Marc Riedel Ph.D. Defense, Electrical Engineering, Caltech November 17, 2003 Combinational Circuits...

Marc RiedelPh.D. Defense, Electrical Engineering, Caltech

November 17, 2003

Combinational Circuitswith Feedback

Combinational Circuits

Logic GateBuilding Block:

1x2x

dx

di

ix

,,1allfor

{0,1}

{0,1}{0,1}: dg

),,( 1 dxxg

Combinational Circuits

Logic GateBuilding Block:

1x2x

dx

),,( 1 dxxg

feed-forward device

Combinational Circuits

“AND” gate

0001

Common Gate:

1x

2x

g0011

0101

1x 2x g

Combinational Circuits

“OR” gate

0011

0101

0111

Common Gate:

1x

2x

g

1x 2x g

Combinational Circuits

“XOR” gate

0011

0101

0110

Common Gate:

1x

2x

g

1x 2x g

),,( 11 mxxf a

),,( 12 mxxf a

),,( 1 mn xxf a

inputs outputs

The current outputs depend only on the current inputs.

Combinational Circuits

1x

2x

mx

mi

ix

,,1allfor

{0,1}

nj

mjf

,,1allfor

{0,1}{0,1}:

combinationallogic

),,( 1 mn xxf a

),,( 11 mxxf a

),,( 12 mxxf a),,( 1 mxxf a

Combinational Circuits

inputs outputs

The current outputs depend only on the current inputs.

1x

2x

mx

combinationallogic

gate

mi

ix

,,1allfor

{0,1}

nj

mjf

,,1allfor

{0,1}{0,1}:

Generally feed-forward (i.e., acyclic) structures.

Combinational Circuits

x

y

x

y

z

z

c

s

Generally feed-forward (i.e., acyclic) structures.

Combinational Circuits

0

1

0

1

1

1

0

1

1

0

1

Feedback

How can we determine the output without knowing the current state?

...

...

...

...

feedback

Feedback

How can we determine the output without knowing the current state?

...

...

...

...

?

?

?

fa a

Example: outputs can be determined in spite of feedback.

x x

Feedback

fa a

0 0

Example: outputs can be determined in spite of feedback.

Feedback

fa a

0 0

00

Example: outputs can be determined in spite of feedback.

Feedback

fa a

Example: outputs can be determined in spite of feedback.

Feedback

x x

fa a

1 1

Example: outputs can be determined in spite of feedback.

Feedback

fa a

1 1

11

There is feedback is a topological sense, but not in an electrical sense.

Example: outputs can be determined in spite of feedback.

Feedback

fa a

Admittedly, this circuit is useless...

Example: outputs can be determined in spite of feedback.

Feedback

x x

x

Rivest’s Circuit

Example due to Rivest:

1x 2x 3x 1x 2x 3x

1fa a 2fa a 3fa a 4fa a 5fa a 6fa a

0 0

Example due to Rivest:

Rivest’s Circuit

2x 3x 2x 3x

1fa a 2fa a 3fa a 4fa a 5fa a 6fa a

1x 1x

Rivest’s Circuit

0 0

0

Example due to Rivest:

2x 3x 2x 3x

1fa a 2fa a 3fa a 4fa a 5fa a 6fa a

Rivest’s Circuit

Example due to Rivest:

1x 2x 3x 1x 2x 3x

1fa a 2fa a 3fa a 4fa a 5fa a 6fa a

0 0

0

1 1

Example due to Rivest:

Rivest’s Circuit

2x 3x 2x 3x

1fa a 2fa a 3fa a 4fa a 5fa a 6fa a

1x 1x

1

1 1

Example due to Rivest:

Rivest’s Circuit

2x 3x 2x 3x

1fa a 2fa a 3fa a 5fa a 6fa a4fa a

4fa a

Example due to Rivest:

1x 1x1 1

Rivest’s Circuit

2x 3x 2x 3x

1fa a 2fa a 3fa a 5fa a 6fa a1

Rivest’s Circuit

Example due to Rivest:

)( 321 xxx )( 312 xxx )( 213 xxx

321 xxx 312 xxx 213 xxx

1x 2x 3x 1x 2x 3x

3 inputs, 6 fan-in two gates. 6 distinct functions, each dependent on all 3 variables.

