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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 1New 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 2New 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 3New 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 4New 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 5New 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 6New 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 7New 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 8New 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 9New 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 10New 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 11New 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).
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 14New 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 15New 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 16New 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 17New 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.
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 19New 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 )
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
Figure 4: Mixed Pseudo Generali zed Kroneck er T ree
SLIDE 23New 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.
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
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 28New 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 29New 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 30New 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 31New 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 32New 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 33New 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 34New 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
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 39New Hiera rchies
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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 40New Hiera rchies
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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 41New Hiera rchies
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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 42New Hiera rchies
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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.
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 45New Hiera rchies
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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 46New Hiera rchies
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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 47New 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 48New Hiera rchies
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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 49New Hiera rchies
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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 50New Hiera rchies
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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 51New Hiera rchies
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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 52New Hiera rchies
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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 53New Hiera rchies
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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 54New Hiera rchies
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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 55New 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 56New Hiera rchies
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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 57New Hiera rchies
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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
