21 Races and State Assignment

7/28/2019 21 Races and State Assignment

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ECE124 Digital Circuits and Systems Page 1

Race Conditions; Critical and Non-Critical Races

A race condition occurs in an asynchronous circuit when 2 or more state variables

change in response to a change in the value of a circuit input.

Unequal circuit delays may imply that the 2 or more state variables may not changesimultaneously this can cause a problem.

Assume two state variables change

If the circuit reaches the same final, stable state regardless of the order in whichthe state variables change, then the race in non-critical.

If the circuit reaches a different final, stable state depending on the order inwhich the state variables change, then the race is critical.

We need to avoid critical races for predictability and to ensure our circuit does theintended function!


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Non-Critical Race

Assume current state is 00, and input changes from 0->1. This requires y1y2 to change

from 00->11.

Depending on circuit delays, several possible transition sequences:

00 -> 11 -> 01 (simultaneous change for y1 and y2).

00 -> 01 (y2 changes first).

00 -> 10 -> 11 -> 01 (y1 changes first)

In all cases, we end up in the same stable state, so the race is non-critical.

00

0 1

00

01

y1y2

curr

state

11

x

11

10

00

01

00

00

01

11


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Critical Race


from 00->11.

Possible transition sequences:

00->11 (y1 and y2 change simultaneously)

00->01->11 (y2 changes first)

00->10 (y1 changes first)

So, depending on which state variable changes first, we can get into a differentstable

statethe race is critical.

00

0 1

00

01

y1y2

curr

state

11

x

11

10

00

11

00

00

11

10


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Why Races Exist

Races are a consequence of how states are assigned when designing a circuit.

That is, races exist in the transition tables, but not in the flow tables.

This should be somewhat clear.

Races are a consequence of state variables changing at the same time andtherefore cant happen until binary state assignment is done.

Flow tables are entirely symbolic, so no problems there

Transition tables have binary values assigned, so potential problems there


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Stability and Cycles

We also need to be careful that our circuit does not become unstable.


from 00->01.

Transitions will be 00->01->11->10->01-> So, our circuit oscillates. If this was not ourintention, we are in trouble.

Should have a stable state for each input condition and make sure we get to a stable

state always.

00

0 1

0001

y1y2

curr

state

01

x

11

10

0011

00

00

10

01


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Race Free State Assignment

We can prevent races (pre-emptive) by performing state assignment such that

transitions from one stable state to another stable state only require one state

variable to change at a time.

Can consider a few different methods for performing race-free state assignment.


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Example

Consider an example flow table

curr

state

next state

x=0 x=1

ab

c

d

ac

c

a

bb

d

d


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Example Transition Diagram

Build a transition diagram that shows transitions from one stable state to another

stable state under a changing input.

Each edgeshows a possible transition. Each node represents a state.

curr

state

next state

x=0 x=1

a

b

c

d

a

c

c

a

b

b

d

d

c

a b

d


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Example State Assignment (A Bad Choice)

Consider assigning binary patterns to states based on the co-ordinates of the state

in the transition diagram:

c

a b

d

00 01

10 11

This is a BAD assignment, since a transition from b->c or d->a requires two state

variables to change simultaneously.


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Example State Assignment (A Good Choice)

Consider shifting states to different co-ordinates:

c

a b

d

This is a GOOD assignment, since a transition from b->c or d->a requires only one state

variables to change.

d

a b

c

00 01

10 11


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Method One for Race Free State Assignment

Given a flow table, try and embed the symbolic states into the co-ordinates of a n-

dimensional cube such that the path from stable state to stable state:

Is direct along a single edge of the cube, or

Goes through newly introduced unstable states along edges of the cube.


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Example

Consider the following flow table:

curr

state

next state

x1x2=00 x1x2=01

ab

c

d

aa

c

c

ab

b

a

x1x2=10 x1x2=11

cd

c

d

bb

d

d


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We can attempt to embed 4 states into a 2-dimensional cube and draw the transition

diagram:

curr

state

next state

x1x2=00 x1x2=01

a

bc

d

a

ac

c

a

bb

a

x1x2=10 x1x2=11

c

dc

d

b

bd

d

If we try, we see that there is no way to re-arrange the states to the corners of thecube to get rid of the diagonals in the transition diagram

No state assignment that has only one state variable changing at a time.

c

a b

d


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Solution is to introduce additional, unstable states and use them as intermediate

states during transitions.

