Symbolic Hazard�Free Minimization

and Encoding of

Asynchronous Finite State Machines

Robert Fuhrer

Columbia University

Bill Lin

IMEC Laboratory

Steven Nowick

Columbia University

�

Contribution

State assignment method yielding hazard�free� low�cost ��level

implementations for asynchronous state machines�

�

Outline

� Introduction

� Background

� Optimal Encoding for Asynchronous FSM�s

� Results

� Conclusions�Future Work

�

Introduction

Optimal State Encoding� Find encoding yielding optimal

implementation of given FSM� according to cost metric�

Well�studied� DeMicheli ��� Devadas ���� Saldanha ���

Dimensions�

� Area vs� speed vs� power

� ��level vs� multi�level logic

� Exact vs� heuristic techniques

� input encoding vs� output encoding � � �

Our focus� of product terms in ��level logic�

exact � heuristic techniques based on input encoding�

Requirements� hazard�free� critical race�free implementation�

�

Introduction �cont��

Optimal State Encoding for Asynchronous FSM�s

Several heuristic encoding techniques exist�

Tracey ���� Tan ���� Saucier ���� Fisher ���� Lam ���

None provides systematic optimal state assignment�

Our research contribution�

� First such work for asynchronous FSM�s

� Deal directly with MIC hazards

� Upper bound on overall logic

� Exactly optimal result for output logic

� Leverage o� of existing synchronous work �KISS

� Use input encoding formulation

�

Asynchronous Finite State Machines

Potential Bene�ts�

� Performance �utilize di�erence between avg � worst�case�

� Avoidance of clock skew

� Low power

Recent successes�

� HP Stetson �Marshall et al� ����

� Low�power infrared communications chip

� Nowick et al� ���

� Cache controller

� Yun et al� ��

� AMD SCSI controller

Recent work� �Nowick ��� �Yun ��� �Davis ��

�

Background� Asynchronous FSM�s

Multiple Input Change �MIC� Machines� Burst�Mode

A

B

C

D

E

F

c+ / z−

a+ c− / z+

c+ / y+

a− / y−

b− c+ / z+

a+ b+ / y+ z−

c− / −−

Initially: abc = 000 yz = 01

� Only speci�ed transitions

� Inputs arrive in any order

Huffman Machine

Inputs Outputs

State

C L

� Generalized fundamental�

mode of operation�

� Hazard�free logic�

�

Background� Asynchronous FSM�s �cont��

Two basic issues confronting asynchronous FSM synthesis�

�� Critical Races

Can cause FSM to settle in wrong state �Tracey ���

_ , _

_ , _

_ , _

11

S1,00

S1,00

S2,10

S4,11

S4,11 S4,00

S3,10

S3,10

S4,00

1000Inputs = a b

S2,10

S2,10

S3 0110

S1

S2

0000

1101

S4,01 0101

_ , _

S4

01

0101

Cause� interference in given input column between�

I� Unstable transition and stable state �fs�� s�� s�g�

�

Background� Asynchronous FSM�s �cont��

�� Critical Races �cont��

_ , _

_ , _

_ , _

11

S1,00

S1,00

S2,10

S4,11

S4,11

1000Inputs = a b

S1 0000_ , _

01

S2,10

S2,10

S4,01

S4,00

S4,00 0101S4

S2

S3

1101

0110

S3,10

S3,100100

Cause� interference in given input column between�

I� Unstable transition and stable state �fs�� s�� s�g�

II� Unstable transition and unstable transition �fs�� s�� s�� s�g�

Solution� judicious state encoding��

Background� Asynchronous FSM�s �cont��

�� Combinational Hazards

Type I� MIC Static HazardsB

A

D1

1

A’

