1. Languages
Given an alphabet , we can make a word or string by concatenating the letters of .
Concatenation of “x” and “y” is “xy”
Typical example: ={0,1}, the possible words over  are the finite bit strings.
A language is a set of words.

2. More about Languages
The empty string  is the unique string with zero length.
Concatenation of two langauges: A•B = { xy | xA and yB }
Typical examples: L = { x | x is a bit string with two zeros } L = { anbn | n  N } L = {1n | n is prime}

3. A Word of Warning
Do not confuse the concatenation of languages with the Cartesian product of sets.
For example, let A = {0,00} then
A•A = { 00, 000, 0000 } with |A•A|=3,
AA = { (0,0), (0,00), (00,0), (00,00) }
with |AA|=4

4. Recognizing Languages
Let L be a language  S
a machine M recognizes L if
“accept”
if and only if xL
xS M
“reject”
if and only if xL

5. Finite Automaton
The most simple machine that is not just a finite list of words.
“Read once”, “no write” procedure.
Typical is its limited memory.
Think cell-phone, elevator door, etc.

6. A Simple Automaton (0) transition rules states starting stateq1 q2 q3 1
0,1
starting state
accepting state

7. A Simple Automaton (1) start accept on input “0110”, the machine goes:q1 q2 q3 1
0,1
start
accept
on input “0110”, the machine goes:
q1  q1  q2  q2  q3 = “reject”

8. A Simple Automaton (2) on input “101”, the machine goes:q1 q2 q3 1
0,1
on input “101”, the machine goes:
q1  q2  q3  q2 = “accept”

9. A Simple Automaton (3) 010: reject 11: accept 010100100100100: acceptq1 q2 q3 1
0,1
010: reject
11: accept
: accept
: reject
: reject

10. Finite Automaton (def.)
A deterministic finite automaton (DFA) M is defined by a 5-tuple M=(Q,,,q0,F)
Q: finite set of states
: finite alphabet
: transition function :QQ
q0Q: start state
FQ: set of accepting states

11. M = (Q,,,q,F) q1 q2 q3 1 0,1 states Q = {q1,q2,q3}
0,1
states Q = {q1,q2,q3}
alphabet  = {0,1}
start state q1
accept states F={q2}
transition function :

12. Recognizing Languages (def)
A finite automaton M = (Q,,,q,F) accepts a string/word w = w1…wn if and only if there is a sequence r0…rn of states in Q such that:
1) r0 = q0
2) (ri,wi+1) = ri+1 for all i = 0,…,n–1
3) rn  F

13. Regular Languages
The language recognized by a finite automaton M is denoted by L(M).
A regular language is a language
for which there exists a recognizing
finite automaton.

14. THANK YOU THT Prepared By:- Preeti Gupta
Deptt. Of Computer Applications
Prepared By:-PP