New Hiera rchies of Rep resentations RM'97, Septemb er 1997 - - PowerPoint PPT Presentation

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New Hiera rchies of Rep resentations RM'97, Septemb er 1997 - - PowerPoint PPT Presentation

New Hiera rchies of Rep resentations RM'97, Septemb er 1997 1 ' $ NEW HIERARCHIES OF AND/EX OR TREES, DECISION DIA GRAMS, LA TTICE DIA GRAMS, CANONICAL F ORMS, AND REGULAR LA YOUTS Ma rek P erk o wski, Lech


slide-1
SLIDE 1 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 1 ' & $ % NEW HIERARCHIES OF AND/EX OR TREES, DECISION DIA GRAMS, LA TTICE DIA GRAMS, CANONICAL F ORMS, AND REGULAR LA YOUTS Ma rek P erk
  • wski,
Lech Jozwiak y, Rolf Drechsler +, P
  • rtland
St. Univ. Dept. Electr. Engn., P
  • rtland,
USA y F acult y
  • f
Electr. Engng., Eindhoven Univ.
  • f
T echn., The Netherlands, + Inst.
  • f
Comp. Sci., Alb ert-Ludwigs Univ., F reiburg in Breisgau, Germany ,
slide-2
SLIDE 2 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 2 ' & $ % MAIN CONTRIBUTIONS OF THIS P APER
  • Generaliz
ati
  • ns
  • f
the Generalized Kroneck er rep resentations.
  • New
canonical AND/EX OR fo rms.
  • Interrelated
hiera rchies
  • f
canonical AND/EX OR trees, decision diagrams, lattice diagrams, canonical fo rms and regula r la y
  • uts.
  • The
new diagrams and fo rms can b e used fo r synthesis
  • f
quasi-mini mum Exclusive Sum
  • f
Pro ducts (ESOP) circuits,
  • Highly
testable multi-level AND/EX OR circuits, that through the "EX OR-related technology mapping" a re adjusted to AND/OR/EX OR custom VLSI, standa rd cell,
  • r
  • ther
technologies.
  • Applicati
  • ns
in Fine Grain Field Programmabl e Gate Arra ys.
slide-3
SLIDE 3 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 3 ' & $ % MAIN CONTRIBUTIONS OF THIS P APER (cont)
  • The
new diagrams can rep resent la rge functions.
  • Lattice
Hiera rchy generalizes and extends the Universal Ak ers Arra y to expansions
  • ther
than Shannon and neighb
  • rho
  • ds
  • ther
then 2-inputs, 2-outputs.
  • These
Lattice Diagrams nd many applicati
  • ns
in la y
  • ut-driven
logic synthesis, pa rticula rly fo r XILINX FPGAs.
slide-4
SLIDE 4 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 4 ' & $ % A CKNO WLEDGE CONTRIBUTIONS OF ZHEGALKIN
  • Zhegalkin
in 1927 discovered the fo rms, no w attributed to Reed and Muller and invented b y them in 1954.
  • His
contributi
  • ns
a re not p rop erly ackno wledged.
  • As
a communit y , it is fair to ackno wledge him, as w e had ackno wledged Davio, Reed and Muller.
  • W
e p rop
  • se
to call all these new fo rms that p rop erly include b
  • th
KRO and GRM, as w ell all the future AND/EX OR fo rms includin g these t w
  • families
concurren tly , the Zhegalkin fo rms.
slide-5
SLIDE 5 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 5 ' & $ % PLAN
  • V
a rious kinds
  • f
trees with multi-va riabl e no des.
  • Review
the concept
  • f
the Generalize d Kroneck er T rees.
  • Generaliz
ed Kroneck er Diagrams and their "pseudo"-lik e generaliza tion s.
  • Generaliz
ed Kroneck er F
  • rms.
  • Extended
Green/Sasao hiera rchie s
  • f
trees, fo rms and Diagrams.
  • New
Hiera rchy
  • f
Zhegalkin Lattice Diagrams.
  • Current
and F uture w
  • rk.
  • Conclusions.
slide-6
SLIDE 6 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 6 ' & $ % KRONECKER F AMILIES OF REPRESENT A TIONS
  • Decision
Diagrams (DDs).
  • Used
in logic synthesis, verication and simulation .
  • Main
internal rep resentati
  • n
  • f
functions,
  • n
which all meaningful
  • p
erations a re executed.
  • DDs
  • riginate
from bina ry decision trees (bina ry expansion trees, Shannon trees), which in turn a re based
  • n