CD0100 1011

00

0

0

0

0 00

01

00

10

11AB

1 1

11

1 11 1

Hazardous�

Background� Asynchronous FSM�s �cont��

�� Combinational Hazards

Type I� MIC Static Hazards

01AB

10

11

00

0000

0

0 0

0

1101CD

1000

1 1

11

1 1 11

A’D

B

A

1

1

B11 1

D

1

required cube Hazard-free

Solution� Some product term must cover each required cube�

��

Background� Asynchronous FSM�s �cont��

�� Combinational Hazards �cont��

Type II� MIC Dynamic Hazards

C

A

C

B

BC00 01 11 10

0

1 0

1

10

0A

10

1

A’

1

B

Hazardous

��

Background� Asynchronous FSM�s �cont��

�� Combinational Hazards �cont��

Type II� MIC Dynamic Hazards

C

A

C

B

BC00 01 11 10

0

1 0

1

10

0A

0

1

11

A’BC’

0 0

privileged cube

Hazard-free

Solution� Implicants must not illegally intersect the privileged

cube of any dynamic transition�

Use only dynamic�hazard�free �DHF� prime implicants�

��

Background� Asynchronous FSM�s �cont��

Exact Hazard�Free ��Level Logic Minimization �Nowick ��

Given� incompletely speci�ed Boolean function �

set of specied input transitions�

�� Generate required cubes

�� Generate DHF prime implicants

�� Solve unate covering problem

Covered objects� required cubes

Covering objects� DHF prime implicants

Limitations�

� BVI only

� Single�output only

��

Background� KISS Optimal State Encoding

Step �� Symbolic Logic Minimization

Goal� Find optimal symbolic cover�

input I

� �

s� s�� � s�� �

s� s�� � s�� �

s� s�� � s�� �

Minimal Symbolic Cover

input present next output

p�� � s�� s� s� �

p�� � s� s� �

p�� � s�� s�� s� s� �

Key Idea� perform symbolic minimization using multi�valued

input �mvi� minimization techniques �Sasao ���

Caveat� Only an approximation for next�state logic�

��

Background� KISS Optimal State Encoding �cont��

Can instantiate symbolic cover with an encoding�

Problem� Instantiated cover incorrect for certain encodings�

Solution� Step �� Encoding Constraints

Constraints allow direct instantiation of minimal symbolic cover�

Derive face embedding constraints� set of N � � dichotomies�

Minimal Symbolic Cover

input present next output

p�� � s�� s� s� �

p�� � s� s� �

p�� � s�� s�� s� s� �

Example� product p� yields seed dichotomy fs�� s�� s�g�

��

Background� KISS Optimal State Encoding �cont��

Step �� Solve Constraints

Solution always exists�

Various exact � heuristic solution methods�

kiss �DeMicheli �� dichot �Saldanha ���

nova �Villa ��� duet �Cieselski ��

Key Result�

Exact constraint solution yields minimum cardinality

output cover �if outputs minimized separately��

However� Approximate solution for next�state logic�

��

Optimal Encoding for Asynchronous FSM�s

Step �� Symbolic Hazard�Free ��Level Logic

Minimization

Unlike KISS� need hazard�free symbolic cover�

Generalize ��level hazard�free bvi algorithm to mvi functions�

� De�ned mvi multiple�input transitions

� De�ned static and dynamic output transitions

� Extended notion of privileged cubes� illegal intersections� etc�

� Generalized hazard�free conditions for mvi functions�

Extended previous algorithm to multiple�output minimization�

��

Step �� Hazard�Free Symbolic Logic Minimization

S1

S2

S3

,1

,1

- , - - , -

,1

,1

Illegal Intersection:

S1,1 S1,0

prime implicant

S1

S2

S3

,1 ,1 S1,0S1,1

- , -

,1 ,1

- , -

No Illegal Intersection:

