:

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**First: find all variables that can produce strings with only terminals**

Round 1: Round 2:

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**Keep only the variables that produce terminal symbols:**

(the rest variables are useless) Remove useless productions

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**Second: Find all variables reachable from Use a Dependency Graph not**

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**Keep only the variables reachable from S**

(the rest variables are useless) Final Grammar Remove useless productions

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

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**Normal Forms for Context-free Grammars**

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**Chomsky Normal Form Each productions has form: or variable variable**

terminal

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Examples: Chomsky Normal Form Not Chomsky Normal Form

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**Convertion to Chomsky Normal Form**

Example: Not Chomsky Normal Form

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**Introduce variables for terminals:**

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**Introduce intermediate variable:**

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**Introduce intermediate variable:**

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**Final grammar in Chomsky Normal Form:**

Initial grammar

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

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The Procedure First remove: Nullable variables Unit productions

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Then, for every symbol : Add production In productions: replace with New variable:

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**Replace any production**

with New intermediate variables:

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**Theorem: For any context-free grammar (which doesn’t produce )**

there is an equivalent grammar in Chomsky Normal Form

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**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

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Greinbach Normal Form All productions have form: symbol variables

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**Observations Greinbach normal forms are very good for parsing**

It is hard to find the Greinbach normal form of any context-free grammar

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Compilers

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Machine Code Program Add v,v,0 cmp v,5 jmplt ELSE THEN: add x, 12,v ELSE: WHILE: cmp x,3 ... v = 5; if (v>5) x = 12 + v; while (x !=3) { x = x - 3; v = 10; } ...... Compiler

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Compiler Lexical analyzer parser input output machine code program

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**A parser knows the grammar**

of the programming language

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**Parser PROGRAM STMT_LIST STMT_LIST STMT; STMT_LIST | STMT;**

STMT EXPR | IF_STMT | WHILE_STMT | { STMT_LIST } EXPR EXPR + EXPR | EXPR - EXPR | ID IF_STMT if (EXPR) then STMT | if (EXPR) then STMT else STMT WHILE_STMT while (EXPR) do STMT

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**The parser finds the derivation**

of a particular input derivation Parser input E => E + E => E + E * E => 10 + E*E => * E => * 5 E -> E + E | E * E | INT * 5

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derivation tree derivation E E => E + E => E + E * E => 10 + E*E => * E => * 5 E + E 10 E E * 2 5

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derivation tree E machine code E + E mult a, 2, 5 add b, 10, a 10 E E * 2 5

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Parsing

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Parser input string derivation grammar

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Example: Parser derivation input ?

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**Exhaustive Search Phase 1: Find derivation of**

All possible derivations of length 1

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Phase 2 Phase 1

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Phase 2 Phase 3

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**Final result of exhaustive search**

(top-down parsing) Parser input derivation

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**Time complexity of exhaustive search**

Suppose there are no productions of the form Number of phases for string : approx. |w|

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For grammar with rules Time for phase 1: possible derivations

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Time for phase 2: possible derivations

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**Time for phase |w| is 2|w|:**

A total of 2|w| possible derivations

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**Extremely bad!!! Total time needed for string : phase 1 phase |w|**

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**For general context-free grammars:**

There exists a parsing algorithm that parses a string in time The CYK parser

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