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Advanced Advanced
Methods based on triangular Methods based on triangular table and table and binate binate covering covering
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1
0
13 4 2
14
6 60 1
21
3
0
2
0
6 0 5 11 0
1
3
Ax
X4X3X2X1123
56
4
ig 5.17bcd
Example 1. Minimize the following Mealy Finite State MachineInput state orcombination of input signals
Internalstate orcombinationof memorysignals
Output states orsignalcombinations
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2
2
3
4
5
6
3, 6 4, 6
2, 4
1, 23, 4 3, 5
2 3 4 51
b)
c)
46
3
56 4
1
There are otherThere are other solutionssolutions
Can I take {1,2,3,5} and {4,6}??
1
0
13 4 2
1
46 6
0 1
2
130
20
6 0 5 11
0
13
Ax
X4X3X2X11
23
56
4
Solution : {1,2}, {3,5}, {4,6}Solution : {1,2}, {3,5}, {4,6}
No, because {1,2,3,5} implies states{3,6} to be in one group.
Solution is to split {1,2,3,5} to {1,2}and {3,5}
{1,2} implies nothing, {3,5}{1,2} implies nothing, {3,5}implies nothing, {4,6} implies {1,2}implies nothing, {4,6} implies {1,2}and {3,5}and {3,5}
I can take all
max cliques butsolution will benot minimal
But how I know to split this way? Heuristics!
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Column Non-cancelled rows Groups of compatible sta
4
3
2
1
{6}
{5,6}
{3,4,5}
{2,3,5}
{4,6}
{3,5} {3,6}
{2,3,5} {2,4}
{1,2,3,5}
= { {4,6} , {1,2,3,5} , {3,6} , {2,4} }
a)
Fig5.17a
Systematic Method of Creating Maximum Compatible GroupsSystematic Method of Creating Maximum Compatible Groups
This method is systematic and createsThis method is systematic and createsall maximum compatible groupsall maximum compatible groups(cliques)(cliques)
In any case creating Maximum Compatible Groups is useful!In any case creating Maximum Compatible Groups is useful!
For small FSMs
you can findthem by visualinspection
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Complete and ClosedComplete and Closed SubgraphSubgraph
CompleteComplete = all state numbershave been used at least one inside
it ClosedClosed = there is no arrow going
out of this graph
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{1,2}
{3,5}
{4,6} {2,3} {2,5}
{1,5}{3,6}
{2,4}{1,3}This way wefound other
solutions.Please drawmachines
Closure graphClosure graph for compatibility pairs
This method selects subsets of maximum cliques in order toThis method selects subsets of maximum cliques in order to
satisfy the completeness and closure conditions for statesatisfy the completeness and closure conditions for statenumbersnumbers
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This is a final stage of state table minimizationThis is a final stage of state table minimizationIt can be done with:
1) ALL groups of compatible states or
2) with the set of closed and closed groups of compatible states
Combining groups of compatibles from the cover to single state
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b)
1
0
13 4 2
14
6 60 1
21
6 60 1
30
20
13
04
60
51
10
60 10 513
02
0
13
Ax
Ax
X4X3X2X1123
53
26
4
46
A = {1,2,3,5}B = {3,6}C = {2,4}
D = {4,6}
{4,6}0
{3,6}1 1 1
{2,4} 2
{3,6}0 01 1
0 0 0
0 0 0
1
1
6
6
6
2
1
3
5
{3,5}{1,2}3
4
X1 X2 X3 X4c)
d) Ax
AB
CD
D0
B 1 1 1C A,C
B0 01 1
0 0 0
0 0 0
1
1
B,D
B,D
B,D
A,C
A
A,B
A
AAA,B
C,D
X1 X2 X3 X4This is non-deterministic FSM,you can make achoice to simplifystate assignment
Combining ALL groups of compatibles from the cover to single state
Now let us go back to fast method, remember that it is not optimal
b f bl f h l
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x
A
B
CD
D0
B1 1 1
C A,C
B0 01 1
0 0 0
0 0 0
1
1
B,D
B,D
B,D
A,C
A
A,B
A
AAA,B
C,D
X1 X2 X3 X4
Select statesfrom groups tocreate largegroups of thesame state
Combining ALL groups of compatibles from the cover to single state
x
A
B
CD
D0
B1 1 1
C A,C
B0 01 1
0 0 0
0 0 0
1
1
B,D
B,D
B,D
A,C
A
A,B
A
AAA,B
C,D
X1 X2 X3 X4
Select Bin wholecolumn
Select Ain wholecolumn
Select C
As you see, it is a good idea to combine FSM minimizationand state assignment. Many methods are based on this idea.
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Creating new table by combiningCreating new table by combining
states from groups of compatiblestates from groups of compatiblestatesstates
The same method of combining states can beapplied to any set of compatible and closed
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23
4
5
6
1 2 3 4 5
3,42,5
3,42,6
3,42,6
3,42,5
1,2
Fig.5.18
Problem for possible homework: Find an FSM table forwhich the following triangular table exists:
E l 3 f FSM Mi i i ti 1 4/ 6/0
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3,46,8
6,8
3,4
1,6v
4,5
3,4
6,8
4,51,8 3,5
1,66,8
1,8
1,6
v v
2,3
23
4567
81 2 3 4 5 6 7
b)
c)
8 1
2
3
45
6
7
d)
{5,6} {1,6}
{4,5}
{2,7} {3} {8}
{1,5}
{5,7}
Fig.5.21.bcd
Example 3 of FSM Minimization 12345678
4/- 6/03/1 8/04/0 8/03/0 1/0-/- 6/05/0 1/03/1 -/-2/1 1/1
1234
8
4/0 1/03/1 8/04/0 8/0
3/0 1/0
2/1 1/1
{11,6}
{22,7}{33}{44,5}
{88}