the fundamental expansion theo rem
  • f
Shannon that is applied in every no de
  • f
a tree.
  • Every
no de is related to
  • ne
input va riable
  • f
the function.
  • "Reduced",
and "Ordered".
  • Disadvantage
  • la
rge functions cannot b e rep resented.
slide-7
SLIDE 7 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 7 ' & $ %
  • Generalise
the concept
  • f
a bina ry tree!!
  • Green/Sasao
hiera rchy
  • f
rep resentati
  • ns
cha racterize s these rep resentation s in an unifo rm w a y that w e will use here in
  • ur
generaliza tion s.
  • All
these rep resentation s can b e used in the rst stage
  • f
logic synthesis
  • the
"technology indep end en t, EX OR synthesis" phase, which is next follo w ed b y the "EX OR-related technology mapping".
slide-8
SLIDE 8 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 8 ' & $ % D A VIO EXP ANSIONS
  • The
hiera rchy is based
  • n
three expansions: f (x 1 ; x 2 ; :::; x n ) = x 1 f (x 2 ; :::; x n )
  • x
1 f 1 (x 2 ; :::; x n ) in sho rt f = x 1 f
  • x
1 f 1 ; called Shannon, (1:1) f (x 1 ; x 2 ; :::; x n ) = 1
  • f
(x 2 ; :::; x n )
  • x
1 f 2 (x 2 ; :::; x n ) in sho rt f = f
  • x
1 f 2 ; called P
  • sitive
Davio, (1:2) f (x 1 ; x 2 ; :::; x n ) = 1
  • f
1 (x 2 ; :::; x n )
  • x
1 f 2 (x 2 ; :::; x n ) in sho rt f = f 1
  • x
1 f 2 ; called Negative Davio, (1:3) where f is f with x 1 replaced b y (negative cofacto r
  • f
va riable x 1 ), f 1 is f with x 1 replaced b y 1 (p
  • sitive
cofacto r
  • f
va riable x 1 ), and f 2 = f
  • f
1 .
slide-9
SLIDE 9 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 9 ' & $ % ORDERED EX OR-BASED REPRESENT A TIONS
  • By
applying recursively expansions (1.1)
  • (1.3)
(o r any subset
  • f
them) to the function va rious t yp es
  • f
bina ry decision trees can b e created.
  • The
concepts
  • f:
{ Shannon T rees, P
  • sitive
Davio T rees, { Negative Davio T rees, { Kroneck er T rees, { Reed-Muller T rees, { Pseudo-Kroneck er T rees, { Pseudo Reed-Muller T rees, { as w ell as
  • f
the co rresp
  • ndin
g decisio n diagrams, { and attened (t w
  • -level)
canonical fo rms.
slide-10
SLIDE 10 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 10 ' & $ % FREE EX OR-BASED REPRESENT A TIONS
  • F
ree Kroneck er T rees use S, pD and nD no des disrega rdi ng any
  • rder
  • f
va riables and expansions (Ho/P erk
  • wski).
  • A
t every tree level, dierent va riables and expansions can
  • ccur.
  • Thus,
the
  • rder
  • f
va riables in every b ranch can b e dierent, and such diagrams a re also called non-o rdered.
  • Simila
rly ,
  • ne
can also dene F ree Bina ry Decision T rees (leading to F ree BDDs) and F ree P
  • sitive
Davio T rees (leadin g to F ree FDDs).
  • F
ree Kroneck er T rees lead to F ree KFDDs (FKFDDs) (Ho/P erk
  • wski).
slide-11
SLIDE 11 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 11 ' & $ % OUR NEW GENERALIZA TIONS
  • Use
bina ry
  • p
erato rs, so useful fo r logic synthesis.
  • Generalized
Kroneck er T rees, F
  • rms
and Decision unify Kroneck er and Generali zed Reed Muller rep resentatio ns (P erk
  • wski/Jozwiak/Drechsl
er ).
  • Here
w e further generalize the Generali zed Kroneck er T rees, F
  • rms
and Decision Diagrams.
  • These
rep resentati
  • ns
a re b etter then all Kroneck er-lik e rep resentations b ecause they did not allo w to create GRM fo rms after attening.
  • W
e create an enhanced Green/Sasao hiera rchy .
  • Because
  • f
the sup erio ri
  • rit
y
  • f
new rep resentations, the circuits a re also never w
  • rse
than the AND/EX OR circuits
  • btained
from the p revious rep resentation s, (includ in g the GKTs).
slide-12
SLIDE 12 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 12 ' & $ %

