1

SPFDs - A new method for SPFDs - A new method for specifying flexibilityspecifying flexibilitySSets of ets of PPairs of airs of FFunctions to be unctions to be

DDistinguishedistinguishedFrom the paper:From the paper:

““A new Method to Express Functional A new Method to Express Functional Permissibilities for LUT Based Permissibilities for LUT Based FPGAsFPGAs and Its and Its Applications Applications ” ”

by S. Yamashita, H. Sawada, and A. Nagoya, by S. Yamashita, H. Sawada, and A. Nagoya, NTT Communications Science Lab., Kyoto, Japan NTT Communications Science Lab., Kyoto, Japan ICCAD’96ICCAD’96

BUTBUT:: this paper is more about a this paper is more about a new kindnew kind of “don’t of “don’t cares”, rather than about application to FPGAscares”, rather than about application to FPGAs

2

Methods for Expressing Methods for Expressing Sets of FunctionsSets of Functions

• Don’t Cares Don’t Cares (=> incompletely specified (=> incompletely specified functions (functions (ISFsISFs))))– Satisfiability DCsSatisfiability DCs– Observability DCsObservability DCs– XDCsXDCs

• ATPG Redundancy RemovalATPG Redundancy Removal• Boolean RelationsBoolean Relations• Sets of Boolean RelationsSets of Boolean Relations• SPFDs. SPFDs. (Actually, they are (Actually, they are setssets of ISFs). of ISFs).

3

ODCs and SPFDsODCs and SPFDs

ODCs:ODCs:– Different ways to compute themDifferent ways to compute them

• Full method (difficult)Full method (difficult)• SubsetsSubsets

– Compatible Sets of Permissible functions Compatible Sets of Permissible functions (CSPFs)(CSPFs)» (Easier to compute)(Easier to compute)

– Compatible observability don’t cares Compatible observability don’t cares ((CODCsCODCs))» see Savoj thesis - generalization of CSPFssee Savoj thesis - generalization of CSPFs

– CSPFs and CODCsCSPFs and CODCs• have advantage that they can be computed fast - from have advantage that they can be computed fast - from

outputs to inputsoutputs to inputs• are independentare independent

4

ODCs and SPFDsODCs and SPFDs

SPFDs:SPFDs:• like like CSPFs/CODCsCSPFs/CODCs

– can be computed fast from outputs to can be computed fast from outputs to inputsinputs

– are independentare independent• but but more powerfulmore powerful and and • not a subsetnot a subset of DCs ( of DCs (they are a sets of they are a sets of

ISFsISFs))

5

ISFsISFsUsually we deal with binary ISFs.Usually we deal with binary ISFs.

– They can be represented as They can be represented as complete bipartite graphscomplete bipartite graphs on on minterms.minterms.

– Vertices are minterms. Set of vertices may be Vertices are minterms. Set of vertices may be subsetsubset of all of all mintermsminterms

– An An edgeedge means that we need to build a function that means that we need to build a function that distinguishesdistinguishes (i.e. has different values) the two minterms on the edge.(i.e. has different values) the two minterms on the edge.

A connected bipartite graph has only A connected bipartite graph has only twotwo colorings, one the colorings, one the onsetonset and the other the and the other the offsetoffset and vice versa ( and vice versa (onsetonset, , offsetoffset))

offsetoffsetonsetonset

6

Building FBuilding F

We start with a complete bipartite graph (ISF) for F. This We start with a complete bipartite graph (ISF) for F. This gives a set of minterms that must be distinguished.gives a set of minterms that must be distinguished.

FF

gg11 gg22 gg33

yy11yy22yy33

xx

YY space is (y space is (y11,y,y22,y,y33). ). Each yEach yii

is a minterm in is a minterm in YY

yy11

yy33

yy22yy44

yy55

7

Building FBuilding FSuppose that the inputs to F are Suppose that the inputs to F are

y = (gy = (g11(x), g(x), g22(x), g(x), g33(x)). (x)).

As long as {yAs long as {y11, y, y22, y, y33} are } are encodedencoded differently from {y differently from {y44, y, y55} by } by g(x), there exists a function F which implements the ISF.g(x), there exists a function F which implements the ISF.

yy11

yy33

yy22yy44

yy55

1 4

2 5

3

001 101 100 001

010 110 101 111

011 010

y y

y y

y

oldold newnew oldold newnew

New F:New F:(101) 1 (001) 0

(110) 1 (111) 0

(010) 1

f f

f f

f

8

Distributing the jobDistributing the job

We can We can distributedistribute the “encoding” job separately to the the “encoding” job separately to the input wires.input wires.

