CSC 4170
Theory of Computation
Finite Automata
Section 1.1
1.1.a
How a finite automaton works
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1.1.b
The language of a machine
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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) =
1.1.c
Formal definition of a finite automaton
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.
1.1.d
Our automaton formalized
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Q:
:
:
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s:
F:
A = (Q, , , s, F)
1.1.e
Formal definition of accepting
M = (Q, , , s, F)
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M accepts the string
u 1 u 2 … un
iff there is a sequence
0
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
u1
u2 …
un
0 1 1 0 0
r1, r2, …,
rn, rn+1
1.1.f
Designing finite automata
Task:
Design an automaton that accepts a bit string iff it contains an even number of “1”s.
1.1.g
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.
1.1.h
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
1.1.i
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
1.1.j
Designing finite automata
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?