P1

X1 X2 X X X 3 4 5

P2 pD pD pD pD pD pD P2 P2 P2

GKT

Figure 1: Generali zed Kroneck er T ree
slide-13
SLIDE 13 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 13 ' & $ % GENERALIZED REED MULLER EXP ANSION WITH CONST ANT COEFFICIENTS
  • f
(x 1 ; : : : ; x n ) = a
  • a
1 ^ x 1
  • a
2 ^ x 2
  • :
: :
  • a
n ^ x n
  • a
12 ^ x 1 ^ x 2
  • a
13 ^ x 1 ^ x 3
  • :
: :
  • a
n1;n ^ x n1 ^ x n
  • :
: :
  • a
12::: n ^ x 1 ^ x 2 ^ x 3 : : : ^ x n (2:1) where a i 's a re either
  • r
1, and ^ x denotes va riable x
  • r
its negation, x.
  • By
assigning a va riable
  • r
a negation
  • f
a va riable to each
  • f
the ^ x i in (2.1) w e create 2 n2 n1 dierent expansion fo rmulas.
  • Each
  • f
them is called a p
  • la
rit y expansion, i.e., an expansion
  • f
a certain p
  • la
rit y .
slide-14
SLIDE 14 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 14 ' & $ % GRM EXP ANSION WITH FUNCTIONAL COEFFICIENTS
  • In
every no de
  • f
GKT, the fo rmula fo r expansion (2.2) is applied which generalizes the fo rmula (2.1) fo r a Generalized Reed-Muller expansion.
  • The
expansion fo rmula (2.2) fo r function f (x 1 ; x 2 ; :::; x m ; :::; x n ) is the same as GRM expansion, (2.1), with resp ect to va riables x 1 ; :::; x m and co ecients a i replaced with subfunctions S F i
  • f
remaining va riables x m+1 ; :::x n : f (x 1 ; x 2 ; :::; x m ; :::; x n ) = S F (x m+1 ; :::; x n )
  • ^
x 1 S F 1 (x m+1 ; :::x n ) ^ x 2 S F 2 (x m+1 ; :::; x n )
  • :
: :
  • ^
x m S F m (x m+1 ; :::; x n )
  • ^
x 1 ^ x 2 S F 12 (x m+1 ; :::; x n ) ^ x 1 ^ x 3 S F 13 (x m+1 ; :::; x n )
  • :
: : ^ x m1 ^ x m S F m1;m (x m+1 ; :::; x n )
  • :
: :
  • ^
x 1 ^ x 2 ^ x 3 : : : ^ x m S F 12:::n (x m+1 ; :::; x n ) (2:2)
slide-15
SLIDE 15 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 15 ' & $ % GRM EXP ANSION (cont) where the so-called Data F unctions S F i a re calculated as co
  • rdinates
  • f
a vecto r CV = M 1
  • FV,
in which: F V (x m+1 ; :::; x n ) is a vecto r
  • f
cofacto rs
  • f
F with resp ect to va riables from the set fx 1 ; :::; x m g.
  • A
2 m
  • 2
m nonsingu l a r matrix M has as its columns all the p ro ducts
  • f
literals ^ x i that have b een used in
  • ne
  • f
2 m2 m1 pa rticula r p
  • la
rit y expansion fo rms sp ecied b y (2.2).
  • The
columns
  • f
the matrix a re linea rl y indep endent with resp ect to the bit-b y-bit exo ring
  • p
eration.
slide-16
SLIDE 16 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 16 ' & $ % Denition 1. The Generalised Kroneck er T ree is a multi-b ranch tree created as follo ws: 1) The set
  • f
all n input va riables is pa rtitioned into disjoint and nonempt y subsets S j such that the union
  • f
all these subsets fo rms the initial set. (If each subset includes just a single va riable, the tree reduces to the sp ecial case
  • f
a KRO tree. If there is
  • nly
  • ne
subset that includes all va riables, the tree co rresp
  • nds
to the sp ecial case
  • f
a GRM.) 2) The sets a re
  • rdered,
each
  • f
them co rresp
  • nds
to a level
  • f
the tree. 3) When the set has
  • ne
va riable: S, nD,
  • r
pD expansion is selected fo r all its no des. 4) When the set is multi-va riable: the fo rmula fo r expansion (2.2) is applied with the same p
  • la
rit y fo r all va riables from S j .
slide-17
SLIDE 17 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 17 ' & $ % PSEUDO GENERALIZED KRONECKER TREE Denition 2. The Pseudo Generalised Kroneck er T ree is a tree with multi-va riabl e expansion no des created as follo ws 1) The set
  • f
all n input va riables is pa rtitioned to disjoint and nonempt y subsets S j such that the union
  • f
these subsets fo rms the initial set (If each subset has just a single va riable, the sp ecial case
  • f
PKRM is considered .) 2) F
  • r
every no de
  • f
the tree, in a level with va riable subset S j , any GRM expansion fo r all va riables from S j can b e p erfo rmed.
slide-18
SLIDE 18 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 18 ' & $ %

nD S pD nD pD S pD nD S level of x1 level of x2, x4 level of x3 f f f 1 x1 x1 1 x2 x4 x2 x4 1 x2 x4 x2 x4 1 x3 1 x3 x3 x3

  • f polarity (x2x4,x4, x2)

=[11,0,1]

  • f polarity (x2x4,x4, x2)

=[00,1,0] GRM(2)-13 expansion GRM(2)-2 expansion

Figure 2: Example
  • f
Pseudo Generalized Kroneck er T ree
slide-19
SLIDE 19 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 19 ' & $ % NOT A TION
  • Numb
er 13 in expansion name (p
  • la
rit y) "GRM(2)-13" is a natural numb er co rresp
  • nding
to the bina ry numb er 1101, called a p
  • la
rit y.
  • The
expansion
  • f
the no de GRM(2)-13 is describ ed b y the follo wing fo rmula: f (x 2 ; x 3 ; x 4 )= S F (f ) 1 (x 3 )x 2 S F (f ) x 2 (x 3 ) x 4 S F (f ) x 4 (x 3 )x 2 x 4 S F (f ) x 2 x 4 (x 3 ) where notation S F (f ) i (X ) denotes function S F i , with a rguments from the set X
  • f
va riables, applied to a rgument function f .
  • The
GRM expansion in no de GRM(2)-2 (fo r p
  • la
rit y [00,1,0]) is describ ed b y the fo rmula: f 1 (x 2 ; x 3 ; x 4 ) =S F (f 1 ) 1 (x 3 )
  • x
2 S F (f 1 ) x 2 (x 3 )
  • x
4 S F (f 1 ) x 4 (x 3 )
  • x
2 x 4 S F (f ) x 2 x 4 (x 3 )
slide-20
SLIDE 20 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 20 ' & $ %

PGKT

P1

X1 X2 X X X 3 4 5

pD pD pD P1 P12 P11 P3 nD S S

Figure 3: Pseudo Generaliz ed Kroneck er T ree
slide-21
SLIDE 21 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 21 ' & $ % MIXED PSEUDO GENERALISED KRONECKER TREE Denition 3. The Mixed Pseudo Generalised Kroneck er T ree (MPGKT) is any multi-b ranc h tree created as follo ws. 1) The set
  • f
all n input va riables is pa rtitioned to disjoint subsets (blo cks) S j
  • f
va riables. 2) F
  • r
every multi-va riab le set, either apply to a ro
  • t
no de in the level an a rbitra ry GRM expansion
  • f
all its va riables
  • r
create a subtree
  • f
single-va riable expansions
  • f
va riables from S j . Ordered sets S j a re assumed. F
  • r
a single-va riabl e level
  • f
the tree, any combinati
  • n
  • f
the S, nD and pD no des can b e applied.
slide-22
SLIDE 22 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 22 ' & $ %

