1. Context-Free Grammars
1. Formal Definition
2. Construction
3. Parse Tree
4. Ambiguity
5. Simplification of CFG
6. CNF & GNF

2. English Grammar sentence  noun_phrase predicate
noun_phrase  article noun
predicate  verb
article  a | an | the
noun  boy | dog
verb  runs | walks
a boy runs a dog walks

3. Context-Free Grammar
A grammar G=( V, T, S, P ) is said to be context-free if all productions in P have the form
A   , where AV, ( V T )*

4. Palindrome Language L={ w | w{0,1}* and w = wR } recursive definition
basis , 0, 1 are palindromes.
induction If w is a palindrome, so is 0w 0 and 1w1.

5. Palindrome Language L={ w | w{0,1}* and w = wR }
definition with grammars or rules
1.  is a palindrome.
2. 0 is a palindrome.
3. 1 is a palindrome.
4. If w is a palindrome, so is 0w0.
5. If w is a palindrome, so is 1w1.

6. CFG & Palindrome Language
L={ w | w{0,1}* and w = wR }
1.  is a P.
1. P  
2. 0 is a P.
2. P  0
3. 1 is a P.
3. P  1
4. If w is a P, so is 0w0.
4. P  0P 0
事儿是一件事儿，只是换了另一种说法。或者采用了另一种语言。
5. If w is a P, so is 1w1.
5. P  1P 1

7. CFG of Palindrome Language
L={ w | w{0,1}* and w = wR }
CFG for palindromes on {0,1}
R = ({S }, {0,1}, P, S ), P is defined as follow
S   , S  0, S  1, S  0S 0, S  1S 1
Compact notation
S   | 0 | 1 | 0S 0 | 1S 1

8. Example 7.1
L={ 0n1n | n  0 }
R = ({S }, {0,1}, P, S ), P is defined as follow
S   | 0S1

9. Example 7.2
L={ 0n1m | n  m }
R = ({S,A,B,C }, {0,1}, P, S ), P is defined as follow
S  AC | CB, C  0C1 | 
A  A0 | 0 , B  1B | 1

10. Example 7.3
L={ w{0,1}* | w contains same number of 0’s
and 1’s }
R = ({S }, {0,1}, P, S ), P is defined as follow
S   | 0S1 | 1S0 |SS

11. Example 7.4
L={w {0,1}*| n0(w)=n1(w) and n0(v)  n1(v)
where v is any prefix of w }
R = ({S }, {0,1}, P, S ), P is defined as follow
S   | 0S1 | SS

12. Derivations and Recursive Inferences
L={a2nbm | n  0, m  0 }
R = ({S,A,B }, {a,b}, P, S ), P is defined as follow
S  AB , A  |aaA , B  | Bb
for w =aabb :
SABaaABaaABbaaBbaaBbbaabb
B Bb
B  Bb
S  AB
A  aaA
A  
B  

13. Context-Free Language
Let G=( V, T, S, P ) be context-free, then
L(G) = {w | w T * and S  w } 

14. Left/Right Most Derivations
L={a2nbm | n  0, m  0 }
S  AB , A  |aaA , B  | Bb
for w =aabb :
SABaaABaaABbaaBbaaBbbaabb
Left most :
SABaaABaaBaaBbaaBbbaabb
Right most :
SABABbABbbAbbaaAbbaabb

15. Parse Tree
Let G = ( V,T , S, P ) be a CFG. A tree is a parse tree for G if :
1. Each interior node is labeled by a variable in V
2. Each leaf is labeled by a symbol in T{}. Any -labeled leaf is the only child of its parent.
3. If an interior node is labeled A, and its children (from left to right) labeled x1,x2, ,xk,
Then A  x1,x2, ,xk  P .

16. Parse Tree Example 7.5 L={ w | w{0,1}* and w = wR }
S   | 0 | 1 | 0S 0 | 1S 1 S S
0 S 0
0 S 0
recursive inferences
derivations
1 S 1
0 S 0  1
w=0110
w=00100

17. Ambiguity G = ({E, I }, {a, b, (, ), +, }, P, E )
EI |E +E |E E|(E), Ia | b
Derivation for w = a a  a :
EEEE+EEI+EEa+EE a+aa 
EEEI+Ea+Ea+EEa+aa 

