synthesis for finite state machines. fsm (finite state machine) optimization state tables state...
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Synthesis For Finite State Machines
FSM (Finite State Machine) Optimization
State tables
State minimization
State assignment
Combinationallogic optimization
net-list
identify and removeequivalent states
assign unique binarycode to each state
use unassigned state-codesas don’t care
FSM Optimization
S2
S1
S3
01-0
00
10-1
0-1-01
-011
11
Combinational Logic
PI PO
PS NS
u1
u2
v1
v2
S4
State Minimization
Goal : identify and remove redundant states
(states which can not be observed from the
FSM I/O behavior)
Why : 1. Reduce number of latches
– assign minimum-length encoding
– only as the logarithm of the number
of states
2. Increase the number of unassigned states
codes
– heuristic to improve state-assignment
and logic-optimization
State Minimization Definition
• Completely-specified state machine
– two states are equivalent if outputs are
identical for all input combinations
Next states are equivalent for all input
combinations
– equivalence of states is an equivalence relation which partitions the states into disjoint equivalence classes
• Incompletely specified state machines
Classical State Minimization
1. Partition states based on input output values
asserted in the state
2. Define the partitions so that all states in a partition transition into the same next-state partition (under corresponding inputs)
Example
Ex :
0 A B 0
1 A C 0
0 B D 0 (A,B,C,D,E,F,H)(G)
1 B E 0
0 C F 0 (A,B,C,E,F,H)(G)(D)
1 C A 0
0 D H 0 (A,C,E,H)(G)(D)(B,F)
1 D G 0
0 E B 0 (A,C,E)(G)(D)(B,F)(H)
1 E C 0
0 F D 0
1 F E 0
0 G F 1
1 G A 0
0 H H 0
1 H A 0
State Assignment
• Assign unique code to each state to produce logic-level description
– utilize unassigned codes effectively as don’t cares
• Choice for S state machine
– minimum-bit encoding
log S– maximum-bit encoding
• one-hot encoding
• using one bit per state
• something in between
• Modern techniques
– hypercube embedding of face constraint derived for collections of states (Kiss,Nova)
– adjacency embedding guided by weights derived between state pairs (Mustang)
Hypercube Embedding Technique
• Observation : one -hot encoding is the easiest
to decode
Am I in state 2,5,12 or 17?
binary : x4’x3’x2’x1x0’(00010) +
x4’x3’x2x1’x0 (00101) +
x4’x3x2x1’x0’(01100) +
x4x3’x2’x1’x0 (10001)
one hot : x2+x5+x12+x17
But one hot uses too many flip flops.
• Exploit this observation
1. two-level minimization after one hot
encoding identifies useful state group for
decoding
2. assigning the states in each group to a single
face of the hypercube allows a single product
term to decode the group to states.
State Group Identification
Ex: state machine
input current-state next state output
0 start S6 00
0 S2 S5 00
0 S3 S5 00
0 S4 S6 00
0 S5 start 10
0 S6 start 01
0 S7 S5 00
1 start S4 01
1 S2 S3 10
1 S3 S7 10
1 S4 S6 10
1 S5 S2 00
1 S6 S2 00
1 S7 S6 00
Symbolic Implicant : represent a transition from
one or more state to a next state under some input
condition.
Representation of Symbolic Implicant
Symbolic cover representation is related to a
multiple-valued logic.
Positional cube notation : a p multiple-valued
logic is represented as P bits
(V1,V2,...,Vp)
Ex: V = 4 for 5-value logic
(00010)
represent a set of values by one string
V = 2 or V = 4
(01010)
Minimization of Multi-valued Logic
Find a minimum multiple-valued-input cover
- espresso
Ex: A minimal multiple-valued-input cover
0 0110001 0000100 00
0 1001000 0000010 00
1 0001001 0000010 10
State Group
Consider the first symbolic implicant
0 0110001 0000100 00
• This implicant shows that input “0” maps
“state-2” or “state-3” or “state-7” into “state-5”
and assert output “00”
• This example shows the effect of symbolic logic minimization is to group together the states that are mapped by some input into the same next-state and assert the same output.
• We call it “state group” if we give encodings to
the states in the state group in adjacent binary
logic and no other states in the group face, then the states group can be implemented as a cube.
Group Face
• group face : the minimal dimension subspace containing the encoding assigned to that group.