Addition: ORMultiplication: AND

Rivest’s Circuit

Individually, each function requires 2 fan-in two gates:

)( 321 xxx

)( 3211 xxxf a 3122 xxxf a )( 2133 xxxf a

3214 xxxf a )( 3125 xxxf a 2136 xxxf a

1x2x

3x

An equivalent feed-forward circuit requires 7 fan-in two gates.

1fa a1x

2x

3x

6fa a

2fa a

5fa a

4fa a

3fa a

1x

2x

3x

2x

3x

A feedback circuit with fewer gates than any equivalent feed-forward circuit.

Rivest’s Circuit

1x 2x 3x 1x 2x 3x

1fa a 2fa a 3fa a 4fa a 5fa a 6fa a

3 inputs, 6 fan-in two gates.

6 distinct functions.

Rivest’s Circuit

n inputs 2n fan-in two gates,

2n distinct functions.

1x nx 1x nx

1fa a

nfa a1nfa a

nf2a a

1fa a...

...

...

...

a

2fa a

Rivest’s Circuit

1n gates

12 n gates

An equivalent feed-forward circuit requires fan-in two gates.

23 n

nf2a

Rivest’s Circuit

n inputs 2n fan-in two gates,

2n distinct functions.

A feedback circuit with the number of gates of any equivalent feed-forward circuit.

32

1x nx 1x nx

1fa a

nfa a1nfa a

nf2a a

Prior Work

• Kautz first discussed the concept of feedback in logic circuits (1970).

• Huffman discussed feedback in threshold networks (1971).

• Rivest presented the first, and only viable, example of a combinational circuit with feedback (1977).

Prior Work

F(X) G(X)e.g., add e.g., shift

Stok discussed feedback at the level of functional units (1992).

Malik (1994) and Shiple et al. (1996) proposed techniques for analysis.

X

G(F(X))

Y

F(G(Y))

Questions

1. Is feedback more than a theoretical curiosity, even a general principle?

322. Can we improve upon the bound of ?

3. Can we optimize real circuits with feedback?

Key Contributions

2. Efficient symbolic algorithm for analysis (both functional and timing).

1. A family of feedback circuits that are asymptotically the size of equivalent feed-forward circuits. 2

1

3. A general methodology for synthesis.

Ph.D. Defense

• Present examples with same property as Rivest’s circuits.

• Illustrate techniques for analysis.

• Focus on synthesis: methodology, examples, optimization results.

Example

43218 )( xxxxf

43215 xxxxf

43214 )( xxxxf

34213 xxxxf

43212 xxxxf

)( 43211 xxxxf

34217 )( xxxxf

43216 xxxxf

a

not symmetrical

4 inputs

8 gates

8 distinct functions

x4

x3

x2

x1

x4

x3

x2

x1

Examples, multiple cycles

x1

x1

x1

x1

x2

x2

x3

x3

321311 2xxxxxxf a

3212 xxxf a

a )( 3123 xxxf

2134 xxxf a

a )( 3215 xxxf

a3216 xxxf a

aa )( 3127 xxxf

aa

1138 xxxf

aa )( 3219 xxxf

, 3 inputs9 gates , 9 distinct functions

Example

x1 x2 x3 x4 x5 x1 x2 x3 x4 x5

1f 2f 3f 4f 5f 6f 7f 8f 9f 10fa

11f 12f 13f 14f 15f 16fa 17f 18f 19f 20fa

, 5 inputs 20 gates , 20 distinct functions.

(“stacked” Rivest circuits)

½ Example

Generalization: family of feedback circuits ½ the size of equivalent feed-forward circuits.

3111 ff a

a

3222 ff a

a

213231333 fffff a

a

Xiiii offunctionsare,,, nxxxX ,,, 21

(sketch)

X

X

X

Analysis

• Functional analysis: determine if the circuit is combinational and if

so, what values appear.

• Timing analysis: determine when the values appear.

Contributions:

1. Symbolic algorithm based on Binary Decision Diagrams.

2. Optimizations based on topology (“first-cut” method).

Analysis

1x 2x 3x 1x 2x 3x0 0 0 00 0

01

Assume gates each have unit delay.

