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Theory of Automata

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Theory of Automata
(CS402)
Theory of Automata
Lecture N0. 23
Reading Material
Introduction to Computer Theory
Chapter 8
Summary
Mealy machines in terms of sequential circuit
Example
Consider the following sequential circuit
A
output
B
input
DELAY
OR
NAND
AND
The following four types of boxes are used in this circuit
NAND box (NOT AND): For the given input, it provides the complement of Boolean AND output.
DELAY box (Flip Flop box): It delays the transmission of signal along the wire by one step (clock pulse).
OR box: For the given input, it provides the Boolean OR output.
AND box: For the given input, it provides the Boolean AND output.
The current in the wire is indicated by 1 and 0 indicates the absence of the current.
There are two points A and B w.r.t. to which following four states of the machine are identified according to the
presence and absence of current at these points i.e.
q0(A=0, B=0) (0,0)
q1 (A=0, B=1) (0,1)
q2 (A=1, B=0) (1,0)
q3 (A=1, B=1) (1,1)
The operation of the circuit is such that the machine changes its state after reading 0 or 1. The
transitions are determined using the following relations
new B = old A
new A = (input) NAND (old A AND old B)
output = (input) OR (old B)
It is to be noted that old A and old B indicate the presence or absence of current at A and B before inputting any
letter. Similarly new A and new B indicate the presence or absence of current after reading certain letter.
At various discrete pulses of a time clock, input is received by the machine and the corresponding output string
is generated.
The transition at the state q0 after reading the letter 0, can be determined, along with the corresponding output
character as under
new B = old A = 0
new A = (input) NAND (old A AND old B)
= 0 NAND ( 0 AND 0) = 0 NAND 0
=1
output = (input) OR (old B) = 0 OR 0 = 0
Thus after reading 0 at q0 new B is 0 and new A is 1 i.e. machine will be at state (1,0) q2 and during this
process it's output character will be 0.
The remaining action of this sequential circuit can be determined as shown by the following suggested transition
table of the corresponding Mealy machine
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Theory of Automata
(CS402)
Inputting 0
Inputting 1
Old state
State
Output
State
Output
q0 (0,0)
(1,0)q2
(1,0)q2
0
1
q1 (0,1)
(1,0)q2
(1,0)q2
1
1
q2 (1,0)
(1,1)q3
(1,1)q3
0
1
q3 (1,1)
(0,1)q1
(1,1)q3
1
1
The corresponding transition diagram may be as follows
q2
0/0, 1/1
0/0, 1/1
0/1
0/1 1/1
q3
q0
1/1
q1
Note: It may be noted that if the string 00 is read at any state, it results in ending in state q3.
Running the string 01101110 on the previous machine, the output string can be determined by the following
table
Input
0
1
1
0
1
1
1
0
States
q0
q2
q3
q1
q2
q3
q1
q2
q3
output
0
1
1
1
1
1
1
0
Following is a note regarding the sequential circuit under consideration
Note
input
output
A
B
DELAY
OR
NAND
AND
It is to be noted that in this sequential circuit, delay box plays an important role in introducing four states of the
machine.
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