ContextFree Languages

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
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 or A ?
  ?.
• 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 or A ?
  ?.
• 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 or A ?
  ?.
• Then the exhaustive search parsing method can
  decide if
• w?L(G) or not.
• Proof no more than w rounds are involved.
• Complexity Pw.

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 w3.
• 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
• ltif-statementgt if ltexpressiongt ltthen_clausegt
  ltelse_clausegt
• ltthen_clausegt then ltstatementgt
• ltelse_clausegt else ltstatementgt

42
Homework

• Exercises 2, 3, 4, 6, 7, 9, 15, 17, 22 of
  Section 5.1 - Linzs book.
• Exercises 1, 2, 5, 6, 8, 10, 11, 12, 13, 15 of
  Section 5.2 - Linzs book.
• Exercises 1, 2, 3 of Section 5.3 - Linzs book.