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Context-Free Languages

Context-Free Languages. Not all languages are regular. L 1 = {a n b n | n  0} is not regular. L 2 = { () , (()) , ((())) , ...} is not regular.  some properties of programming languages. Context-Free Grammars. G = (V, T, S, P) Productions are of the form: A  x

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Context-Free Languages

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  1. Context-Free Languages • Not all languages are regular. • L1 = {anbn | n  0} is not regular. L2 = {(), (()), ((())), ...} is not regular.  some properties of programming languages

  2. Context-Free Grammars G = (V, T, S, P) Productions are of the form: A  x A  V and x  (V  T)*

  3. Context-Free Grammars • A regular language is also a context-free language.

  4. Context-Free Grammars • A regular language is also a context-free language. • Why the name?

  5. Example • G = ({S}, {a, b}, S, P) P = { S  aSa S  bSb S   } S  aSa  aaSaa  aabSbaa  aabbaa

  6. Example • G = ({S}, {a, b}, S, P) P = { S  aSa S  bSb S   } S  aSa  aaSaa  aabSbaa  aabbaa L(G) = {wwR | w  {a, b}*}

  7. Example • G = ({S, A, B}, {a, b}, S, P) P = { S  abB A  aaBb B  bbAa A   } S  abB  abbbAa  abbbaaBba  abbbaabbAaba  abbbaabbaba

  8. Example • L = {anbm | n  m} is context-free. G = (?, {a, b}, S, ?)

  9. Example • L = {anbm | n  m} is context-free. G = (?, {a, b}, S, ?) P = { S  AS1 | S1B S1 aS1b |  A  aA | a B  bB | b }

  10. Example • G = ({S}, {a, b}, S, P) P = { S  aSb | SS | } S  aSb  aaSbb  aaSSbb  aaabSbb  aaababbb

  11. Example • G = ({S}, {a, b}, S, P) P = { S  aSb | SS | } S  aSb  aaSbb  aaSSbb  aaabSbb  aaababbb L(G) = ?

  12. Derivations • G = ({S, A, B}, {a, b}, S, {S  AB, A  aaA, A  , B  Bb, B  }) 1 2 3 4 5 S  AB  aaAB  aaB  aaBb  aab S  AB  ABb  aaABb  aaAb  aab 1 2 3 4 5 1 4 2 5 3

  13. Derivations • Leftmost: in each step the leftmost variable in the sentential form is replaced. • Rightmost: in each step the rightmost variable in the sentential form is replaced.

  14. Example • G = ({S, A, B}, {a, b}, S, {S  AB, A  aaA, A  , B  Bb, B  }) 1 2 3 4 5 S  AB  aaAB  aaB  aaBb  aab leftmost S  AB  ABb  aaABb  aaAb  aab 1 2 3 4 5 1 4 2 5 3

  15. A a b A B c Derivation Trees A  abABc ordered tree

  16. Derivation Trees • Let G = (V, T, S, P) be a context-free grammar. An ordered tree is a derivation tree iff: • The root is labeled S. • Every leaf has a label from T  {}. • Every interior vertex has a label from V. • A vertex has label AV and its children are labeled a1, a2, ..., an iff P contains the production A  a1 a2 ... an. • A leaf labeled  has no siblings.

  17. Derivation Trees • Let G = (V, T, S, P) be a context-free grammar. An ordered tree is a partial derivation tree iff: • The root is labeled S. • Every leaf has a label from T  {}. Every leaf has a label from V T  {}. • Every interior vertex has a label from V. • A vertex has label AV and its children are labeled a1, a2, ..., an iff P contains the production A  a1 a2 ... an. • A leaf labeled  has no siblings.

  18. Derivation Trees The string of symbols obtained by reading the leaves of a tree from left to right (omitting any 's encountered) is called the yield of the tree.

  19. Derivation Trees S S  aAB A  bBb B  A |  yield: abbbb a A B b b B A  b b B 

  20. Derivation Trees S S  aAB A  bBb B  A |  a A B b b B A  b b B partial derivation tree 

  21. Sentential Forms & Derivation Trees • Let G = (V, T, S, P) be a context-free grammar. • wL(G) iff there exists a derivation tree of G whose yield is w. • If tG is a partial derivation tree of G whose root label is S, then the yield of tG is a sentential form of G.