02 01 0102 02

arrival times

Explicit analysis:

0 0 0 00 0

01 02 01 0102 02

0 1 0 10 0

01 11 01 0302 04

1x 2x 3x 1x 2x 3x

Analysis

Explicit analysis:

1x0 1 0 10 0

01 11 01 0302 04

2x 3x 1x 2x 3x

n inputs n2 combinations

Exhaustive evaluation intractable.

Analysis

Explicit analysis:

1x 2x 3x 1x 2x 3x

Symbolic analysis:

1x

31 xx

321 xxx321 xxx

1fa a

01 :

12 :

03 :

14 :

Analysis

654,32 ,,, fffffasimilarly for

Symbolic analysis:

fa a

01

02

11

12

Analysis

),,( 11 nxxc ),,( 12 nxxc

),,( 11 nxxc ),,( 12 nxxc

),,( 1 nxxd undefined

evaluates to 1

evaluates to 0

Symbolic analysis:

fa a

01-7

08-28

11-10

111-21

Analysis

),,( 11 nxxc ),,( 12 nxxc

),,( 11 nxxc ),,( 12 nxxc

),,( 1 nxxd undefined

evaluates to 1

evaluates to 0

range of values

Synthesis

• General methodology: optimize by introducing feedback in the substitution/minimization phase.

• Optimizations are significant and applicable to a wide range of circuits.

Design a circuit to meet a specification.

Example: 7 Segment Display

Inputs a

b

c

d

e

f

g

Output

1001

0001

1110

0110

1010

0010

1100

0100

1000

00000123 xxxx

9

8

7

6

5

4

3

2

1

0

Example: 7 Segment Display

g

f

e

d

c

b

a

)(

)(

))((

))((

))((

)(

))((

20321

10102321

2012103210

102213321

210203321

21310

302321320

xxxxx

xxxxxxxx

xxxxxxxxxx

xxxxxxxxx

xxxxxxxxx

xxxxx

xxxxxxxxx

a

b

c

d

e

f

g

Output

Substitution/Minimization

Basic minimization/restructuring operation: express a function in terms of other functions.

Substitute b into a:

(cost 9)a ))(( 302321320 xxxxxxxxx

(cost 8)

Substitute c into a:(cost 5)

Substitute c, d into a:(cost 4)

a )( 323212 bxxxxxbx

a cxxcx 321

a dccx 1

Substitution/Minimization

Berkeley SISTool

a ))(( 302321320 xxxxxxxxx

},,,{ fdcb

target function

substitutional set

a dccx 1

low-cost expression

Acyclic Substitution

g

f

e

b

a

c

d

Select an acyclic topological ordering:

g

f

e

d

c

b

a

g

f

d

c

b

a

edcaxx 21

dccx 1

xxxxxxxxx 102213321 ))((

dxxxxxx 102320 )(

cdxx 10 )(

Select an acyclic topological ordering:

Cost (literal count): 37

Acyclic Substitution

e 3cxb d

ba f

Acyclic Substitution

Select an acyclic topological ordering:

Nodes at the top benefit little from substitution.

g

f

d

c

b

a

edcaxx 21

dccx 1

xxxxxxxxx 102213321 ))((

dxxxxxx 102320 )(

cdxx 10 )(

e 3cxb d

ba f

Cyclic Substitution

Try substituting every other function into each function:

Not combinational!Cost (literal count): 30

0

1

ex

dccx

fba

geex

bcdx

gxaxex

egxxax

2

3

321

202 f

g

f

d

c

b

a

e

Cost 30Lower bound

Cost 37

Upper boundAcyclic substitution

Unordered substitution

Cyclic solution? Cost 34

Cyclic Substitution

g

f

e

d

c

b

a

Cost (literal count): 34

Combinational solution:

x e0

bxa 3

gxxxax 1023 )(

axxex 321 )( exxxxxx 312320 )(

cxxcx 301

xxxfx 1023 )( f

Cyclic Substitution

Cost (literal count): 34

Combinational solution:

