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Context Free Grammar

Context Free Grammar. Introduction. Why do we want to learn about Context Free Grammars ? Used in many parsers in compilers Yet another compiler-compiler, yacc GNU project parser generator, bison Web application In describing the formats of XML ( eXtensible Markup Language) documents.

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Context Free Grammar

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  1. Context Free Grammar

  2. Introduction • Why do we want to learn about Context Free Grammars? • Used in many parsers in compilers • Yet another compiler-compiler, yacc • GNU project parser generator, bison • Web application • In describing the formats of XML (eXtensible Markup Language) documents

  3. Context Free Grammar • G = (V, T, P, S) • V – a set of variables, e.g. {S, A, B, C, D, E} • T – a set of terminals, e.g. {a, b, c} • P – a set of productions rules • In the form of A  , where AV,  (VT)* e.g. S  aB • S is a special variable called the start symbol

  4. CFG – Example 1 Construct a CFG for the following language : L1={0n1n | n>0} Thought: If  is a string in L1, so is 01, i.e. 0k+11k+1 = 0(0k1k)1 S  01 | 0S1

  5. CFG – Example 2 Construct a CFG for the following language : L2={0i1j | ij and i, j>0} Thought: Similar to L1, but at least one more ‘1’ or at least one more ‘0’ S  AC | CB A  0A | 0 B  B1 | 1 C  0C1 | 

  6. CFG – Example 3 Determine the language generated by the CFG: S  AS |  A  A1 | 0A1 | 01 S generates consecutive ‘A’s A generates 0i1j where i  j Languages: each block of ‘0’s is followed by at least as many ‘1’s

  7. Derivation How a string  is generated by a grammar G Consider the grammar G S  AS |  A  A1 | 0A1 | 01 S  0110011 ?

  8. S A S A A 1 S 0 1 0 A 1  0 1 Derivation Rightmost Derivation Leftmost Derivation Parse Tree or Derivation Tree S  AS 　 A1S 　 011S 　 011AS 　 0110A1S 　 0110011S 　 0110011  S  AS 　 AAS 　 AA  　 A0A1  　 A0011  　 A10011  　 0110011 

  9. Ambiguity Each parse tree has one unique leftmost derivation and one unique rightmost derivation. A grammar is ambiguous if some strings in it have more than one parse trees, i.e., it has more than one leftmost derivations (or more than one rightmost derivations).

  10. Ambiguity • Consider the following grammar G: • S  AS | a | b • A  SS | ab • A string generated by this grammar can have more than one parse trees. Consider the string abb: S S A S A S b S S b a b a b

  11. Ambiguity • As another example, consider the following grammar: E  E + E | E * E | (E) | x | y | z • There are 2 leftmost derivations for x + y + z: E  E + E  x + E  x + E + E  x + y + E  x + y + z E  E + E  E + E + E  x + E + E  x + y + E  x + y + z E E E E E E + + x z y z x y + +

  12. Simplification of CFG • Remove useless variables • Generating Variables • Reachable Variables • Remove -productions, e.g. A  • Remove unit-productions, e.g. AB

  13. Useless Variables • A variable X is useless if: • X does not generate any string of terminals, or • The start symbol, S, cannot generate X. S   X    • where ,   (VT)*, X  V and   T*

  14. Removal of Non-generating Variables • Mark each production of the form: X   where   T* • Repeat • Mark X   where  consists of terminals or variables which are on the left hand side of some marked productions. • Until no new productions is marked. • Remove all unmarked productions.

  15. Example • CFG: S  AB|CA A a B  BC|AB C  aB|b S  CA  A a C  b

  16. Example • CFG: S  aAa|aB A aS|bD B  aBa|b C  abb|DD D  aDa S  aAa|aB  A aS B  aBa|b C  abb

  17. Removal of Non-Reachable Variables • Mark each production of the form: S   where   (VT)* • Repeat • Mark X   where X appears on the right hand side of some marked productions. • Until no new productions is marked. • Remove all unmarked productions.

  18. Example CFG: S  aAa|aB A aS B  aBa|b C  abb S  aAa|aB  A aS B  aBa|b

  19. Example CFG: S  Aab|AB A  a B  bD| bB C  A | B D a S  Aab|AB  A  a B  bD | bB D a

  20. Remove Useless Variables A Bac|bTC|ba T aB|TB B aTB|bBC C TBc|aBC|ac A ba  C ac  A ba

  21. Removal of -productions Step 1: Find nullable variables Step 2: Remove all -productions by replacing some productions

  22. How to find nullable variables ? • Mark all variables A for which there exists a production of the form A  . • Repeat • Mark X for which there exists X   where V* and all symbols in  have been marked. • Until no new variables is marked.

  23. Removal of -Productions • We can remove all -productions (except S   if S is nullable) by rewriting some other productions. • e.g., If X1 and X3 are nullable, we should replace • A  X1X2X3X4 by: • A  X1X2X3X4 | X2X3X4 | X1X2X4 | X2X4

  24. Removal of -productions Example: S  A | B | C A  aAa | B B  bB | bb C  aCaa | D D  baD | abD |  Step1: nullable variables are D, C and S

  25. Removal of -productions(3) Step 2: • eliminate D   by replacing: • D baD | abD into D  baD | abD | ba | ab • eliminate C   by replacing: • C aCaa | D into C  aCaa | D | aaa • S  A | B | C into S  A | B | C |  • The new grammar: • SA | B | C |  • AaAa | B • BbB | bb • CaCaa | D | aaa • DbaD | abD | ba | ab Example:S  A | B | C A  aAa | B B  bB | bb C  aCaa | D D  baD | abD | 

  26. Example S  ABCD A a B   C  ED |  D  BC E  b S  ABCD|ABC|ABD|ACD|AB|AC|AD|A • A a • C  ED|E D  BC|B|C • E  b

  27. Removal of Unit Productions • Unit production: A  B, where A and B  V • 1. Detect cycles of unit productions: • A1  A2  A3  …  A1 • Replaced A1, A2, A3 , …, Ak by any one of them • 2. Detect paths of unit productions: • A1  A2  A3  …, Ak  , where   (VT)*/V • Add productions A1  , A2  , …, Ak   and remove all the unit productions

  28. Removal of Unit Productions Example: S  A | B | C A  aa | B B  bb | C C  cc | A Cycle: A  B  C  A Remove by replace B,C by A S  A | A | A A  aa | A A  bb | A A  cc | A Becomes: S  A A  aa | bb | cc Path: S  A Remove by adding productions S aa | bb | cc

  29. Example CFG: S → A | Bb A → C | a B → aBa | b C → aSa S → Bb | aSa |a → A → a | aSa B → aBa | b C → aSa

  30. Example Substitute

  31. Example Unit productions of form can be removed immediately Remove

  32. Example Substitute

  33. Example Remove repeated productions Final grammar

  34. Thank You

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