1. Lecture 6: Context-Free Languages
虞台文
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智慧型多媒體研究室

2. Content Language Recognizer vs. Language Generator
Context-Free Grammars
Regular Languages vs. Context-Free Languages
Pushdown Automata
Pushdown Automata vs. Context-Free Grammars
Properties of Context-Free Languages
Context-Sensitive Grammars

3. Lecture 6: Context-Free Languages
Language Recognizer vs.
Language Generator
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智慧型多媒體研究室

4. Recognizer vs. Generator
Parsing
Machines
Grammas
Generation

5. Example: Regular Expressions
Is the expression power of r. e. strong enough?
Example: Regular Expressions
A rule (grammar) to generate strings 
Next, generate any number of a’s
or any number b’s
Finally, generate an b
First, generate an a

6. Regular Expressions As Grammars
S, M, A, B
 nonterminals
Production Rules S
 start symbol
a, b
 terminals

7. Regular Expressions As Grammars
Example:

8. Lecture 6: Context-Free Languages
Context-Free Grammars
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9. Definition (V, , P, S) A context-free grammar G is a quadruple where
V : finite set of nonterminals
V   = 
 : finite set of terminals
P : finite set of rules, i.e., subset of V (V )*
S : start symbol, SV
In the form of A
with AV and (V )*

10. Notation Conventions
The capital letters A, B, C, D, E, and S denote nonterminals; S is the start symbol.
The lower-case letters a, b, c, d, e, digits and boldface strings are terminals.
The capital letters X, Y, and Z that may be either terminals or variables.
The lower-case letters u, v, w, x, y, and z denote strings of terminals.
The lower-case Greek letters , , and  denote strings of variables and terminals.

11. Notation Conventions reflexive and transitive closure of
a derivation of G of m from 1.

12. Context-Free Languages
The language generated by
G = (V, , P, S),
denoted as L(G), is

13. Example
L(G)=?
G = (V, , P, E)
E: Expression
T: Term
F: Factor

14. Example
L(G)=? R1 R2 R3 R4 R5 R6 R7

15. Example L(G)=? L(G)=? x1 x1*(x2*x1+x1) x1*(x2*(x1+x1)) x1x2+x1+x2

16. Example L(G)=? G = (V, , P, S) L(G)={anbn|n0} V={S} ={a, b} S  aSb
 aaSbb
 a3Sb3
     anSbn
 anbn

17. Example L(G)=? G = (V, , P, S) L(G)={anbn|n0}
Is the expression power of context-free grammar stronger than regular expression?
V={S}
={a, b}
L(G)={anbn|n0}
S  aSb
S   P=
S  aSb
 aaSbb
 a3Sb3
     anSbn
 anbn

18. Lecture 6: Context-Free Languages
Regular Languages
vs.
Context-Free Languages
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智慧型多媒體研究室

19. Definition  Regular Grammar
A CFG G = (V, , P, S) is regular iff
P  V  *(V{}).
tailing
nonterminal
nonterminal
leading
terminal(s)
RHS
LHS

20. Example G is regular. Consider G = (V, , P, S) with V={S, A, B}
S  bA
S  aB
A  abaS
B  babS
S   P=

21. Regular Grammars vs. Regular Languages
Consider G = (V, , P, S) with
G is regular.
V={S, A, B}
={a, b}
S  bA
 babaS
 · · ·
S  bA
S  aB
A  abaS
B  babS
S   P=
S  aB
 ababS
 · · ·
Regular Expression
L(G)=(abab + baba)*

22. Theorem Let L be a language.
L is regular iff L is generated by a regular grammar.
accepted by a DFA

23. L is regular iff L is generated by a regular grammar.
Theorem
L is regular iff L is generated by a regular grammar.
Pf)
“”
Given a DFA M = (K, , , s, F), we will construct a regular grammar, say, G st. L(M) = L(G).
Consider G = (K, , P, S) st.
From the correspondence btw. M and G, it is easily seen that
. . .
Hence,

24. L is regular iff L is generated by a regular grammar.
Theorem
L is regular iff L is generated by a regular grammar.
Pf)
“”
Given a regular grammar G = (V, , P, S), we will construct an NFA, say, M st. L(G) = L(M).
Consider M = (K, , , s, F) st.
From the correspondence btw. M and G, it is easily seen that
. . .
Hence,

25. Example G is regular. Consider G = (V, , P, S) with V={S, A, B}f B
S  bA
S  aB
A  abaS
B  babS
S   P= a
bab  b S
> A
aba

26. The Chomsky Hierarchy Chomsky Hierarchy Languages Grammars Automaton
Type 0
Recursively enumerable
unrestricted
Turing Machine
Recursive
Decider
Type 1
Context-Sensitive
Linear-Bounded Automaton
Type 2
Context-Free
Push-Down Automaton
Type 3
Regular
NFA or DFA

