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Mealy and Moore MachinesCSE2303 Formal Methods ILecture 8

OverviewMoore MachinesMealy MachinesSequential Circuits

A Moore Machine

a

b

OUT

q0

q1

q2

1

q1

q1

q1

0

q2

q1

q3

0

q3

q3

q1

1

Definition of a Moore MachineA finite set of statesq0, q1, q2, etc.q0 is the start stateAlphabet of input lettersAlphabet of output lettersTransitionsA unique one for each letter and each stateOutput TableA letter for each state

Moore Machine for abaq3/1aababba,bq0/0q1/0q2/0q4/0

Defining a LanguageTo change a FA into a Moore machine which accepts the same languageName each stateName the Start state q0Output 0 in all non-final statesOutput 1 in all Final states.A string is accepted if after it has been completed read in the last letter printed is 1.

A Mealy Machine

Qold

IN

Qnew

OUT

q0

a

q1

1

q0

b

q2

0

q1

a

q1

1

q1

b

q1

1

q2

a

q1

0

q2

b

q2

0

Definition of a Mealy MachineA finite set of statesq0, q1, q2, etc.q0 is the start stateAlphabet of input lettersAlphabet of output lettersTransitionsA unique one for each letter and each stateEach transition also has one output letter

Equivalence of MachinesEvery Moore machine can be turned into a Mealy machine.Every Mealy machine can be turned into a Moore machine.Every regular language can be defined by Moore machine or a Mealy machine.All languages defined by a Moore machine or a Mealy machine are regular.

Sequential CircuitsThe output depends on the sequence of past inputs.

The circuits consist ofa logic gates i.e. AND, OR, NOT, NAND, NORstorage elements i.e. flip-flops, latches.

Clocked Synchronous Sequential CircuitsClockedMeans all the flip-flops employ a clock input.SynchronousMeans all the flip-flops use the same clock signal.

These circuits only change state when a triggering edge or tick occurs on the clock signal.

Logic GatesAND NOTORABABA

A

B

AND

0

0

0

0

1

0

1

0

0

1

1

1

A

B

OR

0

0

0

0

1

1

1

0

1

1

1

1

A

NOT

0

1

1

0

D Flip-FlopStores one bit of informationnew value of Q = old value of DQ* = DDQQCLKNOT Q

Moore CircuitNext State Logic CircuitState MemoryOutput Logic Circuitclock inputclock signalinputoutput

Moore Machineq0/1q1/0q2/0q3/1aaabbba,b

a

b

OUT

q0

q1

q2

1

q1

q1

q1

0

q2

q1

q3

0

q3

q3

q1

1

Mealy CircuitNext State Logic CircuitState MemoryOutput Logic Circuitclock inputclock signalinputoutput

Mealy Machineq0q1q2b/0a/0a/1a/1, b/1b/0

Q

IN

Q*

OUT

q0

a

q1

1

q0

b

q2

0

q1

a

q1

1

q1

b

q1

1

q2

a

q1

0

q2

b

q2

0

D1 = IND2 = IN OR ((Q1 AND Q2) OR (NOT Q2))OUT = Q1 AND Q2Q1* = D1 Q2* = D2INANDOUTORD2clock signalORD1Q1Q2Q2

D1 = IND2 = IN OR ((Q1 AND Q2) OR (NOT Q2))OUT = Q1 AND Q2Q1* = D1 Q2* = D2

Q1

Q2

IN

Q1*

Q2*

OUT

0

0

0

0

1

0

0

0

1

1

1

0

0

1

0

0

0

0

0

1

1

1

1

0

1

0

0

0

1

0

1

0

1

1

1

0

1

1

0

0

1

1

1

1

1

1

1

1

Q1

Q2

q0

0

0

q1

0

1

q2

1

0

q3

1

1

Q

Q1

Q2

IN

Q*

Q1*

Q2*

OUT

q0

0

0

0

q1

0

1

0

q0

0

0

1

q3

1

1

0

q1

0

1

0

q0

0

0

0

q1

0

1

1

q3

1

1

0

q2

1

0

0

q1

0

1

0

q2

1

0

1

q3

1

1

0

q3

1

1

0

q1

0

1

1

q3

1

1

1

q3

1

1

1

Q

IN

Q*

OUT

q0

0

q1

0

q0

1

q3

0

q1

0

q0

0

q1

1

q3

0

q2

0

q1

0

q2

1

q3

0

q3

0

q1

1

q3

1

q3

1

Exerciseq0q1q21/00/00/10/1, 1/11/0Find the corresponding sequential circuit for the following Mealy Machine.

q0q1q21/00/00/10/1, 1/11/0

Q

IN

Q*

OUT

q0

0

q1

1

q0

1

q2

0

q1

0

q1

1

q1

1

q1

1

q2

0

q1

0

q2

1

q2

0

q0q1q21/00/00/10/1, 1/11/0Choose an unique binary code for each state.

Q

A

B

IN

Q*

A*

B*

OUT

q0

0

0

0

q1

0

1

1

q0

0

0

1

q2

1

0

0

q1

0

1

0

q1

0

1

1

q1

0

1

1

q1

0

1

1

q2

1

0

0

q1

0

1

0

q2

1

0

1

q2

1

0

0

A

B

q0

0

0

q1

0

1

q2

1

0

A* = (A B IN ) (A B IN )B* = (A B IN ) (A B IN ) OUT = (A B IN ) (A B IN ) (A B IN )

A

B

IN

A*

B*

OUT

0

0

0

0

1

1

0

0

1

1

0

0

0

1

0

0

1

1

0

1

1

0

1

1

1

0

0

0

1

0

1

0

1

1

0

0

A* = IN (B )B* = A*OUT = (A ) (IN B )

A

B

IN

A*

B*

OUT

0

0

0

0

1

1

0

0

1

1

0

0

0

1

0

0

1

1

0

1

1

0

1

1

1

0

0

0

1

0

1

0

1

1

0

0

A* = IN (B )B* = A*OUT = (A ) (IN B )A*A

PreparationReadChapters 12 of the Text BookRevisionKnow what the following machines are and how to use them.Moore MachineMealy MachineBe able to find the equations for a sequential circuit corresponding to a Mealy machine.