MPGKT

X1

P3 S P3

X2 X3 X X 2 3

pD S nD

X4 X5

P8 P10

Figure 4: Mixed Pseudo Generali zed Kroneck er T ree
slide-23
SLIDE 23 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 23 ' & $ % ORDERED GENERALISED KRONECKER TREE
  • Denition
5. The Ordered Generalised Kroneck er T ree (OGKT) is any multi-b ranc h tree created as a F ree T ree, with the additional constraint that every b ranch has the same
  • rder
  • f
va riables.
  • In
OGKT the sets
  • f
va riables in dierent b ranches ma y have dierent sizes and
  • verlap,
but the
  • rder
must b e the same.
  • F
  • r
instance, in
  • ne
b ranch the rst set is fx ; x 1 ; x 2 g the second set is fx 3 ; x 4 g, and the third set is fx 5 ; x 6 g. In another b ranch the sets a re: fx g, fx 1 g, fx 2 ; x 3 g, and fx 4 ; x 5 ; x 6 g.
slide-24
SLIDE 24 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 24 ' & $ %

P10 P10

X1

P3 S

X2 X X 2 3

pD

X5

nD S

X 3 X4

nD S nD nD

OGKT

X4 X5

Figure 5: Ordered Generalize d Kroneck er T ree
slide-25
SLIDE 25 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 25 ' & $ % FREE GENERALIZED KRONECKER TREES Denition 4. The F ree Generalised Kroneck er T ree (F GKT) is any multi-b ranch tree created as follo ws:
  • F
  • r
every no de
  • f
the tree, an a rbitra ry size subset
  • f
input va riables can b e selected.
  • F
  • r
single va riable sets
  • an
S, pD
  • r
nD no de can b e created, fo r multi-va riabl e sets
  • the
GRM expansion
  • f
its va riables
  • f
any p
  • la
rit y is calculated .
  • The
levels a re no mo re asso ciated with va riables
  • r
their sets, va rious lo cal
  • rders
and pa rtitions
  • f
va riables ma y exist in the b ranches.
slide-26
SLIDE 26 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 26 ' & $ %

