MOORE & MEALY MOORE & MEALY MACHINEMACHINE

LIMITATION OF LIMITATION OF FINITE AUTOMATA MACHINESFINITE AUTOMATA MACHINES

One limitation of the FINITE AUTOMATA is One limitation of the FINITE AUTOMATA is that output is limited to a binary signal that output is limited to a binary signal TRUE(1) or FALSE(0) depending on the TRUE(1) or FALSE(0) depending on the basis of reachability of the final state by the basis of reachability of the final state by the initial state. The only explicit task a machine initial state. The only explicit task a machine has done so far is to recognize a language has done so far is to recognize a language whereas computers can perform calculations whereas computers can perform calculations and convey the result i.e. provide OUTPUT. and convey the result i.e. provide OUTPUT.

FINITE AUTOMATA FINITE AUTOMATA WITH OUTPUTWITH OUTPUT

Moore and Mealy machines remove this Moore and Mealy machines remove this limitation and provide output. They are limitation and provide output. They are FINITE AUTOMATA MACHINE with FINITE AUTOMATA MACHINE with OUTPUT.OUTPUT.

For example We simply want to print out a For example We simply want to print out a copy of the input string.copy of the input string.

DIFFERENCE BETWEENDIFFERENCE BETWEENMEALY AND MOOREMEALY AND MOORE

MACHINESMACHINES In Mealy machine output function Z(t) depends on both In Mealy machine output function Z(t) depends on both

the present state q(t) and the present input x(t). the present state q(t) and the present input x(t).

The expression for Mealy machine is The expression for Mealy machine is Z(t) = Z(t) = λλ[q(t),x(t)] [q(t),x(t)] λλ = output function. = output function. In Moore machine output function Z(t) depends only on In Moore machine output function Z(t) depends only on

the present state and is independent of the current the present state and is independent of the current input.input.

The expression for Moore machine is The expression for Moore machine is Z(t) = Z(t) = λλ[q(t)].[q(t)].

MOORE MACHINEMOORE MACHINE Developed by E.F. Moore in 1956.Developed by E.F. Moore in 1956. It is a finite automata machine with outputIt is a finite automata machine with output The output is associated with each state. The output is associated with each state.

Every state of this machine has a fixed output.Every state of this machine has a fixed output. There is no concept of final state in Moore There is no concept of final state in Moore

machine.machine. It can be represented by Transition table as It can be represented by Transition table as

well as Transition diagram.well as Transition diagram. It is a six tuple machine.It is a six tuple machine.

MATHEMATICAL MATHEMATICAL REPRESENTATIONREPRESENTATION

M = (Q , M = (Q , ΣΣ , , λλ , , ΔΔ , q , qoo , , δδ)) Q = A nonempty finite set of state in M.Q = A nonempty finite set of state in M. ΣΣ = A nonempty finite set of input symbols. = A nonempty finite set of input symbols. ΔΔ = A nonempty finite set of outputs. = A nonempty finite set of outputs. δδ = It is a transition function which takes two = It is a transition function which takes two

arguments input state and input symbol.arguments input state and input symbol. qqo o = = Initial state of M belongs to Q.Initial state of M belongs to Q. λλ = It is a mapping function which maps Q to = It is a mapping function which maps Q to ΔΔ

giving output associated with each state.giving output associated with each state.

REPRESENTATION OFREPRESENTATION OFMOORE MACHINEMOORE MACHINE

Let M be a Moore machine and a1 , a2 , a3 , Let M be a Moore machine and a1 , a2 , a3 , …….a…….ann be input symbols where n>0 then be input symbols where n>0 then output of M is output of M is λλ(q(q11) , ) , λλ(q(q22), ), λλ(q(q33) , ) , λλ(q(q44) ) ………… ………… λλ(q(qnn) , such that ) , such that

δδ(q(qi – 1 , i – 1 , aai i ) = q) = qi i for 1<i<n. for 1<i<n.

