Multi-Valued Logic

1

Multi-Valued LogicMulti-Valued Logic

Up to now…Up to now…two-valuedtwo-valued synthesis synthesis– Binary variables take only values {0, 1}Binary variables take only values {0, 1}

Multi-Valued synthesisMulti-Valued synthesis– Multi-valued variable XMulti-valued variable Xii can take on can take on

values values PPi i = {0,…,|P= {0,…,|Pii|-1} (integers - but |-1} (integers - but no ordering implied)no ordering implied)

– Symbolic variables take values from Symbolic variables take values from symbolic set, e.g. state: {ssymbolic set, e.g. state: {s00,s,s11,…,s,…,snn} or } or X: {a,b,c}.X: {a,b,c}.

2

Multi-Valued LogicMulti-Valued Logic

• Formally: Formally: (sometimes called an (sometimes called an mv-functionmv-function).).

• ProblemProblem: : find the minimum (SOP) form for an find the minimum (SOP) form for an incompletely-specified function of incompletely-specified function of this kindthis kind

• Big News:Big News:Nothing (Nothing (muchmuch) changes) changes

,*}1,0{: 1 nPPF

3

Example “Truth Table”Example “Truth Table”

• PP11={0,1,2}, P={0,1,2}, P22={0,1} ={0,1}

• Here “2” means the value 2 and not {0,1}Here “2” means the value 2 and not {0,1}

f(0,0) = 1f(0,0) = 1 f(2,1) = 1f(2,1) = 1

f(1,0) = 0f(1,0) = 0 f(2,0) = *f(2,0) = * unspecifiedunspecified (don’t cares) (don’t cares)

1 2X X

0 0 1

0 1 1

1 0 0

1 1 1

2 0

2 1 1

4

MV FunctionMV Function

offoff

onon

Don’t Don’t carecare

3X

1X

2X

5

TerminologyTerminology

• Vertex:Vertex:

• Cube:Cube:

• Containment:Containment:

• Implicant:Implicant:

1A point of the space nP P

1A set , n i ic c c c P

( , cubes) iff ,

When c is a vertex, , also say i ic d c d c d i

d c c d

0c, F(v)vc all for that such Cube

6


• Onset minterm:Onset minterm:

• Prime Implicant:Prime Implicant:

• Cover of Cover of F F ::

1Vertex such that 1. If

then each has exactly one member.n

i

v f(v) v (c , ,c )

c

cddc

that such implicant no is there that such Implicant

A set of implicants which together

contain every onset minterm of f

7


• Prime Cover of Prime Cover of F F ::

• Distance of cubes Distance of cubes c,d c,d ::

• Supercube of Supercube of c,d c,d ::

Note:Note: All these definitions are exactly as they All these definitions are exactly as they were in the binary case.were in the binary case.

n

iii dcdc

1

,

prime is elements whoseof each cover,A

1 1, n nc d c d c d

8

Notation-MV LiteralsNotation-MV LiteralsDefinitionDefinition - A multi-valued - A multi-valued literalliteral is a is a binary logic function of the formbinary logic function of the form

wherewhere

X ici

)()( 1 kiici XXX i

iij Pc

DefinitionDefinition - A - A cubecube can can be written as the product of MV-literals:be written as the product of MV-literals:

ncccc 21

ncn

cc XXX 21

21

9

Notation-MV LiteralsNotation-MV Literals

• If cIf cii=P=Pii we may omit from the we may omit from the expression (since =1)expression (since =1)

iPiX

iPiX

• Note analogy to two-valued case:Note analogy to two-valued case:

• Actually, multi-valued notation is superior Actually, multi-valued notation is superior to old (to old (binarybinary).).

{0}

{1}

{0,1} 1

ii

i i

i

X X

X X

X

10

ExampleExample

Can form rows marked Can form rows marked a a (b)(b) as a single mv-cube as a single mv-cube implicantimplicant

ba112

a02

b111

001

ba110

a100

XX 21

0 21

12

Cube {0,2}×{0,1}

Cube {0,1,2}×{1}

{ , }

{ }

a X

b X

FXXX }{},{}{ of implicant an is implicants prime are b and a Both

01

102

01

The following are cube covers of The following are cube covers of FF. . FF22 is a prime cover is a prime cover1 0 0

1 2 1 2

1 0,22 2 1

{ } { } { }

{ } { }

F X X X

F X X

11

Positional NotationPositional Notation

Example:Example: cube cubePP11={A,B,C,D}, P={A,B,C,D}, P22={R,S} (Symbolic) ={R,S} (Symbolic)

{R}B}{A XX 2,

1

A B C D R SA B C D R SCube:Cube:1 1 0 0 1 01 1 0 0 1 0

• A cube does not depend on variable A cube does not depend on variable XXii if it has all if it has all 1’s in the set of columns associated with 1’s in the set of columns associated with XXii . .