topological cycles

g

f

e

d

c

b

a

x e0

bxa 3

gxxxax 1023 )(

axxex 321 )( exxxxxx 312320 )(

cxxcx 301

xxxfx 1023 )( f

Inputs x3, x2, x1, x0

Cost (literal count): 34

ba

ga

e

e

e

c

1

no electrical cycles

Cyclic Substitution

g

f

e

d

c

b

a

x e0

bxa 3

gxxxax 1023 )(

axxex 321 )( exxxxxx 312320 )(

cxxcx 301

xxxfx 1023 )( f

= [0,0,1,0]:

g

f

e

d

c

b

a

Cost (literal count): 34

ba

ga

e

e

e

c

1

1

1

0

1

1

1

0

Cyclic Substitution

no electrical cycles

x e0

bxa 3

gxxxax 1023 )(

axxex 321 )( exxxxxx 312320 )(

cxxcx 301

xxxfx 1023 )( f

Inputs x3, x2, x1, x0 = [0,0,1,0]:

g

f

e

d

c

b

a

Cost (literal count): 34

ba

ga

e

e

e

c

1

1

1

0

1

1

1

0

a

b

c

d

e

f

g

Cyclic Substitution

x e0

bxa 3

gxxxax 1023 )(

axxex 321 )( exxxxxx 312320 )(

cxxcx 301

xxxfx 1023 )( f

Inputs x3, x2, x1, x0 = [0,0,1,0]:

g

f

e

d

c

b

a

Cost (literal count): 34

ba

ga

e

e

e

c

1

1

1

0

1

1

1

0

a

b

c

d

e

f

g

Cyclic Substitution

x e0

bxa 3

gxxxax 1023 )(

axxex 321 )( exxxxxx 312320 )(

cxxcx 301

xxxfx 1023 )( f

Inputs x3, x2, x1, x0 = [0,0,1,0]:

g

f

e

d

c

b

a

Cost (literal count): 34

ba

ga

e

e

e

c

1

1

1

0

1

1

1

0

a

b

c

d

e

f

g

Cyclic Substitution

x e0

bxa 3

gxxxax 1023 )(

axxex 321 )( exxxxxx 312320 )(

cxxcx 301

xxxfx 1023 )( f

Inputs x3, x2, x1, x0 = [0,0,1,0]:

g

f

e

d

c

b

a

Cost (literal count): 34

Cyclic Substitution

x e0

bxa 3

gxxxax 1023 )(

axxex 321 )( exxxxxx 312320 )(

cxxcx 301

xxxfx 1023 )( f

Inputs x3, x2, x1, x0 = [0,1,0,1]:

g

f

e

d

c

b

a

Cost (literal count): 34

Cyclic Substitution

ba

a

a

1

0

c

f

x e0

bxa 3

gxxxax 1023 )(

axxex 321 )( exxxxxx 312320 )(

cxxcx 301

xxxfx 1023 )( f

Inputs x3, x2, x1, x0 = [0,1,0,1]:

no electrical cycles

g

f

e

d

c

b

a

Cost (literal count): 34

1

0

1

0

1

1

1

Cyclic Substitution

ba

a

a

1

0

c

f

x e0

bxa 3

gxxxax 1023 )(

axxex 321 )( exxxxxx 312320 )(

cxxcx 301

xxxfx 1023 )( f

no electrical cycles

Inputs x3, x2, x1, x0 = [0,1,0,1]:

g

f

e

d

c

b

a

Cost (literal count): 34

a

b

c

d

e

f

g

Cyclic Substitution

1

0

1

0

1

1

1ba

a

a

1

0

c

f

x e0

bxa 3

gxxxax 1023 )(

axxex 321 )( exxxxxx 312320 )(

cxxcx 301

xxxfx 1023 )( f

Inputs x3, x2, x1, x0 = [0,1,0,1]:

g

f

e

d

c

b

a

Cost (literal count): 34

Cyclic Substitution

1

0

1

0

1

1

1ba

a

a

1

0

c

f

a

b

d

e

f

g

x e0

bxa 3

gxxxax 1023 )(

axxex 321 )( exxxxxx 312320 )(

cxxcx 301

xxxfx 1023 )( f

Inputs x3, x2, x1, x0 = [0,1,0,1]:

c

Synthesis

Strategy:

• Allow cycles in the substitution phase of logic synthesis. • Find lowest-cost combinational solution.