Ex: 0 010 0**0 group face
0100
0110
Hyper-cube Embedding
c
a b
25
6
12 17
125
6
2 17
state groups :{2,5,12,17}{2,6,17}
wrong!
Hyper-cube Embedding
c
a b
45
2
5
2 4
state groups :{2, 6, 17}{2, 4, 5}
wrong!
6 17
6 17
How to Check if a State Assignment Satisfies the Constraint Matrix?
Step1:
Step2:
Find the group face of the encoding
For all states, check if a state that does not belong to a state group intersects that group face
Example
Constraint matrix A, state encoding S and group-face matrix F
A =011000110010000001001
010110101000001011100
S =
Step1: Group face F = A˙S =1 * *0 ** 0 0
0
Step2: Check encoding of state-6 = [011] Since it does not belong to group 1, 2 and 3,
Encoding of state-6 satisfies the constraint
check [0 1 1] ∩ [1 * *] =
[0 1 1] ∩ [0 * 0] = [0 1 1] ∩ [* 0 0] =
Other State Encoding
If encoding of state-6 = [111], check
Do not satisfy the constraint.
[1 1 1] ∩ [1 * *] = 111 [1 1 1] ∩ [0 * 0] = [1 1 1] ∩ [* 0 0] =
Algorithm for State Assignment
Step 1:
Step 2:
Step 3:
Step 4:
Step 5:
Select an uncoded state (or a state subset).
Determine the encodings for that state (states) satisfying the constraint relation.
If no encoding exists, increase the state code dimension and go to Step 2.
Assign an encoding to the selected state (states).
If all states have been encoded, stop. Else go to Step 1.
Step 3
– Can always increase the coding length by one bit
– New state assignment:
1. For states already assigned, append 0 at the end
2. For the new state, ns,
case1:
case2:
ns does not belong to any state group, encoding of ns = [c | 1] c is any vector
ns belongs to some state group, encoding of ns = [c | 1]c is the encoding of any state that belongs to the state group
Example
Ex: 00100111
A =010110101100
S =
To encode a new state (state-5), we have a new constraint matrix,
A’ =01011 10100 11000
000100010110
S’ =
ns = [1 0 1] or [1 1 1]
For the states already assigned, we have a new encoding,
For the new state (state-5), we have encodings
Hyper-cube Embedding Method
• Advantage :
– use two-level logic minimizer to identify good state group
– almost all of the advantage of one-hot encoding, but fewer state-bit
Adjacency-Based State Assignment
Basic algorithm:
(1) Assign weight w(s,t) to each pair of states
– weight reflects desire of placing states adjacent on the hypercube
(2) Define cost function for assignment of codes
to the states
– penalize weights for the distance between the state codes
eg. w(s,t) * distance(enc(s),enc(t))
(3) Find assignment of codes which minimize
this cost function summed over all pairs of
states.
– heuristic to find an initial solution
– pair-wise interchange (simulated annealing)
to improve solution
Adjacency-Based State Assignment
• Mustang : weight assignment technique based on loosely maximizing common cube factors
How to Assign Weight to State Pair
• Assign weights to state pairs based on ability to extract a common-cube factor if these two states are adjacent on the hyper-cube.
Fan-Out-Oriented (examine present-state pairs)
• Present state pair transition to the same next state
S1 S3
S2
$$$ S1 S2 $$$$$$$ S3 S2 $$$$
Add n to w(S1,S3) because of S2
Fan-Out-Oriented
• present states pair asserts the same output
S2
S3S1
S4
$/j $/j
Add 1 to w(S1 , S3) because of output j
$$$ S1 S2 $$$1$$$$ S3 S4 $$$1$
Fanin-Oriented (exam next state pair)
• The same present state causes transition to next state pair.
$$$ S1 S2 $$$$
$$$ S1 S4 $$$$
Add n/2 to w(S2,S4) because of S1
S1
S2S4
Fanin-Oriented (exam next state pair)
• The same input causes transition to next state pair.
$0$ S1 S2 $$$$
$0$ S3 S4 $$$$
Add 1 to w(S2,S4) because of input i
i i
S1 S3
S2 S4
Which Method Is Better?
• Which is better?
FSMs have no useful two-level
face constraints => adjacency-embedding
FSMs have many two-level
face constraints => face-embedding