1. Synthesis of sequential circuits
Steps in synthesis of a sequential circuit
Specify the FSM (state table or state diagram)
Minimize the states
State assignment/encoding
Specification: PS x NS z S1 S3 1 S5 S2 S4
0/1
0/0 S3 S1 S4
1/1
1/1
0/0
0/1
0/1
1/0
1/1 S2 S5
1/1

2. State minimization (Completely specified FSM)
Completely specified  no don’t cares
Progressively refine a set of equivalence classes, j
For the state machine on the previous slide
0 = all states in one class = {{S1, S2, S3, S4, S5}}
1: distinguished on the basis of output value
1 = {{S1,S2},{S3,S4},{S5}}
2: distinguished if NS for two states in a partition are in different partitions.
Example: For x=1, NS for S3  {S1,S2}; NS for S4 {S5}
2 = {{S1,S2},{S3}{S4},{S5}}
Continue finding j+1 given j in a similar way
Stop when j+1 = j since this implies that k = j for all k > j
In this example, 3 = 2
Final set of equivalence classes = {{S1,S2},{S3}{S4},{S5}}

3. State minimization (Completely specified FSM) – Contd.
Final set of equivalence classes = {{S1,S2},{S3}{S4},{S5}}
State table shown at right
Complexity:
Let ns = # states
At most O(ns) iterations
Each iteration is O(ns)
Total complexity is O(ns2)
A more clever implementation can
achieve O(ns log ns) PS x NS z S1 S3 1 S5 S4

4. State minimization (Incompletely specified FSM)PS x NS z S1 S3 1 S5 d S2 S4
Idea of equivalence does not hold any more
Equivalence involves
Reflexivity
(si <comp> si)
Symmetry
(si <comp> sj)  (sj <comp> si)
Transitivity
(si <comp> sj) and (sj <comp> sk)  (si <comp> sk)
For the state table here, looking at compatibility
only in terms of the output value (similar to
constructing 1 for a completely specified FSM)
s1 <comp> s2, s2 <comp> s3 but s1 is not compatible with s3
Need to work with compatibilities instead of equivalences

5. State minimization (Incompletely specified FSM) – Contd.
Example: state table shown on previous slide
List (possibly) compatible states and clearly incompatible states
List conditions under which states could be compatible
Compatible Incompatible
{S1,S2} {S1,S3}
{S1,S5}  {S3,S4} {S1,S4}
{S2,S3}  {S1,S5} {S2,S5}
{S2,S4}  {S3,S4} {S3,S5}
{S3,S4}  {S2,S4}, {S1,S5} {S4,S5}
{S1,S5} are compatible if {S3,S4} are compatible
Now use list of incompatible states to iteratively strike out states from compatible set and move them to the incompatible set
No such update possible here, although in general it may be possible
Combine now as {S2,S3,S4}  {S1,S5} and {S1,S5}  {S3,S4}: consistent
Result: combine {S2,S3,S4} into one state and {S1,S5} into another

6. State encoding Assign Boolean codes to states Two extremes
One-hot encoding: n bits for n states, only one bit set to 1
Example: S1 = 1000, S2 = 0100, S3 = 0010, S4 = 0001
Large # of state bits (and hence FF’s), simple combinational logic due to extensive don’t care space
Minimum-bit encoding: log2 n bits for n states
Example: S1 = 00, S2 = 01, S3 = 10, S4 = 11
Small # of state bits, possibly more complex combinational logic
More commonly used
Can also be in between the extremes

7. State assignment heuristics
“Fanout oriented”
Same NS from two PS under a given input should be assigned neighboring codes
Example: two inputs X1,X2 and one output Z
Rationale: state table will include
which implies that 0-01  fon(NS2), fon(NS1), fon(Z)
01/1 01 00 11
PS2
PS1 X1 X2
NS2
NS1 Z 1

8. State assignment heuristics – contd.
“Fanin oriented”
If a PS goes to two different NS’s, they should be assigned neighboring codes
Example: two inputs X1,X2 and one output Z
Rationale: state table will include
which implies that 011-  fon(NS1), fon(Z) 01
10/1
11/1 10 11
PS2
PS1 X1 X2
NS2
NS1 Z 1

9. A simple state-assignment algorithm
Considers fanin oriented case only and defines “attractions”
Construct matrices
ns x ns between states
Entry (i,j) = # of transitions from i to j
ns x noutputs between states and outputs
Entry (i,j) = # output transitions with value 1
0/0 S1
1/1
1/1
0/0
1/0
0/0 S3 S2
NS1
NS2
NS3
PS1 1
PS2
PS3 Z
PS1 1
PS2
PS3

10. State assignment algorithm
Attraction metric between states Si and Sj
= Nb PSi PSjT + zi zjT
where Nb = # state bits, PSk = row k of state matrix
zk = row k of output matrix
Attraction12 = 2[110][101]T + [1][0] = 2
Attraction23 = 2[101][101]T + [0][1] = 4
Attraction13 = 2[110][101]T + [1][1] = 3
Build attraction graph
Start with node with max sum of edge attractions
and assign it an encoding (here, S3 is arbitrarily assigned 00)
Assign encodings in order of attraction to neighbors
to reduce Hamming distance (here, S2 = 01, S1 = 10)
Repeat until all states are assigned codes (here, we are done) 10 S1 3 2 4 S3 S2 00 01