P10 P10

X1

P3 S

X X X 2 3

pD

X

nD S

X 3 X

nD S nD nD

X4 X5

FGKT

5 2 4

Figure 6: F ree Generalized Kroneck er T ree
slide-27
SLIDE 27 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 27 ' & $ % ZHEGALKIN F ORMS
  • Zhegalkin
fo rms a re all AND/EX OR fo rms that p rop erly include b
  • th
the KRO and the GRM fo rms.
  • All
Zhegalkin fo rms a re dened here fo r the rst time.
  • The
simplest Zhegalkin fo rm is the Generalized Kroneck er fo rm.
  • Of
course, every pa rticula r GRM
  • r
KRO fo rm, b eing a sp ecial case
  • f
a Generalized Kroneck er fo rm, is in this sense also a Zhegalkin fo rm.
  • W
e p rop
  • se,
ho w ever, that when w e will b e talking ab
  • ut
existing families
  • f
fo rms, w e will k eep the kno wn names without the name \Zhegalkin" (Simil a rl y , every FPRM fo rm is a Kroneck er fo rm, but the Fixed-P
  • la
rit y Reed-Muller family
  • f
fo rms is distinguished from the b roader Kroneck er family
  • f
fo rms).
  • The
fo rms p resented here a re straightfo rw a rd generalization
  • f
the kno wn families.
slide-28
SLIDE 28 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 28 ' & $ % DEFINITION OF ZHEGALKIN F ORMS
  • Denition
7. The canonical AND/EX OR F
  • rm
co rresp
  • ndi
ng to each
  • f
the ab
  • ve
dened t yp es
  • f
trees, the Zhegalkin F
  • rm,
is
  • btained
b y attening the resp ective tree.
  • Flattening
means nding the AND-terms b y follo wing all the paths from the ro
  • t
to all the leafs. This w a y , an AND/EX OR t w
  • level
exp ression is created, that is equivalent to the tree and that is a canonical fo rm.
slide-29
SLIDE 29 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 29 ' & $ % REDUCED ZHEGALKIN F ORMS
  • GKT
families in which fo r the "Kroneck er-t yp e" va riables not all expansion t yp es S, pD and nD a re executed, but
  • nly
a subset
  • f
them.
  • Denition.
The pD/nD Fixed/Mixed Reed-Muller Exp ressions (fo rms) (FMRMEs) a re canonical fo rms
  • btained
analogously to GKE, but instead
  • f
all Shannon, p
  • sitive
and negative Davio expansions,
  • nly
p
  • sitive
and negative Davio expansions a re used fo r the va riables from the "Kroneck er-t yp e" set
  • f
va riables.
  • Because
  • f
the existence
  • f
ecient FPRM minimization algo rithms, calculation
  • f
FMRME fo rms can b e made mo re eciently .
  • Expansions
fo r any non-empt y subset
  • f
fS,pD,nDg expansions fo r "Kroneck er-t yp e" va riables can b e dened; fS,pD,g, fS,nDg, fSg, fpDg, and fnDg.
  • Dene
FMRM T rees and FMRM DDs, analogously to the general cases
  • f
GKTs and GK DDs.
slide-30
SLIDE 30 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 30 ' & $ % GENERALIZED KRONECKER DECISION DIA GRAMS
  • Denition
8. Canonical AND/EX OR Decision Diagrams co rresp
  • ndin
g to each
  • f
the ab
  • ve
dened t yp es
  • f
trees a re
  • btained
b y combining isomo rphic no des (i.e, the no des that co rresp
  • nd
to logically equivalent, tautological, subfunctions) , and removing no des that a re redundan t in the same sense as the
  • ne
discussed fo r GKTs.
  • F
  • r
instance, a no de with t w
  • inputs
  • riginating
from the same no de is removed fo r bina ry no des.
  • Simila
rly , a no de with four inputs
  • riginating
from the same no de is removed fo r quaterna ry no des.
slide-31
SLIDE 31 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 31 ' & $ % GENERALIZED KRONECKER DECISION DIA GRAMS (cont)
  • T
ransfo rmations fo r S, pD and nD no des (Drechsler) a re applied to single-va riabl e no des.
  • The
transfo rmation s fo r multi-va ria bl e no des a re their straightfo rw a rd generaliz ati
  • ns.
  • All
these transfo rmation s generaliz e the OKFDD transfo rmations.
  • This
w a y , the p recise denitions
  • f
all GK decision diagrams and resp ective attened (Zhegalkin) fo rms can b e
  • btained.
slide-32
SLIDE 32 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 32 ' & $ % APPLICA TIONS OF NEW REPRESENT A TIONS (1) Synthesis
  • f
easily testable, t w
  • -level
and multilevel , AND/EX OR circuits. (2) Rep resentation
  • f
la rge functions. (3) Synthesis
  • f
circuits with p redictable and controllab l e timing. (4) Synthesis fo r Fine Grain FPGAs (A tmel) and standa rd FPGAs (XILINX). (5) Exact and app ro ximate ESOP minimizatio n.
slide-33
SLIDE 33 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 33 ' & $ % T yp e
  • f
T ree Exp ression generated Decision Diagram from the tree Generated from T ree Shannon T ree Minterm Expansion Bina ry Decision Diagram (BDD) P
  • sitive
Davio T ree P
  • sitive
P
  • la
rit y F unctional Decision Reed-Muller (PPRM) Diagram (FDD) Kebschull et al Reed-Muller T ree Fixed P
  • la
rit y Reed-Muller Reed-Muller (FPRM) Decision Diagram Kroneck er T ree Kroneck er Expansion (KRO) Ordered Kroneck er F unctional Decision Diagram (OKFDD) P erk
  • wski
et al, Drechsler et al Pseudo Reed-Muller T ree Pseudo Reed-Muller Expansion (PSDRM) Pseudo Reed-Muller Decision Diagram Pseudo Kroneck er T ree Pseudo Kroneck er Pseudo Kroneck er Decision Expansion (PSDKRO) Diagram (PKDD) Sasao F ree Kroneck er T ree F ree Kroneck er F ree Kroneck er Decision Expansion (FKE) Diagram (FKFDD) P erk
  • wski/Ho
T able 1: Relations
  • f
Kno wn Canonical AND/EX OR T rees, Exp ressions and Decision Dia- grams.
slide-34
SLIDE 34 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 34 ' & $ % T yp e
  • f
T ree Exp ression generated Decision Diagram from the tree Generated from T ree pD/nD Fixed/Mixed pD/nD Fixed/Mixed pD/nD Fixed/Mixed RM T ree Expansion Lattice Diagram Generalized Kroneck er Generalized Kroneck er Generalized Kroneck er T ree (GKT) Expansion (GKE) Decision Diagram (GKDD) Pseudo Generalized Pseudo Generalized Pseudo Generalized Kroneck er Kroneck er T ree (PGKT) Kroneck er Expansion (PGKE) Decision Diagram (PGKDD) Mixed Pseudo Generalized Mixed Pseudo Generalized Mixed Pseudo Generalized Kroneck er T ree (MPGKT) Kroneck er Expansion (MPGKE) Kroneck er Decision Diagram (MPGKDD) Ordered Generalized Ordered Generalized Ordered Generalized Kroneck er T ree (OGKT) Kroneck er Expansion (OGKE) Kroneck er Decision Diagram (OGKDD) F ree Generalized F ree Generalized F ree Generalized Kroneck er T ree (F GKT) Kroneck er Expansion (F GKE) Kroneck er Decision Diagram (F GKDD) T able 2: Relations
  • f
New Canonical AND/EX OR T rees, Exp ressions and Decision Diagrams. An example
  • f
Fixed/Mixed rep resentations is in italic, Zhegalkin fo rms a re in b
  • ld.
slide-35
SLIDE 35 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 35 ' & $ %

FGKE ESOP ? ? ? ?

