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Published byEmma Strickland Modified over 2 years ago

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1 Simplifications of Context-Free Grammars

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2 A Substitution Rule Substitute Equivalent grammar

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3 A Substitution Rule Equivalent grammar Substitute

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4 In general: Substitute equivalent grammar

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5 Nullable Variables Nullable Variable:

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6 Removing Nullable Variables Example Grammar: Nullable variable

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7 Substitute Final Grammar

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

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9 Removing Unit Productions Observation: Is removed immediately

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10 Example Grammar:

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11 Substitute

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12 Remove

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13 Substitute

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14 Remove repeated productions Final grammar

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15 Useless Productions Some derivations never terminate... Useless Production

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16 Another grammar: Not reachable from S Useless Production

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

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18 A production is useless if any of its variables is useless Productions useless Variables useless

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19 Removing Useless Productions Example Grammar:

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

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21 Keep only the variables that produce terminal symbols: (the rest variables are useless) Remove useless productions

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

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23 Keep only the variables reachable from S Final Grammar (the rest variables are useless) Remove useless productions

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

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25 Normal Forms for Context-free Grammars

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26 Chomsky Normal Form Each productions has form: variable or terminal

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27 Examples: Not Chomsky Normal Form Chomsky Normal Form

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28 Convertion to Chomsky Normal Form Example: Not Chomsky Normal Form

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29 Introduce variables for terminals:

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30 Introduce intermediate variable:

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31 Introduce intermediate variable:

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32 Final grammar in Chomsky Normal Form: Initial grammar

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

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34 The Procedure First remove: Nullable variables Unit productions

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

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36 Replace any production with New intermediate variables:

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37 Theorem: For any context-free grammar (which doesnt produce ) there is an equivalent grammar in Chomsky Normal Form

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38 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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39 Greinbach Normal Form All productions have form: symbolvariables

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40 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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41 Compilers

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

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43 Lexical analyzer parser Compiler program machine code input output

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44 A parser knows the grammar of the programming language

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45 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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46 The parser finds the derivation of a particular input * 5 Parser E -> E + E | E * E | INT E => E + E => E + E * E => 10 + E*E => * E => * 5 input derivation

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47 10 E 25 E => E + E => E + E * E => 10 + E*E => * E => * 5 derivation derivation tree EE EE + *

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48 10 E 25 derivation tree EE EE + * mult a, 2, 5 add b, 10, a machine code

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49 Parsing

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50 grammar Parser input string derivation

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51 Example: Parser derivation input ?

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52 Exhaustive Search Phase 1: All possible derivations of length 1 Find derivation of

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53

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54 Phase 2 Phase 1

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55 Phase 2 Phase 3

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56 Final result of exhaustive search Parser derivation input (top-down parsing)

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57 Time complexity of exhaustive search Suppose there are no productions of the form Number of phases for string : approx. |w|

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58 Time for phase 1: possible derivations For grammar with rules

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59 Time for phase 2: possible derivations

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60 Time for phase |w| is 2 |w| : A total of 2 |w| possible derivations

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61 Total time needed for string : Extremely bad!!! phase 1 phase 2 phase |w|

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62 For general context-free grammars: There exists a parsing algorithm that parses a string in time The CYK parser

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