I.e., embed the 4-states into the corners of a 3-dimensional cube.

c

a b

d

a b

d

c

e

g

f


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All transitions are made properly by introducing extra unstable states (Note: the

new extra states are always unstable!!!):

a b

d

c

e

g

f

000 001

010

100

110

011

101


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Example Race Free Flow Table

Can now see the original flow table, and an expanded flow table (extra unstable

states) that has a race-free state assignment (see previous page!):

curr

state

next state

x1x2=00 x1x2=01

a

b

c

d

a

a

c

c

a

b

b

a

x1x2=10 x1x2=11

c

d

c

d

b

b

d

d

curr

state

next state

x1x2=00 x1x2=01

a

b

c

d

a

a

c

g

a

b

b

a

x1x2=10 x1x2=11

c

e

c

d

b

b

g

d

e - - d -

f - b - -

g c - - d


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Method Two for Race Free State Assignment

Method useful for flow tables with


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Example

Consider the following flow table, and another (larger table) with equivalent states:

curr

state

next state

x1x2=00 x1x2=01

a

b

c

d

a

a

c

c

a

b

b

a

x1x2=10 x1x2=11

c

d

c

d

b

b

d

d

output

0

1

0

1

curr

state

next state

x1x2=00 x1x2=01 x1x2=10 x1x2=11output

d1 c1 d2 d1 d1 1

d2 c2 d1 d2 d2 1

c1 c1 c2 c1 d1 0

c2 c2 b1 c2 d2 0

b1 a1 b1 b2 b1 1

b2 a2 b2 d2 b2 1

a1 a1 a1 c1 b1 0

a2 a2 a2 a1 b2 0

The flow table with equivalent states permits a race free state assignment.

We can see this by looking at the states on a 3-dimensional cube.


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Example

curr

state

next state

x1x2=00 x1x2=01 x1x2=10 x1x2=11output

d1 c1 d2 d1 d1 1

d2 c2 d1 d2 d2 1

c1 c1 c2 c1 d1 0

c2 c2 b1 c2 d2 0

b1 a1 b1 b2 b1 1

b2 a2 b2 d2 b2 1

a1 a1 a1 c1 b1 0

a2 a2 a2 a1 b2 0

With equivalent states, we always have 1 of the 2 equivalent states directly adjacent

to every other state!

a1 a2

b1

c1

b2

d1

c2 d2


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Method Three for Race Free State Assignment

Can use the idea of one-hot encoding to get a race free state assignment

Given n-states, let the i-th state be encoded as 0010..0 where the 1 is in the i-th

location.

For a transition from i-th stable state to j-th stable state, introduce unstable state with

encoding 0..010..010..0 where the 1s are in the i-th and j-th position.

Again, best illustrated with an example


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Example

Consider the following flow table, with one-hot state assignment:

curr

state

next state

x1x2=00 x1x2=01

a

b

c

d

a

a

c

c

a

b

b

a

x1x2=10 x1x2=11

c

d

c

d

b

b

d

d

0001

0010

0100

1000

Look for transitions and introduce new unstable states, as required using one-hot

encoding scheme.


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Example

currstate

next state

x1x2=00 x1x2=01

a

b

c

d

a

a

c

c

a

b

b

a

x1x2=10 x1x2=11

c

d

c

d

b

b

d

d

0001

0010

0100

1000

curr

state

next state

x1x2=00 x1x2=01

a

b

c

d

a

f

c

c

a

b

h

j

x1x2=10 x1x2=11

e

g

c

d

f

b

i

d

0001

0010

0100

1000e - - c -0101

f a - - b0011

g - - d -1010

h - b - -0110

i c - - d1100

i - a - -1001

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21 Races and State Assignment

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