PPRM FPRM PSRDM GRM

PGKE GKE MPGKE OGKE

FKRO PSDKRO KRO FMRME

Figure 7: Set-theo retical relationshi p among kno wn and new (Zhegalkin) classes
  • f
AND/EX OR canonical fo rms
slide-36
SLIDE 36 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 36 ' & $ % T yp e
  • f
T ree Lattice Diagram Where lattices a re Generated from T ree and t yp e
  • f
regula r la y
  • ut
discussed in mo re detail Shannon T ree Universal Ak ers Arra y Ak ers Pseudo-Symmetric BDD (PSBDD) Chrzano wsk a P
  • sitive
Davio T ree F unctional P erk
  • wski
et al Lattice Diagram Reed-Muller T ree Reed-Muller P erk
  • wski
et al Lattice Diagram Kroneck er T ree Ordered Kroneck er P erk
  • wski
et al Lattice Diagram Pseudo Reed-Muller T ree Pseudo Reed-Muller Lattice Diagram P erk
  • wski
et al Pseudo Kroneck er T ree Pseudo Kroneck er P erk
  • wski
et al Lattice Diagram F ree Kroneck er T ree F ree Kroneck er P erk
  • wski
et al Lattice Diagram pD/nD Fixed/Mixed pD/nD Fixed/Mixed RM T ree RM Lattice Diagram T able 3: Relations
  • f
Canonical AND/EX OR T rees and Lattice Dia- grams (P ART I)
slide-37
SLIDE 37 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 37 ' & $ %
slide-38
SLIDE 38 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 38 ' & $ % T yp e
  • f
T ree Lattice Diagram Where lattices a re Generated from T ree and t yp e
  • f
regula r la y
  • ut
discussed in mo re detail Generalized Kroneck er Generalized Kroneck er simila r, even mo re general diagrams T ree (GKT) Lattice Diagram a re p resented in Lattice Diagram P erk
  • wski/Pierzchala/Drechsler
Pseudo Generalized Pseudo Generalized Kroneck er Example 4.1 with canonical expansion, Kroneck er T ree Lattice see also (PGKT) Diagram P erk
  • wski/Pierzchala/Drechsler
Mixed Pseudo Generalized Mixed Pseudo Generalized Kroneck er T ree (MPGKT) Kroneck er Lattice Diagram Ordered Generalized Ordered Generalized Kroneck er T ree (OGKT) Kroneck er Lattice Diagram F ree Generalized F ree Generalized Kroneck er T ree (F GKT) Kroneck er Lattice Diagram Linea rly Indep endent Linea rly Indep endent P erk
  • wski
Kroneck er T ree Decision Lattice Diagram /Pierzchala/Drechsler Bo
  • lean
T erna ry Bo
  • lean
T erna ry Decision T ree Lattice Diagram P air-Symmetry T erna ry P air-Symmetry T erna ry Example 4.2.
  • Non-canonical
Decision T ree Regula r Lattice Diagram expansion, regula r la y
  • ut
P air-Symmetry T erna ry P air-Symmetry T erna ry Non-canonic expansion, Decision T ree Generalized Lattice Diagram non-regula r la y
  • ut
T able 4: Relations
  • f
Canonical AND/EX OR T rees and Lattice Dia- grams (P ART I I.)
slide-39
SLIDE 39 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 39 ' & $ % CANONICAL AND/EX OR DECISION DIA GRAMS
  • Canonical
AND/EX OR Decision Diagrams a re
  • btained
b y combining isomo rphic no des and removing redundant no des.
  • F
  • r
instance, a no de with t w
  • inputs
  • riginating
from the same no de is removed (Drechsler).
  • Simila
rly , a no de with four inputs
  • riginating
from the same no de is removed.
  • T
ransfo rmations fo r S, pD and nD no des a re applied to single-va riable no des.
  • Their
multi-va riable no de generalizations a re applied to multi-va riable no des.
  • All
these transfo rmations generalize the OKFDD transfo rmations.
  • This
w a y , the denitions
  • f
all attened fo rms and decision diagrams can b e
  • btained.
slide-40
SLIDE 40 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 40 ' & $ % EXAMPLE WHEN PGKT IS BETTER THAN GKT
  • The
PGKT fo r f i has Shannon no de fo r va riable a
  • n
top, GRM no de fo r cofacto r f i a fo r expansion va riables b; c, and GRM no de fo r cofacto r f i a fo r expansion va riables b; c.
  • The
expansion diagram is f i = a (1
  • x
  • b
  • y
  • c
  • z
  • bc
  • v
)
  • a(1
  • v
  • c
  • y
  • b
  • z
  • b
c
  • x):
  • Thus,
it has t w
  • GRM
no des, and 5 no des fo r va riables a; x; y ; z ; v (the va riable-no des fo r va riables x; y ; z ; v a re sha red).
  • Observe
that in the sub diagram fo r f i a the p
  • la
rit y is [11,1,1] and in the sub diagram fo r f i a the p
  • la
rit y is [00,0,0].
slide-41
SLIDE 41 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 41 ' & $ % EXAMPLE WHEN PGKT IS BETTER THAN GKT (cont)
  • It
can b e sho wn, that it is not p
  • ssible
to have a smaller diagram fo r f i a , b ecause each change to another p
  • la
rit y w
  • uld
require exo ring some
  • f
va riables x; y ; z ; v and this w
  • uld
lead some extra exo r no des, which is to mo re no des than four fo r
  • nly
the va riable-no des.
  • Simila
rly , changing the p
  • la
rities in the sub diagram fo r a will alw a ys lead to diagrams with mo re no des than in the ab
  • ve
solution.
  • Since
b
  • th
sub diagrams a re w
  • rse,
the entire diagram is w
  • rse
b y having the same expansion in b
  • th
sub diagrams.
  • Simila
rly it can b e sho wn that the va riable a must b e
  • n
top, and, in general, there exists no GKT that is b etter
  • r
equal to f i.
slide-42
SLIDE 42 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 42 ' & $ % CREA TING NEW HIERARCHY
  • Simila
