1. COSC 3340: Introduction to Theory of Computation
University of Houston
Dr. Verma
Lecture 9
Lecture 9
UofH - COSC Dr. Verma

2. Context Free Languages
Strictly bigger class than regular languages
{anbn | n >= 0}
context-free
languages 
regular languages  
{wwR |w in {a,b}*}
{arithmetic expressions}
Lecture 9
UofH - COSC Dr. Verma

3. A simple grammar for some sentences
<Sentence>  <noun> <verb> <object>
<noun>  Alice | John
<verb>  eats | eat | ate
<object>  apple | orange | mango
The goal is to generate sentences in English over the English alphabet. Example:
<Sentence>  <noun><verb><object>  Alice <verb><object> 
Alice eats<object>  Alice eats apple
Lecture 9
UofH - COSC Dr. Verma

4. Context-free grammars (CFGs)
Informally, a CFG is a finite set of rules.
Each rule is of the form:
<nonterminal symbol>  string over terminals and nonterminals.
Terminals:- symbols that the desired strings should contain.
Example: {a...z, ' ',...}
Nonterminals:- symbols to which rules can be applied.
Example: {<noun>, <verb>, ...}
A special nonterminal is called start symbol.
Example: <Sentence>
Lecture 9
UofH - COSC Dr. Verma

5. A grammar for arithmetic expressions
E  E + E | E * E | (E) | x | y
Start symbol - E.
Terminals?
Ans: {x, y, *, +, (, ), ' '}
Nonterminals?
Ans: {E}
A derivation:
E  E + E  E + E * E  x + E * E  x + x * E  x + x * y
Lecture 9
UofH - COSC Dr. Verma

6. Another grammar for arithmetic expr’s
E  E + T | T
T  T * F | F
F  (E) | x | y
A derivation for x + x * y ?
E E + T T + T F + T x + T 
x + T * F x + F * F x + x * F x + x * y
Why two different grammars for arithmetic expressions?
Lecture 9
UofH - COSC Dr. Verma

7. Context Free Grammar Definition
A CFG G = (V, T, P, S) where V  T = ,
V -- A finite set of symbols called nonterminals
T -- A finite set of symbols called terminals
P is a finite subset of V X (V  T)* called productions or rules
We write A  w whenever (A, w)  P.
S  V -- start symbol
Lecture 9
UofH - COSC Dr. Verma

8. Derivations and L(G) One step derivation: 0 or more steps derivation:
u  v if u = xAy, v = xwy and A  w in P
0 or more steps derivation:
u * v if u = u0  u1  ....  un = v (n  0)
L(G) = { w in T* | S * w }.
A language L is context-free if there is a CFG G with L(G) = L
Lecture 9
UofH - COSC Dr. Verma

9. Example: S  aSb |  Derivation: S  aSb  aaSbb  aabb. L(G) = ?
Ans: {anbn | n  0 }
Lecture 9
UofH - COSC Dr. Verma

10. Parse trees All derivations can be shown in the form of trees.
Order of rule application is lost. S a S b a S b 
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11. Parse trees [contd.] In general, if we apply rule
A  w0w1...wn, then we add nodes for wi as children of node labeled A A w0 w1 w2 wn
  
Lecture 9
UofH - COSC Dr. Verma

12. E  E + E  E + E * E  x + E * E  x + x * E  x + x * y
Lecture 9
UofH - COSC Dr. Verma

13. E E + T T + T F + T x + T x + T * F x + F * F x + x * F x + x * y
Lecture 9
UofH - COSC Dr. Verma

14. Leftmost and Rightmost Derivations
Derivation is leftmost if the nonterminal replaced in every step is the leftmost nonterminal.
Consider E  E + E  E + x.
Is it leftmost derivation?
Derivation is rightmost if the nonterminal replaced in every step is the rightmost nonterminal.
Consider E  E + E  x + E.
Is it rightmost derivation?
Lecture 9
UofH - COSC Dr. Verma

15. Ambiguity
A CFG is ambiguous if there is a string with at least two leftmost derivations
Example:
E  E + E | E * E | (E) | x | y is ambiguous
A CFL is inherently ambiguous if every CFG that generates it is ambiguous.
{anbncm | n, m  0}  {ambncn | n, m  0}
Lecture 9
UofH - COSC Dr. Verma