• Each of the columns of a variable is called a Each of the columns of a variable is called a partpart of of that variable. There is one that variable. There is one partpart for each value a for each value a variable can take.variable can take.

• Extension of Extension of EspressoEspresso notation notation

12


(value=0) (value=1)(value=0) (value=1) 0 1 0 1 11 1 0 1 0 00

1 1 1 1 22

Extension of Extension of EspressoEspresso notation notation

Example:Example: XX1 1 XX2 2 XX33

13

End of lecture 5End of lecture 5

14


XX1 1 XX2 2 XX33

3X

1X

2X

5 Cubes1 2 3 4 5

15

Minimization Problem for Minimization Problem for Multi-Valued LogicMulti-Valued Logic

Given: Given: a cover a cover FF of of and a cover and a cover DD of the don’t- of the don’t-care set d, care set d,

Find: Find: A minimum sum-of-products form for A minimum sum-of-products form for

Same problem as for two-valuedSame problem as for two-valued• Generate primes of (f+d)Generate primes of (f+d)• Generate covering tableGenerate covering table• Solve the covering table Solve the covering table (unate covering problem)(unate covering problem)

Same algorithms as for two-valuedSame algorithms as for two-valued (except for small (except for small details).details).

16

Applications of Multi-Valued Applications of Multi-Valued LogicLogic

Theorem (Hong):Theorem (Hong): minimizing a two-valued (n input) minimizing a two-valued (n input) (m (m output)output) logic function logic function g g is equivalent to minimizing a is equivalent to minimizing a single single binary-outputbinary-output MV-logic function: MV-logic function:

ff : {0,1} : {0,1} {0,1} {0,1} ... ... {0,…,m-1} {0,…,m-1} {0,1} {0,1}

Proof( sketch):Proof( sketch): Let g = {fLet g = {f00,…,f,…,fm-1m-1} be the multiple output function. } be the multiple output function. Consider the Consider the characteristiccharacteristic function f of the multiple output function f of the multiple output function, function, (defined on (n+1) variables with the last one, (defined on (n+1) variables with the last one, yy, , being multi-valued on {0,1,…,m-1} ) being multi-valued on {0,1,…,m-1} ) ::

1

0

( , ) 1 ( ) ( )m

ii

f x y y i f x

17

Applications of Multi-Valued Applications of Multi-Valued LogicLogic

Note:Note: an implicant of g an implicant of g (the multi-output (the multi-output function)function) is a cube c in the is a cube c in the xx-space where -space where each output is turned each output is turned onon only if f only if fii(c)=1. Any (c)=1. Any output not turned on means output not turned on means no information no information (not offset),(not offset), since the each output is the OR since the each output is the OR of all of its input cubes.of all of its input cubes.

XX ff1 1 ff2 2 ff3 3 ff4 4 ff5 5 ff66

x-cube 0 1 0 1 1 0x-cube 0 1 0 1 1 0

gg

18

Other Applications: Other Applications: Encoding ProblemsEncoding Problems

Other Applications:Other Applications:• Input Encoding problemInput Encoding problem

– bit-grouped PLA structurebit-grouped PLA structure

• Output encoding problem?Output encoding problem?– output phase optimization?output phase optimization?

