Finite State Machines with Output

ComputationTheory1 A.L. Mustafa Ghanem Saeed
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ComputerScience Department Collage of Science CihanUniversity
Finite State Machines with Output
(Mealy and Moore Machines)
We shall investigate two different models for FA's with output
capabilities; these are Moore machine and Mealy machine.
Moore Machines:
A Moore machine M is the 5 tuple M = (Q,, Г, Ϩ, q0) where
1. A finite set of states q0, q1, q2, …, where q0 is designed as the
start state.
2. An alphabet of letters for forming the input string Σ= { a, b, c, …}.
3. An alphabet of possible output characters Г = { x, y, z, …}.
4. A transition function Ϩ as in table that shows for each state and
each input letter what state is reached next.
5. Start state as q0.
An output table that shows what character from Г is printed by each state
that is entered.
A Moore machine does not define a language of accepted words, since
every input string creates an output string and there is no such thing as a
final state. The processing is terminated when the last input letter is read
and the last output character is printed.
Example 1:
Input alphabet: Σ = {a, b}
Output alphabet: Г = {0, 1}
Names of states: q0, q1, q2, q3. (q0 = start state)
Q Ϩ(q, a) Ϩ(q, b) Г(Q)
q0 q1 q3 1
q1 q3 q1 0
q2 q0 q3 0
q3 q1 q2 1
Sol:
The Moore machine is:

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Note1:
 The two symbols inside the circle are separated by a slash "/", on the
left side is the nameof the state and on the right is the outputfrom
that state.
 If the inputstring is abab to the Moore machinethen the output will
be 10010.
Note 2:
A Moore machineis very similar to a Finite Automaton (FA), with a few
key differences:
• It has no final states.
• It does not accept or reject input; instead, it generates output from
input.
• Moore machines cannothave nondeterministic states.
Every input gives output not if word belongs to the machine or language
like FA
In each state we stop we have to print out what inside that state
(it's content) so the output will be more than input by one because we
start with start state and print out its content before we trace the input
string
Example 2: construct a Moore machine that take set of all strings over
Alphabet {a, b} as input and print 1 as output for every occurrence of
'aab' as substring.
Sol:

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Input string aaababbaabb
State q0q1q2q2q3q1q0q0q1q2q3q0
Output 000010000010
this machine gives 1 after each aab
aabaabaaababaab
0001001000100001
Note:
 Moore machine use as a string recognizer to give us a mark (1) after
each substring so we design a machine put 0 in all states except the
one after the one represent end of substring aab
 The output of the machine contains 1 for each occurrence of the
substring aab found in the input string.
Example 3: Construct a Moore machine that take set of all strings over
Alphabet {0, 1} as input and produce 'A' as output if input end with (10)
or produce 'B' as output if input end with (11) otherwise produce 'C' .
Sol:
Input alphabet: Σ = {0, 1}
Output alphabet: Г = {A, B, C}
Exam

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ple 4: Construct a Moore machine that take set of all strings over
alphabet {a, b} as input and produce '1' as output if number of a's in
string mod 3=0 otherwise produce '0'.
Sol:
Example 5: Construct a Moore machine that outputs a binary string that
contains a 1 for every double letter substring in an input string composed
of a’s and b’s. For example if abba is the input string, 0010 is the
corresponding output.
Sol:

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Mealy Machines:
A Mealy machine M is the 5 tuple, M = (Q, , ,Г, Ϩ, q0) where:
1. A finite set of states q0,q1,q2,…, where q0 is designated the start
state
2. An alphabet Σ of input letters
3. An alphabet Г of output characters
4. A finite set of transitions that indicate, for each state and letter of
the input alphabet, the state to go to next and the character that is
output.
Note: There are no accept states in Moore and Mealy machine because
they are not a language recognizer, they are an output producer. Its
output will be the same length as its input.
Example 1:
Names of states: q0, q1, q2, q3. (q0 = start state)
Q Ϩ(q, a) Г(q, a) Ϩ(q, b) Г(q, b)
q0 q1 0 q3 0
q1 q3 1 Q2 1
q2 q3 0 q3 1
q3 q3 1 q0 1
Sol:
The Mealy machine is:

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Quick review for Binary system complements
As the binary system has base r = 2. So the two types of complements for
the binary system are 2's complement and 1's complement.
1's complement
The 1's complement of a number is found by changing all 1's to 0's and
all 0's to 1's. This is called as taking complement or 1's complement.
Example of 1's Complement is as follows.
2's complement
The 2's complement of binary number is obtained by adding 1 to the
Least Significant Bit (LSB) of 1's complement of the number.
2's complement = 1's complement + 1
Example of 2's Complement is as follows.
Note:
 Two’s complement is especially important because it allows us to represent signed
numbers in binary.

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 Two’s complement also provides an easier way to subtract numbers using
addition instead of using the longer.
Example 2:
The following Mealy machine prints out the 1's complement of an input
bit string.in other words it flips each digit from a 0 to a 1 or from a 1 to a
0.
Note: If the input is 001010, the output is 110101.This is a case where
the input alphabet and output alphabet are both {0, 1}.
Example 3:
Construct a Mealy machine that take set of all strings over alphabet {0, 1}
as input and produce 'E' as output if number of 1's in string mod 2=0
produce 'O' as output if number of 1's in string mod 2=0.
Sol:
Output alphabet: Г = {E, O}
Note: The machine represented in in example 2, outputs an E if the
number of 1s read so far is even and an O if it is odd; for example, the
translation of 11100101 is OEOOOEEO.
Example 4:
Construct a Mealy machine that take set of all strings over alphabet {0, 1}
and make 2's complement of an input bit string in binary number.
Sol:
Output alphabet: Г = {0,1}

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Note: The machine represented in in example 4, if the input is 1010100,
the output is 0101100. This is a casewhere the input alphabet and output
alphabet are both {0, 1}.
Mealy and Moore machines are equivalent
If Mealy machine capture a model, so does a Moore machine
 Moore model might require more states to model a system but
simpler to understand
 Mealy model might require less states to model but might be more
complex to understand
Note: All examples in Moore machine can be solvein mealy Machine
and vris versa becauseMealy and Mooremachines are equivalent, you
have totry solving themas Homework.

Finite State Machines with Output

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