1. CSC 4170
Theory of Computation
Finite Automata
Section 1.1

2. How a finite automaton works1 q0 q2 1 1 q1

3. The language of a machine
1.1.b 1 q0 q2 1 1 q1
L(M), “the language of M”, or “the language recognized by M”
--- the set all strings that the machine M accepts
What is the language recognized by our automaton A?
L(A) =

4. Formal definition of a finite automaton
1.1.c
A finite automaton is a 5-tuple (Q, , , s, F), where:
Q is a finite set called the states,
 is a finite set called the alphabet,
 is a function of the type Q  Q called the transition function,
s is an element of Q called the start state,
F is a subset of Q called the set of accept states.

5. Our automaton formalized1 q0 q2 Q: : : s: F: 1 1 q1
A = (Q, , , s, F)

6. Formal definition of accepting
M = (Q, , , s, F) 1 q0 q2 1 1 q1
M accepts the string
u1 u2 … un
iff there is a sequence
r1, r2, …, rn, rn+1
of states such that:
r1=s
ri+1 = (ri,ui), for each i with 1 i  n
rn+1  F
u u2 … un
r1, r2, …, rn, rn+1

7. Designing finite automata
Task:
Design an automaton that accepts a bit string iff it contains an even number of “1”s.

8. Designing finite automata
Task:
Design an automaton that accepts a bit string iff the number of “1”s that it contains is
divisible by 3.

9. Designing finite automata
1.1.h
Task: Let L2={w | w is a string of 0s whose length is divisible by 2} and
L3={w | w is a string of 0s whose length is divisible by 3}
Design an automaton that recognizes L2L3

10. Designing finite automata
Task: Let L2={w | w is a string of 0s whose length is divisible by 2} and
L3={w | w is a string of 0s whose length is divisible by 3}
Design an automaton that recognizes L2L3

11. Designing finite automata
1.1.j
Task: Design an automaton that recognizes the language
X={w | w is a string of 0s whose length is divisible neither by 2 nor by 3}
Definition: Let L be a language over an alphabet . The complement of L is the
language {w | w is a string over  such that wL}.
X is the complement of what language?