Note:Note: as long as each edge is distinguished by one of the inputs, as long as each edge is distinguished by one of the inputs, this is sufficient for a new encodingthis is sufficient for a new encoding. .

yy11

yy33

yy22yy44

yy55

yy11

yy22

yy55

yy44

input 1input 1yy22

yy33

yy55

yy44

input 2input 2yy11

yy33

yy44

yy55

input 3input 3

9

Input GraphsInput Graphs

Each of the edges on the input graphs means Each of the edges on the input graphs means that the input signal gthat the input signal gii(x) has to have a (x) has to have a different value for the two nodes.different value for the two nodes.

Note:Note: the graphs are bipartite, but the graphs are bipartite, but not completenot complete and may be and may be disconnecteddisconnected. In the above case, . In the above case, each graph has 4 different legal colorings.each graph has 4 different legal colorings.

yy11

yy22

yy55

yy44

input 1input 1yy22

yy33

yy55

yy44

input 2input 2yy11

yy33

yy44

yy55

input 3input 3

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Assigning ValuesAssigning Values

Now we can assign values to the graphs Now we can assign values to the graphs (color the (color the graphs).graphs).

Thus, for exampleThus, for example

yy11=1*1, y=1*1, y22=00*, y=00*, y33=*10, y=*10, y44=011, y=011, y55=100=100

yy11

yy22

yy55

yy44

input 1input 1yy22

yy33

yy55

yy44

input 2input 2yy11

yy33

yy44

yy55

input 3input 3

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Assigning ValuesAssigning ValuesNote that yNote that y11, y, y22, y, y33, y, y44, y, y55 are all distinctly encoded in this case. But, in general, that is not necessary, are all distinctly encoded in this case. But, in general, that is not necessary,

e.g. could have e.g. could have yy11 yy2 2 yy3 3 all encoded with the same value.all encoded with the same value.

The original ISF can be implemented with either encodingThe original ISF can be implemented with either encodingF(yF(y11) = F(y) = F(y22) = F(y) = F(y33) = 1 and F(y) = 1 and F(y44) = F(y) = F(y55) = 0) = 0

Now just call ESPRESSO to get a minimized F.Now just call ESPRESSO to get a minimized F.

yy11

yy22

yy33

yy44

yy55

yy11

yy22

yy55

yy44

input 1input 1yy22

yy33

yy55

yy44

input 2input 2yy11

yy33

yy44

yy55

input 3input 3

yy11

yy22

yy55

yy44

input 1input 1

yy22

yy33

yy55

yy44

input 2input 2

yy11

yy33

yy44

yy55

input 3input 3

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SPFDsSPFDsNow we generalize this becauseNow we generalize this because

1.1. we canwe can

2.2. it is more powerful it is more powerful (can be applied to multiple outputs and multiple valued variables)(can be applied to multiple outputs and multiple valued variables)

3.3. it improves understandingit improves understanding

Definition 1:Definition 1: A SPFD F(y) on domain Y is an A SPFD F(y) on domain Y is an undirectedundirected graph graph (V,E) where each v(V,E) where each vV is associated with a mintermV is associated with a mintermm = (ym = (y11,...,y,...,ykk) ) Y Y

yy11 yy22

yy66

yy55yy44

yy33

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ImplementsImplements

Definition 2:Definition 2: A function f(y) A function f(y) implementsimplements F(y) = F(y) = (V,E) iff f(y) is a valid coloring of F, i.e.(V,E) iff f(y) is a valid coloring of F, i.e.

(y(y11, y, y22) ) E E iffiff f(y f(y11) ) f(y f(y22), ),

Three ColoringsThree ColoringsMust have at least 3 colors sinceMust have at least 3 colors since

the chromatic number of graph is 3the chromatic number of graph is 3

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CompatibleCompatibleDefinition 3:Definition 3: A mapping y(x) = A mapping y(x) =

(y(y11(x),...,y(x),...,ykk(x)) is (x)) is F(z)-compatibleF(z)-compatible ifif(((z(zii,z,zjj) ) F(z)) F(z)) y(X(zy(X(zii)) )) y(X(z y(X(zjj)) = 0)) = 0where X(zwhere X(zii) = { x | z(x) = z) = { x | z(x) = zi i }.}.

