cyclic combinational circuits theory marc d. riedel california institute of technology marrella...

Cyclic Combinational CircuitsCyclic Combinational CircuitsTheoryTheory

Marc D. RiedelCalifornia Institute of Technology

Marrella splendens Cyclic circuit

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(500 million year old Trilobite) (novel construct)

Theory

Prove that cyclic implementations can have fewer gates than equivalent acyclic ones.

cycliccircuit

acycliccircuit

(optimal)

functions, n variables,m fan-in gatesd

gates n more than gatesn

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

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.

6/7 Construction

Cyclic Circuit: 6 functions, 3 variables, 6 fan-in 2 gates.

AND OR AND OR AND OR

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

1f 2f 3f 4f 5f 6f)( 321 xxx )( 312 xxx )( 213 xxx

321 xxx 312 xxx 213 xxx

Acyclic Circuit: at least 7 fan-in 2 gates.

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

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

Fan-in Lower Bound

An acyclic circuit with

d

v

m

in -fan

ablesinput vari

functionsoutput

Question:

must have at least how many gates?

Fan-in Lower Bound

An acyclic circuit with

d

v

m

in -fan

ablesinput vari

functionsoutput

Claim:

gates. 11

1

md

vmust have at least

Fan-in Lower Bound

At least one output function does not depend upon any other.

To compute this function, consider a tree of fan-in d gates.

1x 2x 3x

variablesd

Fan-in Lower Bound

At least one output function does not depend upon any other.

To compute this function, consider a tree of fan-in d gates.

1x 2x variables1 dd

5x3x 4x

Fan-in Lower Bound

At least one output function does not depend upon any other.

To compute this function, consider a tree of fan-in d gates.

1x 2x s variable)1( kkdin general, with k nodes,

5x3x 4x

Fan-in Lower Bound

At least one output function does not depend upon any other.

To compute this function, consider a tree of fan-in d gates.

1x 2xvkkd )1(set

1

1

d

vkthen

needed nodes 1

1least at

d

v

5x3x 4x

Fan-in Lower Bound

An acyclic circuit with

d

v

m

in -fan

ablesinput vari

functionsoutput

Claim:

gates. 11

1

md

vmust have at least

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

fan-in 2 gates.1mm

To implement a function of variables, we need at least

nfa

1f a

Acyclic Lower Bound

For the other functions, we need at least gates.

1n

1n

2fa a

functions, n variables,m fan-in gates2

“first” function

other functions

1x

2x

mx

fan-in gates2fan-in gatesd

nfa

1f a

Acyclic Lower Bound

2fa a

functions, n variables,m

“first” function

other functions

1x

2x

mx

fan-in 2 gates.Require at least 2 nm fan-in d gates.Require at least 11

1

nd

m

Constructions/Lower Bounds

variables fan-in cyclic acyclicmin. # gates

functions

6 3 2 6 7

2 5 4 2 3

3 12 6 3 5

today’stalk

Best construction (asymptotic): n 2n

6/7 Construction(Rivest’s Circuit)

Cyclic Circuit: 6 functions, 3 variables, 6 fan-in 2 gates.

AND OR AND OR AND OR

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

1f 2f 3f 4f 5f 6f)( 321 xxx )( 312 xxx )( 213 xxx

321 xxx 312 xxx 213 xxx

Acyclic Circuit: at least 7 fan-in 2 gates.

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

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?

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.

21

3. A general methodology for synthesis.

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)

Theory

• Exhibit a cyclic circuit that is optimal in terms of the number of gates, say with C(n) gates, for n variables.

• Prove a lower bound on the size of an acyclic circuit implementing the same functions, say A(n) gates.

Strategy:

Main Result:

)(2

1)( nAnC

6/7 Construction(Rivest’s Circuit)

Cyclic Circuit: 6 functions, 3 variables, 6 fan-in 2 gates.

AND OR AND OR AND OR

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

1f 2f 3f 4f 5f 6f)( 321 xxx )( 312 xxx )( 213 xxx

321 xxx 312 xxx 213 xxx

Acyclic Circuit: at least 7 fan-in 2 gates.

2/3 Construction

Cyclic Circuit: 2 functions, 5 variables, 2 fan-in 4 gates.

cgab decf

cdeabf decabg

a

bc

c

de

Acyclic Circuit: at least 3 fan-in 4 gates.

cdeab decab

a

b

c

c

d

e

1z 2z

2G1G

Single Cycle

Example

czabG 11 czdeG 22

a

b

c

c

d

e

1z 2z

2G1G

5 inputs, 2 fan-in four gates, 2 outputsEach output function depends on all 5 input variables.

Single Cycle

cdeab decab

czabG 11 czdeG 22 abcf 11 g , decf 22 g ,

a

b

c

c

d

e

1z 2z

2G1G

5 inputs, 2 fan-in four gates, 2 outputs.Each output function depends on all 5 input variables.

Single Cycle

Any acyclic implementation would require at least 3 such gates.

cdeab decab

1fa a1x

2x

3x

6fa a

2fa a

5fa a

4fa a

3fa a

1x

2x

3x

2x

3x

Acyclic Circuit: at least 7 fan-in 2 gates.

2/3 Construction

Cyclic Circuit: 2 functions, 5 variables, 2 fan-in 4 gates.

cgab decf

cdeabf decabg

a

bc

c

de

Acyclic Circuit: at least 3 fan-in 4 gates.

3/5 ConstructionCyclic Circuit: 3 functions, 12 variables, 3 fan-in 6 gates.

Acyclic Circuit: at least 5 fan-in 6 gates.

a,c,e,g,i

b,d,f,h,j

a,b,k,l

31 )( faigecaf

32 )( fbjhfdbf

21213 ffbafbfaklf

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