• State encoding problemState encoding problem– Minimize symbolically to get constraints on a Minimize symbolically to get constraints on a

posssible binary encodingposssible binary encoding– solve constraints to derive binary codesolve constraints to derive binary code– Re-minimize binary problemRe-minimize binary problem– Implement in binaryImplement in binary

19

Multi-Valued Minimization Multi-Valued Minimization ExampleExample

1 2 3 1 2 3 1

1 2 3 1 2

2 3 1 2

3 1

3

1 2 3 1 2 3 1

2

0

1 3 1 2

1

3

3

2

2 3 2

X X X X X X X

X X X X X X X X X X X X

X X X X X X X X X X X

X X X X Xg

X

f

f

f

}1,0{}1,0{}2,1,0{}1,0{}1,0{}1,0{:

4321

: PPPPff

0 0 0 1 0 0 1 11 2 3 4 1 2 3 4

0 1 1 0 0 0 21 2 3 4

0 0 1 2 0 1

0 0 0 0 0 1 0 0 0 1 1 01 2 3 4 1 2 3

1

4 1 2 3 4

1 1 1 01 2 3 4

1 0

0 2 1 1 21 2 3 4

1 1

1 2 3 4 1 2

1 2 3 4 1 2 3 4

3 4

X X X X X X X X

X X X X X X X X X

X X X X X X X

X X X X X X X X X X X X

X X X X

X X

X X X

f

X X

X

20

Prime and irredundant SOP of f:Prime and irredundant SOP of f:

(five cubes 1+2+3+4+5)(five cubes 1+2+3+4+5)

Equivalent to:Equivalent to:

Example - after minimizationExample - after minimization

14

13

02

}1,0{1

}2,1{4

}1,0{3

02

01

}2,0{4

03

}1,0{2

01

}2,0{4

13

12

11

}1,0{4

13

12

01

XXXXXXXXXXXX

XXXXXXXXf

31213212

32213211

213213210

432

541

321

XXXXXXXf

XXXXXXXf

XXXXXXXXf

21

Example - after minimizationExample - after minimization

NoteNote: is not a prime of f: is not a prime of f00, but is a , but is a prime of f. Similarly for .prime of f. Similarly for .

321 XXX }2,0{4321 XXXX

321 XXX

ff00 f f11 f f22

22

Shannon CofactorShannon Cofactor

Note:Note: this agrees with “standard” cofactor in the case of this agrees with “standard” cofactor in the case of two-valuedtwo-valuedHintHint: check cases on d: check cases on dii, c, cii, e.g. if d, e.g. if dii=c=cii=1 (i.e. x=1 (i.e. xii in d and c) , then in d and c) , then

(c(cdd))i i == cci i ddi i = 2 = {0,1}= 2 = {0,1}

Rationale:Rationale: Only Only carecare about value of c on subspace given about value of c on subspace given by by dd. . ( d is don’t care)( d is don’t care)

Cofactor of cubeCofactor of cube cc with respect to cubewith respect to cube dd (c(cdd))

Note:Note:

otherwise

if

nnd dcdc

dcc

11

\i i id P d

23

Shannon Cofactor - ExampleShannon Cofactor - Example

Example:Example: space is {0,1} space is {0,1} {0,1,2} {0,1,2}

01

}2,1,0{2

01

}2,0{2

01 2

2}1,0{

1XXXXX XX

Cofactor of cover with respect to cube Cofactor of cover with respect to cube dd is ismccF 1

1 md d dF c c

Note:Note: Cofactor of a cover with respect to Cofactor of a cover with respect to another cover is not defined.another cover is not defined.

24

Shannon Cofactor-ExampleShannon Cofactor-Example

}1{

3}1{

2}2,1{

1}1{

3}0{

2}1,0{

1

}1{3

}1{2

}2{1

}0{3

}0{2

}0{1

}1{3

}0{2

}0{1

XXXXXX

XXXXXXXXXf dd

F = (f,r) and F = (f,r) and cubecube d = X d = X11{0,2}{0,2}

Consider the generalized cofactor: Consider the generalized cofactor: Co(F,d) = (fd,Co(F,d) = (fd,d, rd)d, rd)

Note:Note: We keep all the onset We keep all the onset (not in(not indd ) ) and project the care and project the care onset fd toonset fd tod. Also, as in the binary case, butd. Also, as in the binary case, butdf fØ

FF

Co(F,d)Co(F,d) ffdd

dd

ddf df

25

Shannon Cofactor Expansion Shannon Cofactor Expansion Theorem (General Case)Theorem (General Case)

Theorem:Theorem: Let f be any function and {c Let f be any function and {c11, …, c, …, ctt}} any set any set of cubes which of cubes which partitionpartition the input space: the input space:

jicc

c

ji

t

i

i

for

and 11

ThenThen

t

ic

iifcf

1

26

We immediately have:We immediately have:

Shannon Cofactor Expansion Shannon Cofactor Expansion Theorem (General Case)Theorem (General Case)

1 iff 1 for every , and icf f i

i.e. most Shannon cofactor results continue to i.e. most Shannon cofactor results continue to hold. However, hold. However, notenote , but , but

1

( )i

ti

ci

f c f

cc

f f cc

c f c f

27

Recursive Paradigm: Recursive Paradigm: Multi-Valued VersionMulti-Valued Version

1

1

Tautology( )

if 1 return true;

if 0 return false;

Choose ,..., such that

1

for (i 1,...,t)

if (

(don't need for

Tautol

this)

ogy( i

t

i j

t i

i

c

i j

f

f

f

c c

c c

c

f

c c

) false)

return false;

return true;

28

Still Open:Still Open:• Unate leaves Unate leaves (what does unateness mean?)(what does unateness mean?)• Splitting choice Splitting choice (i.e. which { c(i.e. which { ci i })})• Unate ReductionUnate Reduction

Recursive Paradigm: MV Recursive Paradigm: MV versionversion

29

Definition 1:Definition 1: f is said to be f is said to be weakly unateweakly unate in X in Xii if if there exists some value there exists some value jj, such that changing X, such that changing Xii from value = j to something else, does not cause from value = j to something else, does not cause f to decrease.f to decrease.

• Analog to unateness in two-valued case set j=0 Analog to unateness in two-valued case set j=0 and get monotone increasing; set j=1 and get and get monotone increasing; set j=1 and get monotone decreasingmonotone decreasing

In general:In general: detecting unateness is hard detecting unateness is hard (obviously)(obviously)

Special case:Special case: unate cover unate cover

Unateness: Multi-ValuedUnateness: Multi-Valued

( , , ) ( , , ), i if X j f X k k j

30

Definition 2:Definition 2: A cover F= c A cover F= c1 1 +…+ c+…+ ctt is said to be is said to be weakly unateweakly unate in X in Xii iff there is some iff there is some jj such that, for each such that, for each cube cube cckk, either:, either:

(monotone increasing from (monotone increasing from value j in variable Xvalue j in variable X ii))

Weakly-Unate CoverWeakly-Unate Cover

or

(universal set)

ki

ki i

j c

c P j (Xj (Xii))

31

Analogy to two-value:Analogy to two-value:• Rewrite (binary to MV)Rewrite (binary to MV)

ExampleExample


}1,0{

}1{

}0{

2 i

ii

ii

XXXXX

112

100

112

100

ii XX

112

011

011

ii XX

Here j=1 i.e. monotone increasing from Here j=1 i.e. monotone increasing from j=1 (monotone j=1 (monotone decreasingdecreasing in X in Xi i ))

Here j=0 i.e. monotone increasing from Here j=0 i.e. monotone increasing from j=0 (monotone j=0 (monotone increasingincreasing in X in Xi i ))

32

Easy to detect:Easy to detect:

Unate variables are those for whichUnate variables are those for which

(Just looking for a column with all 0’s, except for rows of all 1’s)(Just looking for a column with all 0’s, except for rows of all 1’s)


k

for each variable

for each cube c

(start with all values)

(if not all va if

\

lues )

(remove any val ue in )

i

i i

ki i

ki i i

i

k

X

v P

c P

v c cv

P

iv

33

1. throw out rows of all 1’s1. throw out rows of all 1’s

2. Look for column of all 0’s2. Look for column of all 0’s


j (Xj (Xii))

cc11 01010 01010 cc22 00100 00100 . . ........ 01.. 01.. cct-1t-1 11111 11111 cctt 11111 11111

34

F is F is weakly-unateweakly-unate in every variable. in every variable.

ExampleExample

}4,3,2,1,0{321

}3,2,0{3

}4,3,2,1,0{2

}4{1

}3,1,0{3

}4,1{2

}3,2{1

}4,3,2,1,0{3

}2{2

}3,1{1

}3,1{3

}4,3{2

}2,1{1

}3,2,1,0{3

}4{2

}4,3,2,1,0{1

PPPXXX

XXXXXXXXXXXXf

}4{},0{ 321 vvv

XX1 1 XX2 2 XX33

cc11 cc22 cc33 cc44 cc55

35

Theorem 1:Theorem 1: Let {c Let {c11, …, c, …, ctt} be a cube } be a cube partitionpartition as in Shannon as in Shannon expansion theorem. Then:expansion theorem. Then:

Proof:Proof: follows two-valued case exactly. follows two-valued case exactly.