27. The Chomsky Hierarchy Non-recursively enumerable
Context-sensitive
Context-free
Regular

28. Lecture 6: Context-Free Languages
Pushdown Automata
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29. Stack or Pushdown store
Basic Concept
How to accept language L={wwR: w*}? a b
Finite
Control
Reading head
Input
tape
Stack or Pushdown store

30. Definition (K, , , , s, F) A pushdown automata is a sextuple where
K : a finite set of states
 : a finite set of input symbols
 : a finite set of stack symbols
 : the transition function K **  K *
s K : the initial state
F K : the set of final states

31. Definition  More Precise Version
A PDA is possibly nondeterministic.
Definition  More Precise Version
A pushdown automata is a sextuple
(K, , , , s, F)
where
K : a finite set of states
 : a finite set of input symbols
 : a finite set of stack symbols
 : the transition function K **  2K *
s K : the initial state
F K : the set of final states

32. The Transition Function of PDA
K **  K *
current state
symbol(s) under reading head
symbol(s) on the top of stack
next state
stack symbol(s) replaced

33. The Transition Function of PDA
:K **  K *
The Transition Function of PDA
(p, v, ) (q, ) v w u p q
finite
control   v w u p q
finite
control  

34. Stack Operations (p, v, ) (q, ) Case 1:   ,   
replacement
push
Case 2:   ,  = 
Case 4:  = ,  = 
Stack Operations
pop
none
(p, v, ) (q, ) v w u p q
finite
control   v w u p q
finite
control  

35. Yields in One Step
if (p, v, ) (q, )

36. in some (including zero) steps
Yields *
in some (including zero) steps if

37. Languages Accepted by PDA’s
M = (K, , , , s, F)

38. Example
A deterministic PDA

39. Example t1 t2 t3 t4 t5 state input stack transition abbcbba s  -f 3 ba f 5 a f 5  f 4

40. How to design a PDA to accept L = {wwR}?
Example
state
input
stack
transition
How to design a PDA to accept L = {wwR}? t1 t2 t3 t4 t5
abbcbba s  -
bbcbba s a 1
bcbba s ba 2
cbba s
bba 2
bba f 3 ba f 5 a f 5  f 4

41. Example t1 t2 t3 t4 t5 state input stack transition s abbbba  - sf
bba
bba 3 f ba ba 5 f a a 5 f   4

42. A nondeterministic PDA
Example t1 t2 t3 t4 t5
A nondeterministic PDA

43. Lecture 6: Context-Free Languages
Pushdown Automata
vs.
Context-Free Grammars
大同大學資工所
智慧型多媒體研究室

44. Leftmost and Rightmost Derivations
Given a context-free grammar there may be several ways of rewriting its nonterminals to arrive at precisely the same string.
Two systematic ways of derivations are
Leftmost derivation
Rightmost derivation

45. Example V={S, A, B}, ={a, b, c} Consider G = (V, , P, S) with
Note that there are other derivations that are neither leftmost nor rightmost.
Example
V={S, A, B}, ={a, b, c}
Consider G = (V, , P, S) with
S  aABb
A  bAb
A  c
B  aB
B  b P=
Rightmost Derivation
Leftmost Derivation
S  aABb
 abAbBb
 abcbBb
 abcbaBb
 abcbabb L
S  aABb
 aAaBb
 aAabb
 abAbabb
 abcbabb R

46. See textbook for the proof
Lemma
For any context-free grammar G = (V, , P, S) and any string w *,
See textbook for the proof

47.  Context-Free Grammar
Theorem
The class of languages accepted by pushdown automata is exactly the class of context-free languages.
Pf)
Context-Free Grammar
 Pushdown Automaton
G = (V, , P, S)
M = ?
Pushdown Automaton
 Context-Free Grammar
M = (K, , , , s, F)
G = ?

48. Theorem Pf) G = (V, , P, S)  M = ({p, q}, , V , , p, {q}) M = ?
The class of languages accepted by pushdown automata is exactly the class of context-free languages.
Theorem
Pf)
Context-Free Grammar
 Pushdown Automaton
G = (V, , P, S)
 M = ({p, q}, , V , , p, {q})
M = ?
Transition Function
Stack Symbols
Input Symbols
Start State
Final State
States

49. One can show that L(M)=L(G).
The class of languages accepted by pushdown automata is exactly the class of context-free languages.
Theorem
Pf)
Context-Free Grammar
 Pushdown Automaton
G = (V, , P, S)
 M = ({p, q}, , V, , p, {q})
M = ?
One can show that L(M)=L(G).
(see text)

50.  Context-Free Grammar
The class of languages accepted by pushdown automata is exactly the class of context-free languages.
Theorem
Pf)
Pushdown Automaton
 Context-Free Grammar
M = (K, , , , s, F)
G = ?
see text

51. Lecture 6: Context-Free Languages
Properties of
Context-Free Languages
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52. Review  Properties of Regular Languages
The regular languages are closed under:
union;
concatenation;
Keene star;
complementation;
intersection.

53. Properties of Context-Free Languages
The context-free languages are closed under:
union;
concatenation;
Keene star;
complementation;
intersection. ? ? ? ? ?

