This document discusses finite automata and regular languages. It begins by introducing finite state machines as the simplest computational model due to their extremely limited memory. Examples of finite state machines in everyday devices like automatic doors, elevators, and calculators are provided. The document then presents a formal definition of a finite automaton as a 5-tuple consisting of a finite set of states, a finite input alphabet, a transition function, a start state, and a set of accept states. An example three-state finite automaton M1 is defined formally using this 5-tuple notation. The language recognized by M1 is described as the set of strings containing at least one 1 and an even number of 0s following the last 1.

1. Theory of Computation - Lecture 3
Regular Languages

What is a computer?
Complicated, we need idealized computer for
managing mathematical theories...
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
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Hence: Computational Models.
Comp. Models vary, depending on the
features we want them to focus on.
Simplest Comp. Model is Finite State
Machine, or Finite Automaton
Dr. Maamoun Ahmed

2. Finite Automaton


Good models for computers with extremely
limited memories.
With such limited memories, what can such
computers do? Give examples of such
computers!
…..

3. Example – Automatic Door
Door
Front pad
Rear pad

4. FSM/FA of Auto-door
FRONT
REAR
BOTH
REAR
BOTH
NIETHER
FRONT
CLOSED
OPEN
NEITHER

5. Cont.

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How many bits required for auto-door
implementation?
More examples: Elevator, what are the
states and inputs?
Dishwasher, Washing machine,
Thermostats, digital watches and
calculators, etc....
Finite Automata was in mind when all of
above were designed!

6. Closer look at Finite Automata

Abstracting, application-independent:
Example: A finite automaton called M1 with 3
DD SSt
states:
iai tata
g ete
a
1
0
q1
q2
q3
0,1
0
rgr
am
am
1

7. Closer Look – Cont.

Start State

Accept State

Transitions
- When 1101 is entered from the input to Start State
(one by one, L2R), the output will be either
Accept/Reject or for now, Yes/No respectively.

Will 1101 output Accept or Reject in our
example?
What about 1? 01?11?101001001?
What about 100? 0100? 1000000?

8. 


What strings will M1 accept? What will it
reject?
Can you describe the language consisting
of all strings that M1 accepts?
…..Go on!

9. Formal definition of a FA
Why do we need a formal definition?
1- It is precise: Uncertain whether some FA allowed
to have 0 accept states or not? Formal definition
helps you decide.
2- It provides Notations, and they help you think
and express your thoughts clearly.

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Formal definition says that FA is a list of five
objects: Set of states, input alphabet, rules for
moving, start state, and accept states
What do we call a list of 5 elements ?

10. Formal definition of FA – cont.


We define a FA to be a 5-tuple consisting
of these five parts.
Transition function ( δ) defines the rules of
moving.
Ex.: 2 states x, y. and an arrow from x to y
with a label=1, then moving from x to y is
ruled by reading 1 as an input, which can
be written as : δ(x,1)=y

11. Formal definition of FA

A finite automaton is a 5-tuple (Q, ∑ , δ, q0,
F) where:

Q is a finite set called the states

∑ is a finite set called the alphabet

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δ:Q X ∑ → Q is the transition function
q0 ∈ Q is the start state, and
F ⊆ Q is the set of accept
states(sometimes called Final states).

12. Example

Go back to slide no.6 (M1 FA), we can
describe M1 formally by writing
M1 = (Q, ∑, δ, q1, F), where:
1. Q = {q1, q2, q3}
2. ∑ = {0,1}
3. δ is described as
0
1
q1
q1
q2
q2
q3
q2
q3
q2
q2

13. Example – cont.
4. q1 is the start state, and
5. F = {q2}
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
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Note: A is a set of all strings that machine M
accepts, we say: A is the language of machine
M and write L(M)=A. We say: M recognizes A.
A machine may accept several strings, but it
always recognizes only one language.
If the machine accepts no strings, it still
recognizes one language, the empty language ϕ

14. Example – cont.

So, in our example let:
A = {w | w contains at least one 1 and an
even number of 0s follows the last 1}.
Then L(M1) = A, or equivalently, M1
recognizes A.

Examples of FA (1.7 – 1.15) pages 37-40