Application to TautologyApplication to Tautology

1 iff 1 for every kcf f k

1 21 2

,

1

tt

c c c

i j

i

f c f c f c f

c c i j

c

(1)(1)

36

Theorem 2:Theorem 2: Let f be Let f be weakly unateweakly unate in variable x in variable xi i from from value j. Then:value j. Then:

Analogous to for monotone increasing Analogous to for monotone increasing (from 0).(from 0).

Monotone TheoremMonotone Theorem

iXXPkff k

ij

i all for }{}{

xxff

Proof:Proof:

We first show that is independent of . Let be any

cover for . Then is a cover for . Each cube is

either or has all 1's in the (When cofactoring bycube

part.

{k}i

{k} {k} {k}i i i

lix

l l

x x x

i

f x {c }

f {c } f c

x , 0 , 1 all 1's, because we add 1's

to all values \ ).

{k} l li ik ik

i

x c cP {k}

(2)(2)

37

Monotone TheoremMonotone Theorem

1

0

Let be an arbitrary point in the space excluding .

Let . Since is weakly unate from ,

. By Shannon expansion

so

i

{l}i

i

{j} {k}i i

p{l}i x

l

v x

u x v,w x v f j

f(u) f(w)

f x f

{ {

(since is indepen

( ) ( ) ( ) ( )

. Since is an arbitrary

point,

dent

t

of

h

)

en

j} k}i i

{j} {k}i

}

i

{ki

ix

x x

x x

f x

f u f v f w f v

v

f f

38

Theorem 3:Theorem 3: (unate reduction)(unate reduction) f is weakly-unate in X f is weakly-unate in Xii, , and the “unate value” is j. Then f = 1 iff and the “unate value” is j. Then f = 1 iff

Weakly Unate Reduction Weakly Unate Reduction TheoremTheorem

Proof:Proof:

{ }

{

1{ }

0

Write

from (1), 1 iff 1 for every , and hence f 1 implies 1

If 1, then by (2) 1 and hence 1

( )

( ) ,

and so

i

ki

{k} { j }i i

j } {k} { j } {k}i ii i

pk

i Xk

x x

x xx x

f X f

f f k f

f f f f

f 1 by (1).

1{ j }iX

f

39

Note this is exactly two-valued theorem:

1 1 1

except that here we are doing one unate variable at a time.

Also note that

01101

0101 ...

i

A XA B

T B

X

F

{0}

0

0... ... ...11111

0011011111

11111

11111

because does not intersect with the first 3 cubes.

i

i

X

{ }i

X

F

x

Weakly Unate Reduction Weakly Unate Reduction TheoremTheorem

40

Definition 3:Definition 3: Cover c Cover c1 1 + … + c+ … + ctt is weakly-unate iff it is weakly-unate iff it is weakly-unate in is weakly-unate in allall variables. variables.

Theorem 4 :Theorem 4 : cc11+ …+c+ …+ctt weakly-unate then c weakly-unate then c11+ …+c+ …+ctt =1 iff c=1 iff cjj=1 for =1 for somesome cube j. cube j.

Proof.Proof. Follows from reduction theorem. Follows from reduction theorem.

Thus for weakly unate cover, can tell immediately.Thus for weakly unate cover, can tell immediately.

Tautology for Weakly Unate Tautology for Weakly Unate CoverCover

111 means nPPj j

nc c X X

1111110110011

111110101011

111100110101

1100110110011

101110101011

010100110101

Vertex Vertex 1000 0100 01001000 0100 0100 not covered. not covered.

41

Reduction in One StepReduction in One Step

11Form cube where

(i.e.if is "binate")

(unate part o

(Unate reduction. Gets r

f variable

id of cubes with 0 in any of

{ } par

)

1 iff 1

ts)

nμ μn

i i ii

i i

i

c

c X X

P v Xμ

v X

v

f f

c exactly as in two-valued algorithmc exactly as in two-valued algorithm

c is cube of unate variables, e.g.c is cube of unate variables, e.g.

then Athen Acc=0. Hence f=0. Hence fcc=(T B).=(T B).