rly it can b e sho wn that:
  • free
GKT is b etter than the
  • rdered
GKT.
  • .........
  • all
p
  • ssible
changes
  • f
  • rders
in the subtrees to the same
  • rder
in b
  • th
subtrees will alw a ys result in higher total no de costs.
  • Examples
  • f
fo rms with sp ecial p rop erties, lik e sup erio rit y
  • f
GKE with resp ect to PSDKRO, PGKE with resp ect to GKE, MPGKE with resp ect to PGKE, and so
  • n,
can b e created.
  • This
w a y , the hiera rchy from Fig. 4 has b een also created.
slide-43
SLIDE 43 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 43 ' & $ % APPLICA TIONS
  • Create
very fast cub e-based ESOP minimizer s based
  • n
the idea
  • f
sepa rating va riables to "Kroneck er-t yp e" va riables and "GRM-t yp e va riables ".
  • The
concepts p resented here can b e applied fo r the synthesis
  • f
easily testable t w
  • -level
AND/EX OR circuits.
  • There
a re many
  • ther
new canonical expansion fo rms resulting from the attening
  • f
the new trees.
  • F
urthermo re, va riants
  • f
non-o rdered sets S j can also b e considered.
slide-44
SLIDE 44 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 44 ' & $ % APPLICA TIONS (cont)
  • Much
resea rch
  • n
minimal ESOP circuits.
  • So
fa r, the exact solution can b e
  • btained
  • nly
fo r a very small numb er
  • f
input va riables.
  • The
qualit y
  • f
AND/EX OR circuits from these expansions should b e b etter that those co rresp
  • ndin
g to GRMs
  • r
(Pseudo) KROs b ecause the sea rch space
  • f
GKTs is much la rger than the sea rch space
  • f
the GRM expansions.
  • The
size fo r which exact solutions can b e found remains an
  • p
ern p roblem.
  • Qualit
y
  • f
sea rch heuristics to pa rtition and to
  • rder
the input va riables.
slide-45
SLIDE 45 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 45 ' & $ % ZHEGALKIN LA TTICE DIA GRAMS AND REGULAR LA TTICE LA YOUTS
  • Ak
ers intro duc ed the so-called Universal Ak ers Arra ys in 1972.
  • They
a re regula r lattices and lo
  • k
lik e BDDs fo r symmetric functions.
  • It
w as p roven that every bina ry function can b e realized with such structure, but an exp
  • nentia
l numb er
  • f
levels w as necessa ry (which means, the control va riables in diagonal buses w ere rep eated very many times).
  • Sometimes
the
  • nly
w a y to implemen t a function is to rep eat the same va riable subsequently, without
  • ther
va riables intersp ersed .
slide-46
SLIDE 46 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 46 ' & $ % ZHEGALKIN LA TTICE DIA GRAMS AND REGULAR LA TTICE LA YOUTS (cont)
  • The
a rra ys
  • f
Ak ers w ere universal and they w ere unnecessa rily la rge, b ecause they w ere calculate d
  • nce
fo r all fo r the w
  • rst
case functions.
  • No
ecient p ro cedure s fo r nding
  • rder
  • f
(rep eated ) va riables w ere given, and some functions w ere next sho wn fo r which this app roach is very inecient.
slide-47
SLIDE 47 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 47 ' & $ % LA TTICE DIA GRAMS
  • Because
  • f
the p rogress
  • f
ha rdw a re and soft w a re technologies since 1972,
  • ur
app roach is quite dierent from that
  • f
Ak ers.
  • W
e do not w ant to design a universal a rra y fo r all functions, b ecause this w
  • uld
b e very inecient fo r nea rly all functions.
  • Instead
w e create a logic/l a y
  • ut
functions' generato r that gives ecient results fo r many real-life functions, not
  • nly
symmetrical.
  • As
sho wn b y Ross et al that, in contrast to the randomly generated "w
  • rst-case"
functions, 98%
  • f
functions from real-life a re decomp
  • sab
le.
  • Therefo
re, the functions a re either decomp
  • sable
to the easy realizabl e functions,
  • r
they do not exist in p ractice.
slide-48
SLIDE 48 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 48 ' & $ % LA TTICE DIA GRAMS (cont)
  • Analysis
  • f
realization s
  • f
a rithmetic, symmetric, unate and standa rd b enchma rk functions and new technologies (Concurren t Logic, A tmel).
  • Our
generaliza tion s
  • f
expansions : 1. S,pD and nD expansions. 2. All Linea rly-Indep en de nt expansions, the Bo
  • lean
T erna ry expansions, all Zhegalkin expansions. 3. All Kroneck er, Pseudo-Kroneck er, Mixed, and
  • ther
Decision Diagram concepts that a re used in Reed-Muller logic.
slide-49
SLIDE 49 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 49 ' & $ % LA TTICE DIA GRAMS (cont)
  • Mo
re p
  • w
erful neighb
  • rho
  • d
geometries.
  • 2x2
Lattices, 3x3, 4x4 lattices.
  • In
essence, the 2x2,3x3,4x4 and 8x8 neighb
  • rho
  • d
s a re used in patents and publish ed w
  • rks.
  • W
e allo w to mix control va riables in diagonal buses. This p ermits to realize F ree diagrams.
slide-50
SLIDE 50 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 50 ' & $ % LA TTICE DIA GRAMS (cont)
  • Calculation
  • f
data input functions to lattice no des fo r any t yp e
  • f
expansions and any lattice neighb
  • rh
  • d
s is p erfo rmed b y the same technique
  • f
solving logic equations fo r a given structure, as
  • ne
used fo r Linea rly Indep enden t logic.
  • In
contrast to LI logic, the structural equations can have