X(zX(zjj))

y(x)y(x)

Y-spaceY-space

PrimaryPrimaryInputInputSpaceSpace

z(x)z(x)

X(zX(zii))

y(x)y(x)

z(x)z(x)

zziizzjj

F(z)F(z)Z-spaceZ-space

gg

Boolean NetworkBoolean Network

Y-spaceY-space

Z-spaceZ-space

x-space (x-space (primary inputsprimary inputs))

15

Re-implementing a Re-implementing a FunctionFunctionTheorem 1:Theorem 1: If y(x) is F(z)-compatible, then there If y(x) is F(z)-compatible, then there existsexists a function g(y) which implements a function g(y) which implements

F(z).F(z).

Proof.Proof.Translate SPFD F(z) into F(y). Thus for each (zTranslate SPFD F(z) into F(y). Thus for each (z ii,, zzjj) ) F(z)), put an edge for every pair (y F(z)), put an edge for every pair (ykk,, yyll) ) where ywhere ykk y(X(z y(X(zii)) and y)) and yll y(X(z y(X(zjj)). Since y(X(z)). Since y(X(zii)) )) y(X(z y(X(zjj)) = 0, there is no self edge in )) = 0, there is no self edge in F(y). Hence a coloring exists. F(y). Hence a coloring exists.

X(zX(zjj))

y(x)y(x)

Y-spaceY-space

PrimaryPrimaryInputInputSpaceSpace

z(x)z(x)

X(zX(zii))

y(x)y(x)

z(x)z(x)

zziizzjj

F(z)F(z)Z-spaceZ-space

16

Re-implementing a Re-implementing a FunctionFunction

ColorColor F(y) and implement the F(y) and implement the corresponding ISF by some function g(y).corresponding ISF by some function g(y).

Each y(X(zEach y(X(zii)) creates an )) creates an equivalenceequivalence class class which can all be colored with the same which can all be colored with the same color, color, (but different colors can be used if (but different colors can be used if preferred.)preferred.)

17

Re-implementing the Re-implementing the Inputs to a FunctionInputs to a Function

We want to build new functions We want to build new functions y’(x)y’(x) at the inputs to G at the inputs to G..

GG

Boolean NetworkBoolean Network

Y-spaceY-space

Z-spaceZ-space

x-space (primary inputs)x-space (primary inputs)

18

Re-implementing the Re-implementing the Inputs to a FunctionInputs to a Function

For new functions y’(x), just need for each edge in For new functions y’(x), just need for each edge in F(y), F(y), at least one input that has different values at least one input that has different values for the two minterms of the vertices of the edgefor the two minterms of the vertices of the edge

Theorem 2:Theorem 2: y’(x) is F(z)-compatible if y’(x) is F(z)-compatible if

(z(zii,z,zjj))F, F, k, k, (((x(xii,x,xjj) ) (X(z (X(zii),X(z),X(zjj))), y’))), y’kk(x(xii) ) y’ y’kk(x(xjj))

Proof.Proof. Since y’ Since y’kk(x(xii) ) y’ y’kk(x(xjj), then y’(x), then y’(xii) ) y’(x y’(xjj). ). Hence y’(X(zHence y’(X(zii)) )) y’(X(z y’(X(zjj)) = 0.)) = 0.

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Re-using Old TopologyRe-using Old TopologyNoteNote: up to now we could have assigned any edge of the node SPFD to any of the inputs: up to now we could have assigned any edge of the node SPFD to any of the inputs

An efficient way of building a new mapping y’(x) is to use part of the An efficient way of building a new mapping y’(x) is to use part of the oldold implementation. implementation.

GG

Boolean NetworkBoolean Network

Y-spaceY-space

Z-spaceZ-space

x-space (primary inputs)x-space (primary inputs)

ww11 ww22 ww33

20

Re-using Old TopologyRe-using Old Topology

Construct SPFDs {FConstruct SPFDs {F11(y),...,F(y),...,Frr(y)} (y)} (for each local (for each local

input)input) such that such that

1.1. FFk k (y) = { (y(y) = { (yii,y,yjj) ) F(y) only if } F(y) only if }

i.e. the ki.e. the kthth input currently distinguishes (y input currently distinguishes (yii,y,yjj) for all those x ) for all those x

which map to ywhich map to yii, or to y, or to yjj

2.2. (y(yii,y,yjj) ) F(y), F(y), k, (yk, (yii,y,yjj) ) F Fkk(y)(y)

i.e. each edge is assigned to one of the inputsi.e. each edge is assigned to one of the inputs

yyj

k

i

k

21

Efficient Re-Efficient Re-implementationimplementationTheorem 3:Theorem 3: Suppose the current implementation of the global functions Suppose the current implementation of the global functions

at y, y(x), are implemented with sets of intermediate variables at y, y(x), are implemented with sets of intermediate variables ww = = ((ww11,...,,...,wwrr),),

i.e. y(x) = (fi.e. y(x) = (f11((ww11(x)),...,f(x)),...,frr((wwrr(x))).(x))).