BT

XAf

0110102122021201

cA

42

Revised TautologyRevised Tautology

1

Tautology( )

while

if 1 return true;

if 0 return false;

Choose ,.

( is weakly unate in some variables)

(some cube partition of space)..,

for ( 1 )

c

t

F

F F

F

F

c c

i ,...,t

F

if (Tautology( ) false)

return false;

return true;

icf

Left open:Left open: how to split?how to split?

i.e. how to choose ci.e. how to choose c11, …, c, …, ctt where c where ci i c cj j = = , and , and ccii =1 =1..

43

““Split by value” Split by value”

– Gets rid of variable XGets rid of variable Xii in a single step. in a single step.

Methods of SplittingMethods of Splitting

}0{iX

}1{iX }2{

iX

}3{iX

44

““Split by parts”Split by parts”q, s partition Pq, s partition Pi i (e.g. q={0,1}, s={2,3}(e.g. q={0,1}, s={2,3}

– May get to unate leaves May get to unate leaves (somewhat)(somewhat) more quickly more quickly– More freedom to choose good partitions -don’t need to More freedom to choose good partitions -don’t need to

entirely eliminate variable Xentirely eliminate variable Xii at a node before splitting on X at a node before splitting on Xkk..

In practice, “split by parts” is usedIn practice, “split by parts” is used

Methods of SplittingMethods of Splitting

qiX s

iXq s

ii

q si i XX

f X f X f

45

Cover F = Cover F = 1 1 +…+ +…+ |F||F|

Goal:Goal: get to weakly unate leaves as fast as possible get to weakly unate leaves as fast as possible

Definition 4:Definition 4: ActiveActive value of variable X value of variable X ii::

(Any value (Any value k of k of X Xii with all 1’s in column is with all 1’s in column is notnot activeactive))

Choose variable with most active valuesChoose variable with most active values

(Note: all (Note: all inactiveinactive values can be equivalently grouped values can be equivalently grouped into into oneone value.) value.)

Choice of Splitting VariableChoice of Splitting Variable

{ | ( ) such that }kij k j

46

Tie breaks Tie breaks (|F| is number of cubes)(|F| is number of cubes)– Variables Variables ii maximizing maximizing

(“Smallest” variable = most 0’s in columns)(“Smallest” variable = most 0’s in columns)

– Variables minimizingVariables minimizing

(least “2’s”)(least “2’s”)

Choice of Splitting VariableChoice of Splitting Variable

| | | |

1 1

| \ | | |F F

j ji i i

j j

P

||

1

F

j

jiiP

47

Cover F=cCover F=c11+…+ c+…+ ctt, variable X, variable Xii

• Goal: Like to find Goal: Like to find partitionpartition q, s of P q, s of Pii such that: such that:

is minimized.is minimized.

• Hard problem! Use heuristicHard problem! Use heuristic• ““Fast to compute” more important than quality...Fast to compute” more important than quality...

Choice of PartitionChoice of Partition

| | | |q sii XX

F F

qiX s

iX

F

qiX

F siX

Fs not actives not active q not activeq not active

Heuristic:Heuristic:

• m m activeactive values in X values in Xii

• q gets first m/2 active q gets first m/2 active values, s the restvalues, s the rest

This reduces the number This reduces the number of active values on of active values on each side by halfeach side by half

48

End of lecture 6End of lecture 6

49

Weakly-unate good enough for Weakly-unate good enough for tautologytautology based based algorithms, but…algorithms, but…

• F weakly-unate F weakly-unate F Fcc weakly-unate weakly-unate

ExampleExample: F is weakly unate cover.: F is weakly unate cover.

Strongly Unate FunctionsStrongly Unate Functions

XX1 1 XX2 2 XX3 3 XX3 3

10 11 11 111 10 11 11 111 FF = 11 10 10 100 = 11 10 10 100 11 11 10 010 11 11 10 010

cc = 11 11 10 110 = 11 11 10 110

10 11 11 111 10 11 11 111 FFcc = 11 10 11 101 = 11 10 11 101 11 11 11 011 11 11 11 011

FFcc is not weakly unate in X is not weakly unate in X33. (But in . (But in this example, fthis example, fcc is!) is!)

