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Simplifications of Context-Free Grammars

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Presentation on theme: "Simplifications of Context-Free Grammars"— Presentation transcript:

1 Simplifications of Context-Free Grammars

2 A Substitution Rule Equivalent grammar Substitute

3 A Substitution Rule Substitute Equivalent grammar

4 In general: Substitute equivalent grammar

5 Nullable Variables Nullable Variable:

6 Removing Nullable Variables
Example Grammar: Nullable variable

7 Final Grammar Substitute

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

9 Removing Unit Productions
Observation: Is removed immediately

10 Example Grammar:

11 Substitute

12 Remove

13 Substitute

14 Remove repeated productions
Final grammar

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

16 Another grammar: Useless Production Not reachable from S

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

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

19 Removing Useless Productions
Example Grammar:

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

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

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

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

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

25 Normal Forms for Context-free Grammars

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

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 variable:

31 Introduce intermediate variable:

32 Final grammar in Chomsky Normal Form:
Initial grammar

33 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

34 The Procedure First remove: Nullable variables Unit productions

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

36 Replace any production
with New intermediate variables:

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

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

39 Greinbach Normal Form All productions have form: symbol variables

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

41 Compilers

42 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

43 Compiler Lexical analyzer parser input output machine code program

44 A parser knows the grammar
of the programming language


46 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

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

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

49 Parsing

50 Parser input string derivation grammar

51 Example: Parser derivation input ?

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


54 Phase 2 Phase 1

55 Phase 2 Phase 3

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

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

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

59 Time for phase 2: possible derivations

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

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

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

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