  • ne,
many ,
  • r
no solutions.
  • When
there a re many solutions, the
  • ne
evaluated as b est is tak en.
  • When
there a re no solutions, the backtrack to another structure, another expansions,
  • r
another blo cks
  • f
input va riables is executed.
slide-51
SLIDE 51 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 51 ' & $ % LA TTICE DIA GRAMS (cont)
  • Selection
  • f
the
  • rder
  • f
(usually rep eated) va riables is done using the concept
  • f
the b est sepa ration
  • f
most dierent-valu e minterms, using the Rep eated V a riable Maps.
  • V
a riable
  • rdering
(rep eati ng) and va riable pa rtitioni ng.
  • In
contrast to the w
  • rst-case
randomly generated functions, fo r real-life b enchma rk functions
  • nly
few rep etition s
  • f
va riables a re enough.
  • It
is esp eciall y easy to symmetricize the w eakly sp ecied functions and Bo
  • lean
relations.
slide-52
SLIDE 52 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 52 ' & $ % CONCLUSIONS AND FUTURE W ORK The generaliza tio n
  • f
the DDs can b e done in several w a ys, which can b e combined together to create va rious new rep resentation s.
  • D1.
Creating new and mo re p
  • w
erful expansion t yp es fo r no des (also non-canonica l) , thus depa rting from the standa rd S, pD, nD set
  • f
expansions.
  • D2.
Allo wing fo r several t yp es
  • f
expansions in every level
  • f
the expansion tree,
  • D3.
Allo wing mo re freedom in
  • rdering
va riables in b ranches
  • f
the tree (includ in g the case
  • f
no
  • rdering
at all),
slide-53
SLIDE 53 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 53 ' & $ % CONCLUSIONS AND FUTURE W ORK. (cont)
  • D4.
Combining not
  • nly
isomo rphic no des in trees to create Directed Acyclic Graphs but also combining a rbitra ry no des together, thus mo difying functions realized b y these no des. No des a re combined using joining
  • p
erations from [? , ?, ?]. This app roach, in general, leads to the need
  • f
rep etitions
  • f
va riables (as in Regula r and Generalize d Lattices).
  • D5.
Creating generalized expansions fo r sets
  • f
va riables, instead fo r single va riables.
slide-54
SLIDE 54 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 54 ' & $ % CONCLUSIONS AND FUTURE W ORK. (cont)
  • Some
  • f
these generalize d t yp es
  • f
decision diagram rep resentation s have already b een intro duc ed, investigated theo reticall y , and implemente d in CAD to
  • ls
to mention
  • nly
those used fo r Exo r Logic.
  • Many
mo re, ho w ever, remain still to b e analysed and evaluated exp erimenta ll y .
  • Many
t yp es that ma y result from the ab
  • ve
dimensions
  • f
generaliza tion have not even b een dened y et.
  • Considering
the Green/Sasao hiera rchy , the new rep resentations will b e not w
  • rse
than the kno wn
  • nes
in terms
  • f
complexit y .
slide-55
SLIDE 55 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 55 ' & $ % CONCLUSIONS AND FUTURE W ORK. (cont)
  • W
e exp ect the new rep resentation s to nd applicati
  • ns
in logic synthesis fo r ESOP circuits, ne grain FPGAs, and rep resentation
  • f
la rge functions.
  • Symmetric
functions.
  • Zhegalkin
Lattices a re sup erio r to Universal Ak ers Arra ys when realizin g all totally symmetric functions, pa rtially symmetric functions,
  • r
easily symmetrizabl e functions fo r which
  • nly
few va riables require rep etitions in the structure.
  • They
should b e combined with Ashenurst/Curtis decomp
  • sitions
, fo r the la y
  • ut-driven
synthesisk the realization
  • f
a rbitra ry functions.
slide-56
SLIDE 56 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 56 ' & $ % CONCLUSIONS AND FUTURE W ORK. (cont)
  • All
these resea rches have application s to submicron technology
  • They
can b e all used fo r custom logic/la y
  • ut
synthesis.
  • T
  • develop
new t yp es
  • f
FPGAs and synthesis fo r FPGAs.
  • K*BMDs
(Drechsler ,Bec k er) use a rithmetic
  • p
eration s
  • f
addition and multipli cati
  • n
instead
  • f
bina ry logic
  • p
erations, and nd applicati
  • ns
in verication
  • f
digital systems, and fo r solving general p roblems in discrete mathematics.
  • Here
another app roach to the generaliz ati
  • n
  • f
Kroneck er diagrams w as p rop
  • sed.
  • All
ideas here can b e further extended to multiple-val ue d logic, integers, and w
  • rd-level
diagrams.
slide-57
SLIDE 57 New Hiera rchies
  • f
Rep resentations RM'97, Septemb er 1997 57 ' & $ % CONCLUSIONS AND FUTURE W ORK. (cont)
  • The
diagrams and lattices intro duc ed here can also b e generaliz ed to integer a rithmetic (and also fo r rational a rithmetic realized with continuous logic), where + and * a rithmetic
  • p
erato rs and mo re general linea rly indep en den t
  • p
erations w
  • uld
b e used.
  • These
"w
  • rd-level"
expansions can b e derived.
  • The
w
  • rd-level
expansions together with the generaliz ati
  • n
t yp es D1-D5 can b e used to create trees, fo rms, diagrams, lattices, and la y
  • uts.
  • Lattices
have b een also p rop
  • sed
fo r continuous logic and MV logic (P erk
  • wski,Pierzchal
a,D rech sl er) .