(However, I think this also holds (However, I think this also holds for f and ffor f and fcc as well as well

i.e. f can be weakly unatei.e. f can be weakly unate

in a variable but fin a variable but fc c may not be).may not be).

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F weakly-unate F weakly-unate does not implydoes not imply every prime of f every prime of f essential. essential. ExampleExample: f = { p1,p2,p3,p4,p5 }: f = { p1,p2,p3,p4,p5 }

p1,…, p5 are all primes.p1,…, p5 are all primes.– P1 essentialP1 essential– p2 nonessentialp2 nonessential– p3 essentialp3 essential– p4 nonessentialp4 nonessential– p5 essentialp5 essential


p1p1 11111 00001 11110 11111 00001 11110

p2p2 01100 00011 01010 01100 00011 01010

p3p3 01010 00100 11111 01010 00100 11111

p4p4 00110 01001 11010 00110 01001 11010

p5p5 00001 11111 10110 00001 11111 10110

Weakly unate in all variablesWeakly unate in all variables

A column of all 1’s indicates a A column of all 1’s indicates a value that is value that is not activenot active..

5 4 3 2 1Cubes 5

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f weakly-unate f weakly-unate doesdoes not not implyimply ff weakly-unate. weakly-unate.

All these are primes.All these are primes.

Need stronger condition...Need stronger condition...


00110 01000 0010100110 01000 00101

11111 00001 0000111111 00001 00001

00001 11110 0100100001 11110 01001

f f == 01100 00010 1010101100 00010 10101

11000 11000 1111111000 11000 11111

10110 10100 1111110110 10100 11111

10010 10010 1111110010 10010 11111

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Stongly Unate functionsStongly Unate functions

Definition 5:Definition 5: f is f is strongly-unatestrongly-unate in X in Xii iff there is some iff there is some

total order total order << on P on Pii such that, for j<k in P such that, for j<k in Pii

Thus “increasing” XThus “increasing” Xii (from value j to value k, if j<k)(from value j to value k, if j<k) doesn’t decrease f.doesn’t decrease f.

}{}{ ki

ji XX

ff

Example:Example: strongly unate cover strongly unate cover (order is from left to right i.e. (order is from left to right i.e. 1<2<3 on 1<2<3 on allall variables) variables)

Can detect strongly unate cover by the existence of a value Can detect strongly unate cover by the existence of a value order for each Porder for each Pii, where the 0’s are on the left for all cubes., where the 0’s are on the left for all cubes.

111 011 1111111 011 1111

f f == 011 001 1111011 001 1111

001 111 0011001 111 0011

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• f strongly-unate f strongly-unate f weakly-unate f weakly-unate• f strongly-unate f strongly-unate ff strongly-unate strongly-unate

• f strongly-unate f strongly-unate f fcc strongly-unate strongly-unate

• f strongly-unate f strongly-unate every prime of f essential.. every prime of f essential..

But:But:• Weakly-unate applies to a cover more oftenWeakly-unate applies to a cover more often

– easier to computeeasier to compute– good enough for good enough for tautologytautology based algorithms based algorithms

Strongly unate Strongly unate notnot used in two-level logic used in two-level logic minimization algorithms (so far).minimization algorithms (so far).

Strongly Unate Functions: Strongly Unate Functions: PropositionsPropositions

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Use basic logic synthesis algorithmUse basic logic synthesis algorithm

1.1. Generate all primesGenerate all primes

2.2. Form covering tableForm covering table

3.3. Solve covering tableSolve covering table

Steps (2)-(3) are same as in binary caseSteps (2)-(3) are same as in binary case

Exact MinimizationExact Minimization

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Prime GenerationPrime GenerationTheorem 5Theorem 5 (Prime merging)(Prime merging) Let f be any function, Let f be any function,

l, r be any cubes such that ll, r be any cubes such that lr=r= and l+r=1. and l+r=1.Then the primes of f are the Then the primes of f are the maximalmaximal cubes among: cubes among:– the primes of lfthe primes of lfl l = = l l primes of f primes of fll, and, and

– the primes of rfthe primes of rfrr = = r r primes of f primes of frr, and, and

– the cubes of the cubes of ccllccrr where c where cll primes of lf primes of lfl l and cand crr primes of primes of rfrfrr..