21213

321321

312321

)(

)(

)(

xxxxxc

xxxxxxb

xxxxxxa

Collapsed:

Cost: 17

321

1321

323

xxaxc

cxxxxb

xxbxa

Solution:

Cost: 13

“Break-Down” approach

• Exclude edges• Search performed outside space

of combinational solutions

cost 12

cost 13 cost 12

cost 13combinational

cost 14

Branch and Bound

“Build-Up” approach

• Include edges• Search performed inside space

of combinational solutions

cost 17

cost 16cost 15not combinational

cost 14

Branch and Bound

cost 13best solution

Implementation: CYCLIFY Program

• Incorporated synthesis methodology in a general logic synthesis environment (Berkeley SIS package).

• Trials on wide range of circuits– randomly generated– benchmarks– industrial designs.

• Consistently successful at finding superior cyclic solutions.

Benchmark Circuits

Cost (literals in factored form) of Berkeley SIS Simplify vs. Cyclify

Circuit # Inputs # Outputs Berkeley Simplify Caltech Cyclify Improvementdc1 4 7 39 34 12.80%ex6 8 11 85 76 10.60%p82 5 14 104 90 13.50%t4 12 8 109 89 18.30%bbsse 11 11 118 106 10.20%sse 11 11 118 106 10.20%5xp1 7 10 123 109 11.40%s386 11 11 131 113 13.70%dk17 10 11 160 136 15.00%apla 10 12 185 131 29.20%tms 8 16 185 158 14.60%cse 11 11 212 177 16.50%clip 9 5 213 189 11.30%m2 8 16 231 207 10.40%s510 25 13 260 227 12.70%t1 21 23 273 206 24.50%ex1 13 24 309 276 10.70%exp 8 18 320 262 18.10%

Benchmarks

Example: EXP circuit

Cyclic Solution (Caltech CYCLIFY): cost 262

Acyclic Solution (Berkeley SIS): cost 320

cost measured by the literal count in the substitute/minimize phase

Discussion

• A new definition for the term “combinational circuit”: a directed, possibly cyclic, collection of logic gates.

• Most circuits can be optimized with feedback.

• Optimizations are significant.

Paradigm shift:

Current Work

• Implement more sophisticated search heuristics (e.g., simulated annealing).

• Extend ideas to a decomposition and technology mapping phases of synthesis.

• Address optimization of circuits for delay with feedback.

Future Directions

inputs outputsdata structure

Structured Network Representations

databases, biological systems,...

Binary Decision Diagrams

• Introduced by Lee (1959).• Popularized by Bryant (1986).

Graph-based Representation of Boolean Functions

• compact (functions of 50 variables)• efficient (linear time manipluation)

Widely used; has had a significant impact on the CAD industry.

0

1

fa

10

1x

2x

3x

Binary Decision Diagrams

0

1

fa

10

1x

2x

3x1 1 1 1

0000111

0011001

0101010

0000011

1x 2x 3x f

0 1BDDs generally defined as Directed Acyclic Graphs

Graph-based Representation of Boolean Functions

Binary Decision Diagrams

Short described a cyclic structure for a BDD variant (1960).We suggest cycles are a general phenomenon.

1fa a 2fa a 3fa a 4fa a 5fa a 6fa a

1x 2x 3x 1x 2x 3x

00 1 01 1

Binary Decision Diagrams

Short described a cyclic structure for a BDD variant (1960).We suggest cycles are a general phenomenon.

1fa a 2fa a 3fa a 4fa a 5fa a 6fa a

1x 2x 3x 1x 2x 3x

00 1 01 1

0 * * 0

1fa a1x 2x 3x

0

1 0 1 1

1

Binary Decision Diagrams

Short described a cyclic structure for a BDD variant (1960).We suggest cycles are a general phenomenon.

1fa a 2fa a 3fa a 4fa a 5fa a 6fa a

1x 2x 3x 1x 2x 3x

00 1 01 1

)( 3211 xxxf a 3122 xxxf a )( 2133 xxxf a

3214 xxxf a )( 3125 xxxf a 2136 xxxf a

Binary Decision Diagrams

Short described a cyclic structure for a BDD variant (1960).We suggest cycles are a general phenomenon.

1fa a 2fa a 3fa a 4fa a 5fa a 6fa a

1x 2x 3x 1x 2x 3x

00 1 01 1

Future research awaits...