1. Formal Language & Automata Theory
Shyamanta M Hazarika
Computer Sc. & Engineering
Tezpur University

2. Regular Expressions A regular expression defines a language.
Regular expressions are defined through union,
concatenation, and closure.
Union: L1  L2  {x | x in L1 or x in L2}.
Concatenation: L1L2  {xy | x in L1 and y in L2}.
(Kleene) Closure: L*  i=0.. Li, where L0 = {}, Li =LLi-1.
Example: {01, 10}* = {, 01, 10, 0110, 0101, 1010, …}.

3. Formal Definition Definition: Regular expressions (over ).
1)  is a regular expression denoting  (empty set).
2)  is a regular expression denoting {}.
3) For each a  , a is a regular expression denoting {a}.
4) If E (F) is a reg. exp. denoting language L(E) (L(F)),
then (E + F) is a reg. exp. denoting L(E)  L(F),
(EF) ………………………….L(E)L(F),
(E*) ………………………….L(E)*.
We will omit parenthesis if no confusion.

4. Example
00(0 + 1)*01 + 
= {x | x =  or x, |x|  4, begins with 00 and ends
with 01}
(An alphabet of {0,1} is assumed here, and in many
other examples as well.)

5. Finite Automata and Reg. Exps.
Regular expressions define the same class of languages as Finite Automaton.
NFA
DFA
-NFA
Reg. Exp.

6. Theorem
Theorem: If R is a regular expression, then L(R) = L(E) for some NFA E with -moves.
Proof:
We construct E so that it has one final state and no
transitions out of that state.
We do this by induction on the number of operators in R.

7. Basis Only three cases: Case 1: R = . Case 2: R = . Case 3: R = a.q0
Start q0 qf
Start q0 qf
Start a

8. Inductive Step Case 1: R = R1 + R2.
By the induction hypothesis, machines E1 and E2 exist
for R1 and R2, respectively. Construct E as follows: q1 f1 E1  q0
Start f0 q2 f2 E2

9. Inductive Step (Continued)
Case 2: R = R1R2.
Start q1 f1 E1 q2 E2 f2 
Case 3: R = R1*.  q1 f1 E1   q0 f0
Start 

10. Example
0(0 + 1)* 0: 1:
0 + 1:
Start 1
Start    1 
Start

11. Example (Continued) 1 
0 + 1:
(0 + 1)*:
Start 1 
Start

12. Example (Continued)      (0 + 1)*:   0(0 + 1)*:           
(0 + 1)*:
0(0 + 1)*:  1 
Start         1 
Start

13. Example (Continued)  0(0 + 1)*:             1 
Start
Note: A much simpler machine exists.