(( stands for stands for consensusconsensus))

1 11

( )n

n nii

ic d c d c d c d

NoteNote: this easily specializes to binary theorem when all values are binary. : this easily specializes to binary theorem when all values are binary.

How do we get all the primes at a leaf? (see Rudell paper)How do we get all the primes at a leaf? (see Rudell paper)

NoteNote: if c and d are distance 2, then result is : if c and d are distance 2, then result is

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Use Espresso-IIUse Espresso-II• Irredundant, Reduce, LastgaspIrredundant, Reduce, Lastgasp

(Unchanged because (Unchanged because tautologytautology based) based)• Essential Primes essentially unchangedEssential Primes essentially unchanged

(minor technical differences in expression)(minor technical differences in expression)• ExpandExpand

Minor difference from Espresso-IIMinor difference from Espresso-IICan’t use blocking matrix BCan’t use blocking matrix Bcc when expanding when expanding

cube c.cube c.– May be able to expand cMay be able to expand cii even if column j is in a minimal even if column j is in a minimal

column cover of Bcolumn cover of Bcc. .

Heuristic MinimizationHeuristic Minimization

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Expand example:Expand example:Consider the case jConsider the case jccii, j, jrrii

kk for each cube k of the offset, j can be for each cube k of the offset, j can be added to cadded to cii even when i is in a minimal column cover of B even when i is in a minimal column cover of Bcc. Adding j . Adding j to cto cii leaves c leaves cii r rii

kk = = unchanged. unchanged.

This implies that This implies that BBcckiki = 1 because c= 1 because cii r rii

kk = = . But we still may be . But we still may be expand to change to expand to change to

Heuristic MinimizationHeuristic Minimization

...X...Xi i ......

11100c

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Multi-value EXPANDMulti-value EXPANDWe build the blocking We build the blocking functionfunction g(y) using the cover of g(y) using the cover of

the offset R = { rthe offset R = { r1 1 +…+ r+…+ r|R| |R| }. To expand cube c, let }. To expand cube c, let variable yvariable ykk

jj denote that in positional notation, the denote that in positional notation, the expanded cube has a 1 in value k of variable xexpanded cube has a 1 in value k of variable x jj . . Then intersects rThen intersects rii if if

Hence, intersects the offset is given by the functionHence, intersects the offset is given by the function

n

j

p

k

kj

kj

ij

yr1

1

0

1)(

||

1 1

1

0

)()(R

i

n

j

p

k

kj

kj

ij

yryg

c c

c

Note: Note: this is monotone this is monotone increasingincreasing in y in y

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Multi-value EXPANDMulti-value EXPAND

Its complement g(y) Its complement g(y) (monotone decreasing)(monotone decreasing) gives all legitimate expansions of c. A prime gives all legitimate expansions of c. A prime of g gives a prime expansion of c. of g gives a prime expansion of c.

Note:Note: g is a binary function, and is unate. It is g is a binary function, and is unate. It is convenient to obtainconvenient to obtaingg(y) in SOP form by (y) in SOP form by complementing (complementing (twicetwice))

using our unate complementer.using our unate complementer.

n

j

p

k

kj

kj

ij

yr1

1

0

)(

1| |

1 1 0

( ) ( )jpR n

i k kj j

i j k

g y r y

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• Adding multi-valued permits Adding multi-valued permits – minimizing multiple-output functions, minimizing multiple-output functions, – solving encoding problems, and solving encoding problems, and – other applications which are naturally multi-valued.other applications which are naturally multi-valued.

• Minimization fundamentals basically unchangedMinimization fundamentals basically unchanged• Details changed inDetails changed in

– ConsensusConsensus– All-prime generation (minor)All-prime generation (minor)– Unateness Unateness (2 kinds now - largest change)(2 kinds now - largest change)– Essential primes Essential primes (to accommodate consensus)(to accommodate consensus)– Expand Expand (blocking matrix no longer an efficiency gain)(blocking matrix no longer an efficiency gain)

• Most details in paper by Rudell and Sangiovanni on Most details in paper by Rudell and Sangiovanni on MV optimization.MV optimization.

ConclusionsConclusions

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Multi-Valued Logic