18. Ambiguity parse-tree for w = a a  a : E E E  E E  E E + E I I

19. Removing Ambiguity EI | E+E | EE | (E) , Ia | b
ET|E+T, TF|TF, FI|(E) , Ia|b|Ia|Ib
Left most derivation for w = a a  a :
EETT+TF+T I+T a+Ta+TF
a+FFa+IFa+aFa+aIa+aa
ETTT(E)T(E+T)T(a+a)a 

20. Inherent Ambiguity What is inherent ambiguity
A CFL L is said to be inherently ambiguous if every grammar that generates it is ambiguous.
Example 7.6 Let L={ w | w{0,1}* and n0(w)=n1(w) }
L is not inherently ambiguous ,because there is an unambiguous CFG :
S   | 0S1 | 1S0 | 0S11S0 | 1S00S1

21. Example 7.7
L={anbncmdm | n1, m1} {anbmcmdn | n1, m1}
The CFG for L is :
SAB | C , AaAb | ab, BcBd | cd
CaCd | aDd, DbDc | bc
Let w= abcd , there are two left most derivations
SAB abB abcd
SC aDd abcd

22. Simplification of CFG Why & what :
SA | B, A1CA | 1DE | , B1CB | 1DF,
C1CC | 1DG | 0G, D1CD | 1DH | 0H,
E0A, F0B, G, H1
-productions
unit productions
useless symbols and productions

23. Eliminating -productions*
Variable A is said to be nullable if A  .
Let G=(V,T,P,S) is a CFG
If A → P, then A is nullable.
If A → A1 A2  AkP, and Ai → P for i=1, ,k
then A is nullable.
Example 7.8 G : SAB, AaAA|, BbBB|
A  A is nullable.
SAB  S is nullable.
B  B is nullable.

24. Eliminating unit productions
Example 7.9 G : SA|B|0S1, A0A|0, B1B|1
S 0A|0|1B|1|0S1
A0A|0
B1B|1

25. Eliminating useless productions
A symbol X is useful for a grammar G=(V,T,P,S),
if there is a derivation for some wT* * *
S  X  w *
A symbol X is generating if X  w for some wT* *
A symbol X is reachable if S  X for {,}(VT)*

26. Example G : SAB|a, Ab
S and A are generating , B is not.
Eliminate B, that eliminate SAB, leaving
Sa, Ab
Now only S is reachable. So there leaves Sa.
If eliminate non-reachable symbol first :
SAB|a, Ab  SAB|a, Ab
Then eliminate non-generating symbol :
SAB|a, Ab  Sa, Ab

27. Example 7.11 G : SA | B, A1CA | 1DE | 
B1CB | 1DF, C1CC| 1DG | 0G, D1CD | 1DH | 0H,
E0A, F0B, G, H1
eliminating -productions
the only one : A
SA | B, A1CA | 1C | 1DE, B1CB | 1DF,
C1CC | 1DG | 0G, D1CD | 1DH | 0H,
E0A|0, F0B, G, H1

28. SA | B, A1CA | 1C | 1DE, B1CB | 1DF,
C1CC | 1DG | 0G, D1CD | 1DH | 0H,
E0A|0, F0B, G, H1
eliminating unit productions
the only two : SA and SB
S1CA | 1C | 1DE|1CB | 1DF,
A1CA | 1C | 1DE, B1CB | 1DF,
C1CC | 1DG | 0G, D1CD|1DH | 0H,
E0A|0, F0B, G, H1

29. S1CA | 1C | 1DE|1CB | 1DF,
A1CA | 1C | 1DE, B1CB | 1DF,
C1CC | 1DG | 0G, D1CD|1DH | 0H,
E0A|0, F0B, G, H1
eliminating useless symbols and productions
S1DE ,A1DE , D1DH | 0H , E0A | 0, H1

30. Chomsky Normal Form(CNF)
1. A BC ;
2. A a .
S1DE ,A1DE , D1DH | 0H , E0A | 0, H1
Chomsky normal form :
SIE ,AIE, DIH|EH, EEA|0, IHD, H1
DIH|FH, EFA|0, F0

31. Chomsky Normal Form(CNF)
Example 7.12 Convert following grammar to CNF
SABa , Aaab , BAc

32. Greibach Normal Form(GNF)
A  ax , where aT , xV
Example 7.13 Convert following grammar to GNF
SAB , AaA|bB|b , Bb
Example 7.14 Convert following grammar to GNF
S01S1|00

33. ? ? ?
eliminating -productions :   L ?
Greibach normal form :
A  a advantage ?
Chomsky normal form :
A  a |BC advantage ?
left recursiveness
A  A shortage ?