1. COSC 3340: Introduction to Theory of Computation
University of Houston
Dr. Verma
Lecture 10
Lecture 10
UofH - COSC Dr. Verma

2. Chomsky Normal Form (CNF)
Rules of CFG G are in one of two forms:
(i) A  a
(ii) A  BC, B  S and C  S
+ Only one rule of the form S   is allowed if  in L(G).
Easier to reason with in proofs
Leads to more efficient algorithms
Credited to Prof. Noam Chomsky at MIT
Reading Assignment: Converting a CFG to CNF.
Lecture 10
UofH - COSC Dr. Verma

3. Exercises Are the following CFG's in CNF? (i) S  aSb | 
(ii) S  aS | Sb | 
(iii) S  AS | SB | 
A  a
B  b
(iv) S  AS | SB
A  a | 
Lecture 10
UofH - COSC Dr. Verma

4. Closure properties of CFL's
CFL's are closed under:
(i) Union
(ii) Concatenation
(iii) Kleene Star
What about intersection and complement?
Lecture 10
UofH - COSC Dr. Verma

5. The setting L1 = L(G1) where G1 = (V1, T, P1, S1) L2 = L(G2) where
Assume wlog that V1  V2 = 
Lecture 10
UofH - COSC Dr. Verma

6. Closure under Union -- Example
L1 = { anbn | n  0 }
L2 = { bnan | n  0 }
G1 ?
Ans: S1  aS1b | 
G2 ?
Ans: S2  bS2a | 
How to make grammar for L1  L2 ?
Ans: Idea: Add new start symbol S and rules S  S1 | S2
Lecture 10
UofH - COSC Dr. Verma

7. Closure under Union General construction
Let G = (V, T, P, S) where
V = V1  V2  { S },
S  V1  V2
P = P1  P2  { S  S1 | S2 }
Lecture 10
UofH - COSC Dr. Verma

8. Closure under concatenation Example
L1 = { anbn | n  0 }
L2 = { bnan | n  0 }
Is L1L2 = { anb{2n}an | n >= 0 } ?
Ans: No! It is { anb{n+m}am | n, m  0 }
How to make grammar for L1L2 ?
Idea: Add new start symbol and rule S  S1S2
Lecture 10
UofH - COSC Dr. Verma

9. Closure under concatenation General construction
Let G = (V, T, P, S) where
V = V1  V2  { S },
S  V1  V2
P = P1  P2  { S  S1S2 }
Lecture 10
UofH - COSC Dr. Verma

10. Closure under kleene star Examples
L1 = {anbn | n  0}
What is (L1)*?
Ans: { a{n1}b{n1} ... a{nk}b{nk} | k  0 and ni  0 for all i }
L2 = { a{n2} | n  1 }
What is (L2)*?
Ans: a*. Why?
How to make grammar for (L1)*?
Idea: Add new start symbol S and rules S  SS1 | .
Lecture 10
UofH - COSC Dr. Verma

11. Closure under kleene star General construction
Let G = (V, T, P, S) where
V = V1  { S },
S  V1
P = P1  { S  SS1 | }
Lecture 10
UofH - COSC Dr. Verma

12. Tips for Designing CFG's
Use closure properties -- divide and conquer
Analyze strings – Is order important? Number? Do we need recursion?
Flat versus hierarchical?
Are any possibilities (strings) missing?
Is the grammar generating too many strings?
Lecture 10
UofH - COSC Dr. Verma