DHF prime implicant

��

Optimal Encoding for Asynchronous FSM�s �cont��

Step �� Encoding Constraints

KISS face embedding constraints inadequate�

Asynchronous FSM de�nes OFF�set during state transitions�

Inputs = a b

0101

0110

1101

0000S1

S2

S3

S4 S1,00

S1,00

S2,10

S2,10

S4,01 S4,11

S4,11

S2,10 S3,10

S3,10

10110100

_ , _

_ , _ _ , _

_ , _ S4,00

S4,00

Goal� Prevent instantiated implicants from hitting OFF�set�

�

Optimal Encoding for Asynchronous FSM�s �cont��

Step �� Encoding Constraints �cont��

KISS face embedding constraints inadequate�

Asynchronous FSM de�nes OFF�set during state transitions�

00Inputs = a b01 11 10

, -, 1, 0

, 0

, 1

, 0 , 1

S4, 0

S4, 0

, -

, - , -

, 1

, 1

, 1

, 1

, 1

0000

0101

0100

S4

S1

S2 1101

0110S3

Goal� Prevent instantiated implicants from hitting OFF�set�

Example� column �� � dichotomy fs�� s�� s�� s�g�

��

Optimal Encoding for Asynchronous FSM�s �cont��

Step �� Encoding Constraints �cont��

Observation�

� Critical race�free dichotomies� �� �� �� �

� Face embedding dichotomies� N � �

Asynchronous optimality dichotomies� N � �� N � ��

Subsume both KISS optimality and critical race�free constraints�

��

Optimal Encoding for Asynchronous FSM�s �cont��

Step �� Constraint Solution

Exact solution� used dichot �Saldanha ���

Observation� More constraints than in synchronous case�

Increased code length � increased logic complexity�

Heuristic solution� Fixed code length via simulated annealing�

Idea� Satisfy maximum of constraints possible�

Successfully used in synchronous methods �e�g� Villa� Lin��

��

Optimal Encoding for Asynchronous FSM�s �cont��

Step �� Constraint Solution �cont��

Asynchronous Correctness Requirement�

Critical race�free constraints must be satis�ed�

Solution� Partition constraints into � classes�

�� compulsory constraints �for correctness�

�� optional constraints �for optimality�

Assign su�ciently large weights to compulsory constraints to

ensure satisfaction�

��

Theoretical Results

�� Overall logic�

� Unlike KISS� instantiated cover may be incorrect

� need added cubes to avoid next�state hazards

� Cardinality j �C j� O�j C j � u�

u � of unstable state transitions

� Upper bound on logic complexity� due to input encoding

�� Output logic� exactly optimal over all CRF codes

if outputs minimized separately

Important for burst�mode� where input�output latency

determines operating speed���

Experimental Results

Up to ��� improvement in overall logic with no increase in code

length� using heuristic constraint satisfaction�

heuristic exact base�crf

DESIGN I�S�O bits cubes bits cubes bits cubes

sbuf�read�ctl ����� � � � � � �

pscsi�ircv ����� � � � �� � �

pscsi�trcv ����� � � � �� � ��

sscsi�trcv�csm ���� � �� � �� � ��

pscsi�trcv�bm ����� � �� � � � ��

rf�control ���� � �� � � � �

sscsi�tsend�csm ���� � �� � � ��

it�control ��� � � � � � �

pe�send�ifc ��� � �� � �� � ��

pscsi�isend ����� � �� � �� � ��

sscsi�trcv�bm ���� � �� �� � ��

sscsi�tsend�bm ��� � �� � � � ��

sscsi�isend�bm ���� � �� �� � ��

sd�control ������� �� � �� � �

stetson�p� ������� � �� � �� � ��

stetson�p� �������� � � �� � �

��

Conclusions

� First systematic optimal encoding method for asynchronous

FSM�s

� Symbolic hazard�free ��level logic minimization

� Extended encoding constraints for asynchronous FSM�s

� Constraint solution�

� Compulsory vs� optional constraints

� Exactly optimal output logic

� Signi�cant improvement on industrial examples

��

Future Work

� Improve results of annealing algorithm

� Extend to output encoding formulation

� Generalized symbolic transitions

� larger machine class� extended burst�mode

� relax operating constraints��