Then for each k, any coloring of FThen for each k, any coloring of Fk k (y) can be implemented with some (y) can be implemented with some function f’function f’k k ((wwkk(x)).(x)).(i.e. the new fanins of G can be implemented with the same inputs (i.e. the new fanins of G can be implemented with the same inputs ww11,...,,...,wwrr as the as the old fanins)old fanins)

Thus a new set of input functions y’(x) can be obtained asThus a new set of input functions y’(x) can be obtained asy’ = (f’y’ = (f’11((ww11(x)),...,(f’(x)),...,(f’rr((wwrr(x)))(x)))

i.e. a set of functions of the i.e. a set of functions of the currentcurrent intermediate inputs to f intermediate inputs to f11,...,f,...,frr..

GGY-spaceY-space

ww11 ww22 wwrr……

22

End of lecture 15End of lecture 15

23

Efficient Re-Efficient Re-implementationimplementation

Proof.Proof. Consider F Consider Fkk(y) for some input k. The current (y) for some input k. The current ffk k implements Fimplements Fkk(y) since for (y(y) since for (yii,y,yjj) ) F Fkk, f, fkk(y(yii) ) ffkk(y(yjj), i.e.), i.e.

ffkk(X(y(X(yii)) )) f fkk(X(y(X(yjj)) = 0)) = 0Thus for all xThus for all xii X(y X(yii), x), xjj X(y X(yjj), we must have ), we must have wwkk(x(xii) ) wwkk(x(xjj). Otherwise, f). Otherwise, fkk((wwkk(x(xii)) = f)) = fkk((wwkk(x(xjj)) )) contradicting that fcontradicting that fkk(y(yii) ) f fkk(y(yjj). Since w). Since wkk distinguishes xdistinguishes xii from x from xj j (i.e. the information is (i.e. the information is available at the inputs of node k)available at the inputs of node k), functions , functions f’f’kk((wwkk) exist which distinguish x) exist which distinguish xii and x and xjj..

24

How Do We Use ThisHow Do We Use This

Note:Note: since y(x) is F(z)-compatible, then so is since y(x) is F(z)-compatible, then so is yy++ = (y(x),y = (y(x),yk+1k+1(x)).(x)).

Theorem 4:Theorem 4: Wire yWire y11 can be removed if and only ifcan be removed if and only if(y(y22,...,y,...,yk+1k+1) is F(z)-compatible.) is F(z)-compatible.

GG

Boolean NetworkBoolean Network

y-spacey-space

z-spacez-space

x-space (primary inputs)x-space (primary inputs)

25

Conclusions and futureConclusions and future

• SPFDs offer SPFDs offer greater choicegreater choice in implementing a in implementing a circuitcircuit– Can change both a node and its inputsCan change both a node and its inputs– An SPFD can be colored in many waysAn SPFD can be colored in many ways

• New way to obtain New way to obtain BooleanBoolean divisors divisors• Implementation using Implementation using MDDsMDDs in progress in progress• Implementation using Implementation using SATSAT in progress in progress• Useful for DSM for Useful for DSM for rewiringrewiring for noise avoidance for noise avoidance

or better layoutor better layout• Can be used to define “don’t care wires”Can be used to define “don’t care wires”

26

Ruggedizing SPFD Ruggedizing SPFD computationcomputation

Currently done using Currently done using BDDsBDDs– Works for medium sized circuitsWorks for medium sized circuits

Have developed a method using Have developed a method using SATSAT– Implementation almost doneImplementation almost done

Z-spaceZ-space

Y-spaceY-space

SPFD(SPFD(ZZ))

SPFD(SPFD(YY))

Primary inputs XPrimary inputs X

Image computation:Image computation: which minterms in Zwhich minterms in Z are mapped into whichare mapped into which minterms in Y?minterms in Y?

z -> yz -> yif exists x such thatif exists x such thatffZZ(x) = z and f(x) = z and fYY(x) = (x) =

yy

27

MV-SPFDs and Wire MV-SPFDs and Wire RemovalRemoval

An MV-SPFD represents the An MV-SPFD represents the informationinformation content content of a wire in a network.of a wire in a network.– the MV-SPFD of each fanin to a node, gives the the MV-SPFD of each fanin to a node, gives the

information that it information that it must providemust provide. .