54. Properties of Context-Free Languages
The context-free languages are closed under:
union;
concatenation;
Keene star.

55. Properties of Context-Free Languages
The context-free languages are closed under:
union;
concatenation;
Keene star.

56. Properties of Context-Free Languages
The context-free languages are closed under:
union;
concatenation;
Keene star.

57. Review: The Pumping Theorem for Regular Languages
Let L be an infinite regular set. Then there are strings u,v, and w st. |v|>0 and uvnwL for all n  0.

58. What CFG’s Can? What CFG’s cannot?
Which of the following languages are context-free?

59. The Pumping Theorem for Context-Free Languages
See textbook for the proof.
The Pumping Theorem for Context-Free Languages
Let G be context-free gramma. Then there is a number K, depending on G, such that any w  L(G) with |w|>K can be rewritten as w = uvxyz in such a way that either v or y is nonempty and uvnxynz  L(G) for all n  0.

60. Theorem is not context-free. Pf) u v x y Let
Let G be context-free gramma. Then there is a number K, depending on G, such that any w  L(G) with |w|>K can be rewritten as w = uvxyz in such a way that either v or y is nonempty and uvnxynz  L(G) for all n  0.
Theorem
is not context-free.
Pf)
Let u v x y
It is impossible for us to rewrite w as w = uvxyz such that |vy|>0 and uvnxynz  L(G) for all n  0.

61. Theorem
The context-free languages is not closed under intersection or complementation.

62.  CFL’s are not closed under intersection.
Theorem
The context-free languages is not closed under intersection or complementation.
Pf)
Facts:
context-free
 not context-free
 CFL’s are not closed under intersection.
Fact:
If CFL’s are closed under complementation,
they would be also closed under intersection.

63. Theorem: Algorithmic Properties
See textbook for the proof.
Theorem: Algorithmic Properties
There are algorithms for answering the following questions about context-free grammars:
Given a grammar G, and a string w, is wL(G)?
Is L(G)=?
How?
What is compiler-compiler?

64. The Chomsky Hierarchy Chomsky Hierarchy Languages Grammars Automaton
Type 0
Recursively enumerable
unrestricted
Turing Machine
Recursive
Decider
Type 1
Context-Sensitive
Linear-Bounded Automaton
Type 2
Context-Free
Push-Down Automaton
Type 3
Regular
NFA or DFA

65. The Chomsky Hierarchy Non-recursively enumerable
Context-sensitive
Context-free
Regular

66. Lecture 6: Context-Free Languages
Context-Sensitive Grammars
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智慧型多媒體研究室

67. Definition  Context-Free Grammar (Review)
A context-free grammar G is a quadruple
(V, , P, S)
where
V : finite set of nonterminals
V   = 
 : finite set of terminals
P : finite set of rules in the form of A
with AV and (V )*
S : start symbol, SV

68. Definition  Context-Sensitive Grammar (I)
A context-sensitive grammar G is a quadruple
A context-free grammar G is a quadruple
(V, , P, S)
where
V : finite set of nonterminals
V   = 
 : finite set of terminals
P : finite set of rules in the form of A
with AV and (V )*
  
with , (V )+ and ||  ||.
S : start symbol, SV

69. Definition  Context-Sensitive Grammar (II)
A context-sensitive grammar G is a quadruple
A context-free grammar G is a quadruple
(V, , P, S)
where
V : finite set of nonterminals
V   = 
 : finite set of terminals
P : finite set of rules in the form of A
with AV and (V )*
  
A  
with , (V )*, AV and (V )+.
with , (V )+ and ||  ||.
S : start symbol, SV

70. Definition  Context-Sensitive Grammar (II)
A language L is said to be context-sensitive if there exists a context-sensitive grammar G, such that
L = L(G) or
L = L(G){}

71. Is it context-sensitive?
Review
is not context-free.
Is it context-sensitive?

72. Example
Consider G = (V, , P, S) with
V={S, A, B}
={a, b, c}
L(G)=?

73. Example
Consider G = (V, , P, S) with
V={S, A, B}
={a, b, c}

74. Example
Consider G = (V, , P, S) with
V={S, A, B}
={a, b, c}

75. The Chomsky Hierarchy Chomsky Hierarchy Languages Grammars Automaton
Type 0
Recursively enumerable
unrestricted
Turing Machine
Recursive
Decider
Type 1
Context-Sensitive
Linear-Bounded Automaton
Type 2
Context-Free
Push-Down Automaton
Type 3
Regular
NFA or DFA

76. The Chomsky Hierarchy Non-recursively enumerable
Context-sensitive
Context-free
Regular

77. Exercises
Construct pushdown automata for the following languages over the alphabet {a, b}:
{w | w has an equal number of a’s and b’s }
{w | the length of w is odd and its middle symbol is a b}
Prove that the following languages are not context free:
{aibjck | 0 < i < j < k}
{wwRw | w  {a,b}*}
Construct context-free grammars that define the following languages over {a, b}:
{ambn | m  n}
{aibjck | i + j = k}