EXAMPLE OF EXAMPLE OF MOORE MACHINEMOORE MACHINE

qq00 qq33 qq11 00

qq11 qq11 qq22 11

qq22 qq22 qq33 00

qq33 qq33 qq00 00

Present Present statestate

inputinputa=0 a=1a=0 a=1

outputoutputTRANSITION TABLE

VALUES OF VALUES OF Q, Q, ΣΣ, , ΔΔ

Q = {Q = {qq0 0 ,, qq1 1 ,, qq2 2 ,, qq3 3 }} ΣΣ = { 0 , 1 } = { 0 , 1 } ΔΔ = { 0 , 1 } = { 0 , 1 } λλ(q(q00) = 0 , ) = 0 , λλ(q(q11) = 1 , ) = 1 , λλ(q(q22) = 0 , ) = 0 , λλ(q(q33) = 0 ) = 0

TRANSITION DIAGRAM

q0

q2 q3

q1

0

0

11

1

00 0

1

01

0

0

PROCESSING OF STRINGPROCESSING OF STRINGTHROUGHTHROUGH

MOORE MACHINEMOORE MACHINE Let the input string be w = 0111 and Output Let the input string be w = 0111 and Output

string will be string will be єєw’ = 00010.w’ = 00010.

q2q01110 01000

q3 q1q0

|w| = 4|w’| = 5

Since for the input of null string the output is 0

єє

A Moore machine that counts the A Moore machine that counts the occurrence of substring “aab” in occurrence of substring “aab” in

the input strungthe input strung Let the input string be w = “aaabbbaabaa”Let the input string be w = “aaabbbaabaa” M = (Q , M = (Q , ΣΣ , , λλ , , ΔΔ , q , qoo , , δδ)) Q = {Q = { qq00,, qq1 1 ,, qq2 2 ,, qq3 3 }} ΣΣ = { a , b } = { a , b } ΔΔ = { 0 , 1 } = { 0 , 1 } λλ(q(q0 0 ) = 0 , ) = 0 , λλ(q(q1 1 ) = 0 , ) = 0 , λλ(q(q2 2 ) = 0 , ) = 0 , λλ(q(q33) = 1 ) = 1

q0

q1

q2

q3

ba

bb

a

a

ab

TRANSITION DIAGRAM

1

00

0

a a a b b b a a b a a

q3q0baaa 10000

q1 q2q2

q3q0baab 10000

qo q2q1

q1

b

0q20

a

a

0 0 0 0 1 0 0 0 0 1 0 0

RESULTRESULT The occurrence of 1’s in the output string is The occurrence of 1’s in the output string is

2 so the number of times “aab” appears in 2 so the number of times “aab” appears in the input string is 2.the input string is 2.

MEALY MACHINEMEALY MACHINE Independently developed by G.H. Mealy in Independently developed by G.H. Mealy in

19551955 Output is associated with each transition.Output is associated with each transition. Output is fixed for a particular input symbol.Output is fixed for a particular input symbol. It can be represented by Transition table as It can be represented by Transition table as

well as Transition diagram.well as Transition diagram. It is a six tuple machine.It is a six tuple machine.

MATHEMATICAL MATHEMATICAL REPRESENTATIONREPRESENTATION

M = (Q , M = (Q , ΣΣ , , λλ , , ΔΔ , q , qoo , , δδ)) Q = A nonempty finite set of state in M.Q = A nonempty finite set of state in M. ΣΣ = A nonempty finite set of input symbols. = A nonempty finite set of input symbols. ΔΔ = A nonempty finite set of outputs. = A nonempty finite set of outputs. δδ = It is a transition function which takes two = It is a transition function which takes two

arguments input state and input symbol.arguments input state and input symbol. qqo o = = Initial state of M. Initial state of M. λλ = It is a mapping function which maps Q * = It is a mapping function which maps Q * ΣΣ to to ΔΔ

giving output associated with each transition.giving output associated with each transition.