The sum The sum totaltotal of the information content of a of the information content of a node’s fanin wires must node’s fanin wires must covercover the information the information required of the output of the node.required of the output of the node.

We determine a minimal subset of fanins atWe determine a minimal subset of fanins at each each node which cover the required information by node which cover the required information by solving a solving a covering problemcovering problem..

28

Wire Wire Removal and Removal and

Wire Wire SubstitutionSubstitution

SPFDSPFDoutout

SPFDin2

SPFDin1

SPFDSPFDin3in3

SPFDSPFDin4in4

SPFDSPFDin5in5

SPFDSPFDoutout SPFD SPFDin3in3SPFDSPFDin4in4SPFDSPFDin5in5

Wire removal: suppose

Then, can remove in1 and in2

Important:Because wires arechanged, the internallogic of a block maychange.

SPFDSPFDin4in4 SPFD SPFDin6in6

Wire substitution: suppose

Then, can replace in4 by in6

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Don’t Care WiresDon’t Care WiresEach input pin Each input pin will have a will have a setset of sources, of sources, SS

(among which we can choose any one).(among which we can choose any one).

Each source Each source (driver of a wire)(driver of a wire) distinguishesdistinguishes a a setset of pairs of minterms of pairs of minterms (SPFD)(SPFD) in the primary in the primary input spaceinput space

Wire Wire is assigned as a is assigned as a candidatecandidate source for source for input pin input pin if: if:

SPFDSPFD SPFD SPFD

30

Compatible sets of don’t Compatible sets of don’t care wirescare wires

Want to be able to select one source from Want to be able to select one source from each set each set SS independentindependent of the selection of the selection made from any other set made from any other set SS

’’ ( (compatible compatible

set of candidate sets of sourcesset of candidate sets of sources))

TheoremTheorem: A set of sets {: A set of sets {SS} of candidate } of candidate wires is wires is compatiblecompatible if if– it can never produce a it can never produce a cycliccyclic network network– for each wire for each wire SSSPFDSPFD SPFD SPFD

31

Choosing best subset of wires

AWC:AWC: alternate wire choicealternate wire choice problem problem Given:Given: a a pointpoint placement of pins and a set placement of pins and a set

of candidate wires for each input pinof candidate wires for each input pinFind:Find: a a selectionselection of wires for each input of wires for each input

pin that minimizes the total half-pin that minimizes the total half-perimeters of the nets’ bounding boxes.perimeters of the nets’ bounding boxes.

(The problem is NP-complete)(The problem is NP-complete)

32

Placement using don’t care wires

Current placement by Current placement by

Mincut/Annealing Mincut/Annealing or by aor by a Force Force

Directed Method:Directed Method:

– wire choiceswire choices considered in move to considered in move to

determine cost of movedetermine cost of move

• use semi-greedy use semi-greedy AWCAWC algorithm during algorithm during

each placement moveeach placement move

33

Experiments – total wire Experiments – total wire length length

(placement, no routing)(placement, no routing)Initial PLA Initial PLA netlistnetlist

Alternate wires Alternate wires (13.5%)(13.5%)

Re-synthesisRe-synthesis

PlacePlace

Wire Wire choiceschoices

13.7%13.7%

Wire removalWire removal

and Re-and Re-synthesissynthesis

PlacePlace

8.5%8.5%

Wire removalWire removal

PlacePlace

14.4%14.4%

PlacePlace 6.6%6.6%PlacePlace

0%0%

Max Alternates Max Alternates (15%)(15%)

PlacePlace 10.2%10.2%

Notes: 13.5% -> 15% in alternate wires results in 6.6% -> 10.2% improvement in wire length

Wire removal currently does not use placement information

Wire removalWire removal

PlacePlace

? %? %

34

Primary outputs: information required (in terms of PIs)

??

????

????

??

??

??

??

InformationArrives - PIs

A block logic function exists iff the information at the inputs covers the information required at the output

Function inside block only needs to provide the information required of its fanouts

??

Wireplanning before Wireplanning before decompositiondecomposition

35

Information Flow TheoremInformation Flow TheoremTheorem:Theorem: A set of logic functions A set of logic functions

exists for the blocks exists for the blocks if and onlyif and only if if there exists a path from input there exists a path from input IIk k to to output output OOjj whenever whenever OOj j depends on depends on IIkk. .

??

????

????

??

??

??

?? Open question: Howdo you make the assignment of SPFDedges so that the resulting logic in theblocks is as simple aspossible?

Open question: Howdo you make the assignment of SPFDedges so that the resulting logic in theblocks is as simple aspossible?