fsm minimization

7/30/2019 FSM minimization

1/13

Advanced Advanced

Methods based on triangular Methods based on triangular table and table and binate binate covering covering


2/13

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


3/13


4/13

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!


5/13

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


6/13

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


7/13

{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


8/13

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


9/13

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


10/13

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.


11/13

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


12/13

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


13/13

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}

fsm minimization

Documents