REPRESENTATION OFREPRESENTATION OFMEALY MACHINEMEALY MACHINE

Let M be a Mealy machine and a1 , a2 , a3 , Let M be a Mealy machine and a1 , a2 , a3 , …….a…….ann be input symbols where n>0 then be input symbols where n>0 then output of M is output of M is λλ(q(q0 0 , , aa11) , ) , λλ(q(q1 1 , , aa22) ) λλ(q(q2 2 , , aa33) ) λλ(q(q3 3 , , aa44) ) λλ(q(q4 4 , , aa55)………… )………… λλ(q(qn - 1 n - 1 , , aann), such ), such that that

δδ(q(qi – 1 , i – 1 , aai i ) = q) = qi i for 1<i<n. for 1<i<n.

EXAMPLE OFEXAMPLE OFMEALY MACHINEMEALY MACHINE

qq11 qq3 3 00 qq22 00qq22 qq1 1 11 qq44 00qq33 qq2 2 11 qq11 11

qq44 qq4 4 11 qq33 00

Present Present statestate

Input a = 0 Input a = 0 State outputState output

Input a = 1Input a = 1State State outputoutput

TRANSITION TABLE

Values of Values of Q , Q , ΣΣ, , ΔΔ

Q = {Q = { qq1 1 ,, qq2 2 ,, qq3 3 ,, qq4 4 }} ΣΣ = { 0 , 1 } = { 0 , 1 } ΔΔ = { 0 , 1 } = { 0 , 1 } λλ(q(q1 1 , 0, 0) = 0 , ) = 0 , λλ(q(q2 2 , 0, 0) = 1 , ) = 1 , λλ(q(q3 3 , 0, 0) = 1 , ) = 1 ,

λλ(q(q4 4 , 0, 0) = 1) = 1 λλ(q(q1 1 , 1, 1) = 0 , ) = 0 , λλ(q(q2 2 , 1, 1) = 0 , ) = 0 , λλ(q(q3 3 , 1, 1) = 1 , ) = 1 ,

λλ(q(q4 4 , 1, 1) = 0) = 0

q1

q3 q4

q2

TRANSITION DIAGRAM

1/0

0/1

0/1

1/0

1/1 0/0

1/0

0/1

PROCESSING OF STRINGPROCESSING OF STRINGTHROUGHTHROUGH

MEALY MACHINEMEALY MACHINE Let the input string be w = 0111 and Output Let the input string be w = 0111 and Output

string will be w’ = 0100.string will be w’ = 0100.

q4q11/01/01/10/0 q3 q2q1

|w| = 4|w’| = 4

TRANSFORMING A MOORE MACHINE INTOTRANSFORMING A MOORE MACHINE INTOA MEALY MACHINEA MEALY MACHINE

q1

q1

P

c

b d

a

d

c/P

b/P

a/P

MOOREMACHINE

MEALYMACHINE

EXAMPLE

q0 q3

q2

q1a

b

b

a0 1

0

1

b

a

a/b

MOORE MACHINE

CONVERT INTO MEALY MACHINE

q0 q3

q2

q1a/1

b/0

b/0

a/1

b/1

a/0

a/1 , b/1

MEALY MACHINE

TRANSFORMING A MEALY MACHINE INTOTRANSFORMING A MEALY MACHINE INTOA MOORE MACHINEA MOORE MACHINE

q1

q11 q12

b/0

a/0

b/1 b/1a/1

10

ba

ba/1 a/1

b/1 b/1

MOORE MACHINE

MEALY MACHINE

EXAMPLE

q0

q1

q3

q2

a/0 a/1

a/0b/1b/0

a/1 b/0

b/1

MEALY MACHINE

q02q3

q22

q21

q1

q01

aa

ab

b

ab a

b

1b a

0

a

1

1

0

0

b

MOORE MACHINE