Final fa part1

Finite Automata
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What are machines?
–

●

What are constituents of a machine?
–

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Automation of various steps.
Each step takes an input and produces an output.

Machine can be defined as
–

Set of states which take input in specified format and
produce desired output.

Finite Automata
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●

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Finite Automata (FA) is a language recognizing
machine.
Accepts all words of the language for which it is
built.
Rejects all other words.

Finite Automata
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Five tuple machine defined as M = (Q, ∑, ∂, Q0, F)
–

Q: Set of states such as Q0, Q1, Q2, Q3..... Qn.

–

∑: Alphabet of the language for which the machine is built such
as {a, b} or {0, 1}.

–

∂: Transition function or Next Move function. It maps Q X ∑ ->
Q.

–

∂ (Qi, a) = Qj where
– 'Qi' -> current state
– 'a' -> current input symbol
– 'Qj' -> next state
Q0 -> start state.

–

F -> collection of final states.

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Finite Automata
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Example of a Switch to “switch on” a fan.
–

Q = {OFF, ON}

–

∑ = {push}

–

∂ has two rules
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●

∂ (OFF, push) = ON.
∂ (ON, push) = OFF.

–

Q0 = {OFF}

–

F = {ON}

Finite Automata
●

FA is a collection of 3 things:
–

A finite set of states, one of them is the start state
and some (maybe) none as the final states.

–

An alphabet ∑, which consists of all possible input
letters.

–

A finite set of transition rules which direct from each
state and for each particular input letter which state
to go next.

Finite Automata
We discuss an example here. Following are two
representations of an FA:
●
●

Transition Table
Transition Graph
Transition Table
a

b

-x

y

z

y

x

z

+z

z

z

Transition Graph

Finite Automata
Depiction of initial and final states for an FA can be defined in several ways:
●

Use a '-' symbol for initial state and a '+' symbol for final state.

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Use a '->' symbol for initial state and double encircle the final state.

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Write 'start' for the initial state and 'final' for the final state.

Finite Automata
FA has
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3 states labelled x, y and z

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∑ = {a,b}

●

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State x is designated as the
start state.
State z is designated as the
final state.

Finite Automata
The transition table as well as the transition diagrams clearly define the ∂ function.
It is a depiction of the following rules:
Rule 1: If we are in state x and we encounter an input 'a' then we go to state y.
Rule 2: If we are in state x and we encounter an input 'b' then we go to state z.
Rule 3: If we are in state y and we encounter an input 'a' then we go to state x.
Rule 4: If we are in state y and we encounter an input 'b' then we go to state z.
Rule 5: If we are in state z and we encounter an input 'a' then we remain in state z.
Rule 6: If we are in state z and we encounter an input 'b' then we remain in state z.

Transition Graph

Transition Table
a

b

-x

y

z

y

x

z

+z

z

z

Finite Automata
●

To understand the language
dry run the input
–

“aaa”

–

“abba”

Finite Automata
●

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If after complete string traversal, we do not reach the “final state”,
the string is then REJECTED by the FA.
If after complete string traversal, we reach the “final state”, the string
is then ACCEPTED by the FA.
Thus, FA is a language recognizer, which accepts all the
words in the language for which it is defined and rejects all the
other strings.

Finite Automata
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What language does this FA represents?

Regular Expression for the language of
the FA: (a + b)* b (a + b)*

Finite Automata
●

A finite automata is
–
–

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“Deterministic” in nature.
Finite

Note: By convention we say that the null word (λ) starts as
well as ends in the initial state.

Finite Automata
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Analyse the FA given below

Regular Expression for the language of the
FA: (a + b) (a + b)*

Finite Automata
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What is the language of the FA given below?

Regular Expression for the language of the
FA: (a + b)*

Finite Automata
●

What is the language accepted by the FA which has no final
state?

Such FA's accept “no language”.

Finite Automata
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An interesting example. Identify the language.

This FA also accepts “no language”.

Finite Automata
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What is the language of the FA given below?

This FA accepts all words of even length over the alphabet ∑
= {a,b}.

Finite Automata
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Draw an FA which accepts all the words in the language a (a
+ b)* where ∑ = {a,b}.

First draw a diagram which accepts the minimum
length word which is 'a'.

Finite Automata
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Now complete all the other transitions.
For words starting from 'b', we
introduce a new state y, which is a
“trap state” .
Trap State
–

On reaching trap state, we can never
go back to the other states of the FA
and accept the word.

–

Reaching a trap state means we
“reject” the word.

–

We have only one trap state for the
whole FA.

–

We need to complete the transitions
of the trap state too.

Finite Automata
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●

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Draw an FA which accepts all the words
that have a double 'a' or a double 'b' in
them where ∑ = {a,b}.
Two minimum length words 'aa' and
'bb'.
First draw an FA which accepts these
two words.
All the states signify something:
–

State 1: Neither an 'a' nor a 'b' has
occurred.

–

State 2: One 'a' has occurred.

–

State 3: One 'b' has occurred.

–

State 4: Either two a's or two b's have
occurred in continuation.

Finite Automata
●

Now complete the remaining transitions.

Finite Automata
●

Examine the following FA and find out the language
represented by it.

FA accepts all the words which have their third letter as
'b'. The regular expression is (a+b) (a+b) (b) (a+b)*.

Finite Automata
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Construct an FA that accepts only the word 'baa' where ∑ =
{a,b}.

First draw the word 'baa'.
Now complete the FA.

Finite Automata

Note that there is only one “trap state” for the whole FA.

Finite Automata
●

Draw an FA which accepts only the word λ where ∑ =
{a,b}.

Now complete the transitions.

Finite Automata
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FA which accepts only the word λ where ∑ = {a,b}.

Finite Automata
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Draw an FA for all the words which end in 'a' where ∑ =
{a,b}.

Draw the minimum word i.e. 'a'.

Finite Automata
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Complete the transitions.

Finite Automata
●

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Draw an FA for all the words which have a double 'a' in
them where ∑ = {a,b}.

Significance of each state:
–

State 1: Zero a's are encountered consecutively.

–

State 2: One 'a' is encountered consecutively.

–

State 3: Double 'a' has been encountered.

Finite Automata
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Now complete all the transitions.

Finite Automata
●

Draw an FA for all the words with different first and last
letters where ∑ = {a,b}.

Final fa part1

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Final fa part1