  22. Membership and Parsing • Membership algorithm: tells if wL(G). • Parsing: finding a sequence of productions by which wL(G) is derived.

  23. Membership and Parsing S  SS | aSb | bSa |  w = aabb

  24. Exhaustive Search Parsing • Top-down parsing. • S  SS | aSb | bSa |  w = aabb 1. S  SS S  SS  SSS S  aSb  aSSb 2. S  aSb S  SS  aSbS S  aSb  aaSbb 3. S  bSa S  SS  bSaS S  aSb  abSab 4. S   S  SS  S S  aSb  ab S  aSb  aaSbb  aabb

  25. Exhaustive Search Parsing • It is not efficient. • If w  L(G) then it may never terminate, due to: A  B A  

  26. Exhaustive Search Parsing Suppose G = (V, T, S, P) is a context-free grammar which does not have any rule of the form A  B orA  . Then the exhaustive search parsing method can decide if wL(G) or not.

  27. Exhaustive Search Parsing Suppose G = (V, T, S, P) is a context-free grammar which does not have any rule of the form A  B orA  . Then the exhaustive search parsing method can decide if wL(G) or not. Proof: no more than |w| rounds are involved.

  28. Exhaustive Search Parsing Suppose G = (V, T, S, P) is a context-free grammar which does not have any rule of the form A  B orA  . Then the exhaustive search parsing method can decide if wL(G) or not. Proof: no more than |w| rounds are involved. Complexity:|P||w|.

  29. Theorem • For every context-free grammar G, there exists an algorithm that parses any wL(G) in a number of steps proportional to |w|3. • Not satisfactory as linear time parsing algorithm (proportional to the length of a string)

  30. Simple Grammars (S-Grammars) G = (V, T, S, P) Productions are of the form: A  ax A  V, a  T, and x  V*, and any pair (A,a) can occur in at most one rule.

  31. Simple Grammars (S-Grammars) Any wL(G) can be parsed in at most |w| steps. w = a1a2 ... an S  a1A1A2 ... Am  a1a2B1B2 ... BkA1A2 ... Am

  32. Simple Grammars (S-Grammars) Many features of Pascal-like programming languages can be expressed with s-grammars.

  33. Ambiguity A context-free grammar G is said to be ambiguous if there exits some wL(G) that has at least two distinct derivation trees.

  34. Ambiguity S  aSb | SS| w = aabb S S S S a b a b S S  a b a b S S  

  35. Ambiguity • G = ({E, I}, {a, b, c, +, *, (, )}, E, P) w = a + b * c E  I E  E + E E  E * E E  (E) I  a | b | c

  36. Ambiguity • G = ({E, T, F, I}, {a, b, c, +, *, (, )}, E, P) w = a + b * c E  T | E + T T  F | T * F E  E * E F  I | (E) I  a | b | c

  37. Ambiguity If L is a context-free language for which there exists an unambiguous grammar, then L is said to be unambiguous.

  38. Ambiguity If L is a context-free language for which there exists an unambiguous grammar, then L is said to be unambiguous. If every grammar that generates L is ambiguous, then L is called inherently ambiguous.

  39. Ambiguity • L = {anbncm}  {anbmcm} is inherently ambiguous. L = L1  L2 S  S1 | S2 S1 S1c | A S2 aS2 | B A  aAb |  B  bBc | 

  40. CFGs and Programming Languages • Important uses of formal languages: • to define precisely a programming language. • to construct an efficient translator for it. • RLs are used for simple patterns, while CFLs for more complicated aspects.

  41. CFGs and Programming Languages • S-grammars: <if-statement> ::= if <expression> <then_clause> <else_clause> <then_clause> ::= then <statement> <else_clause> ::= else <statement>

  42. Homework • Exercises: 2, 3, 4, 6, 7, 9, 15, 17, 22 of Section 5.1 - Linz’s book. • Exercises: 1, 2, 5, 6, 8, 10, 11, 12, 13, 15 of Section 5.2 - Linz’s book. • Exercises: 1, 2, 3 of Section 5.3 - Linz’s book.

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