Risk decomposition and ring theory, lattice techniques and universal algebras, and unary functions are considered

Livingston, D.

Stoughton, J. W.

English

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v7	 David Livingston
and
John W. Stoughton, Principal Investigator
	 -
Final Report
For the period September,, 1980 to September 5, 1981
iR
Prepared for
	
L _ 	 National Aeronautics and SpFade Administration
Langley Research Centerh..	 Hampton, Virginia
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! Task Autho-itition No. 3
°	 -	 Jamie°^ A. Drs^t, Technical Monitor
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^.Ar	 (NASA-CR-16345 .8) INVESTIGATION OF	 N83_34414
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FUNC TIGNS FOR REMOTE SENSING APPLICATIONS 	 Unclas
tFinal Report, 5 Sep. _(old Dominion Univ..,	 G3/43 42015	 l
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DEPARTMENT OF ELECTRICAL ENGINEERING
SCHOOL OF ENGINEERING
OLD DOMINION UNIVERSITY
NORFOLK, V RGINU,
INVESTIGATION OF ANALYTICAL METHODS FOR EFFICIENT
PARTITIONING OF ON-'BOARD PROCESSING FUNCTIONS FOR
REMOTE SENSING APPLICATIONS
By
David Livingston
and
John W. Stoughton, Principal Investigator
Final Report
For the period September 5, 1980 to September 5, 1981
Prepared for the
National Aeronautics and Space Administration
Langley Research Center
Hampton, Virginia
Under
Research Grant NAS1-15648
Task Authorization No„ 43
James A. Dorst, Technical Monitor
Contracts Branch/Acquisition Division
Submitted by the
Old Dominion University Research Foundation
P.O. Box 6369
Norfolk, Virginia
September 1983
kINVESTIGATION OF ANALYTICAL METHODS FOR EFFICIENT
PARTITIONING OF ON-BOARD PROCESSING FUNCTIONS FOR
REMOTE SENSING APPLICATIONS
By
David Livingston) and John W. Stoughton
INTRODUCTION
This 'final report details the results developed to fulfill the require-
ments for Task NAS1-15648-43 [ 1] . It is divided into four sections:
Section I Risk Decomposition
Section II Universal Algebras
Section III Unary Functions
Section IV Current Topics
Section I generalizes the results reported in the previous report and
presents some new examples based on ring theory. Section II further gener-
alizes the use of lattice techniques to any system with algebraic structure.
The third section deals with a specific algebra called unary functions.
Areas which are currently under consideration are briefly described in the
final section. It is recommended that the previous report be reviewed
since the functions for this report are described therein.
SECTION I. RING DECOMPOSITION
The decomposition of multiplication and addition processes may be ob-
tained simultaneously if the algebraic structure under consideration has
the form of a ring. As in group decomposition, it is necessary to obtain
the lattice of substructures. The substructures of rings are called ideals
and are defined as follows:
Let {R, +, •} be a ring where R is a set which is closed under addition
and multiplication. {I, +, •} is an 'ideal if and only if:
l Research Assis^' taut, currently with I.B.M. Watson Research Center
2 Associate Professor, Department of Electrical Engineering, Old Dominion
#:	 University, Norfolk, Virginia.
ORIGINAL PAG12 6O
1) I<R,	 OF POOR QUALITY
2) fi t + } is a subgroup of f R, ,4 }, and
3) For all aeI and all be R, a • b	 c, where cel.
The set of all ideals belonging to a ring form a lattice under the par-
tial ordering of inclusion. As in the case of groups, equivalence classes
or partitions may be obtained by finding the additive cosets of each ideal.
A lattice of the resulting partitions may be constructed from which the
decomposition structure and element representations can be obtained.
Example 1. Integers Modulo Eight
The addition and multiplication tables are:
+	 0	 1	 2	 3	 4	 5	 6	 7
0	 0	 1	 2	 3	 4	 5	 6	 7
1	 1	 2	 3	 4	 5	 6	 7	 0
2	 2	 3	 4	 5	 6	 7	 0	 1
3	 3	 4	 5	 6	 7	 0	 1	 2
4	 4	 5	 6	 7	 0	 1	 2	 3
5	 5	 6	 7	 0	 1	 2	 3	 4
6	 6	 7	 0	 1	 2	 3	 4	 5
7	 7	 0	 1	 2	 3	 4	 5	 6
and
•	 0	 1	 2	 3	 4	 5	 6	 7
0	 0	 0 0	 0 0	 0	 0 0
1	 0	 1	 2	 3	 4	 5	 6	 7
2 0	 2 4 6 0 2 4 6
3	 0	 3	 6	 1	 4	 7	 2	 5
4 0 4 0 4 0 4 0 4
5	 0	 5	 2	 7	 4	 1	 6	 3
6	 0	 6	 4	 2 0	 6	 4	 2
a
7	 0	 7	 6	 5	 4	 3	 2	 1.
a	 _ 2
From these tables the following ideals may be found:
oRrorrvAL PAC2 GSl
1) The ring itself
	 OF POOR QUALITY
2) +	 0 2 4 6	 6	 0 2 4 6
0 0 2 4 6	 0	 0 0 0 0
2 2 4 6 0	 2 0 4 0 4
4 4 6 0 2	 4 0 0 0 0
6	 6 0 2 4	 6	 0 4 0 4
3) +	 0 4	 0 4
0 0 4
	
0 0 0
4 4 0	 4 0 0
4) +	 0	 0
0	 0	 0	 0
The partitions associated with each ideal and its corresponding lattice
are:
1) { 0,	 1,	 3,	 4,	 5,	 6,	 71	 = 111
	
111
2) { 0, 2, 4, 6; -S;-5, 71	 = 112
	 112	 Ti
3) { 0, 4,	 1,	 5;	 2,	 6;	 3, 71	
= 113	 113	 T2
4) { 0; 1; 2; 3; 4; 5; 6; 7}	 = 11¢	 11
Two partitions T 1 and T2	 may be created such that 11 2 • T 1 • T2 + 113•T2
110 .
T i = { 0, 1, 4, 5; 2, 3, 6, 71 and T2 = {0, 1, 2, 3; 4, 5, 6, 7 1,
By choosing T 1 and T 2 as such, a pipeline or serial structure is obtained:
[112	 Ti	 T2
This decomposition structure may be used for both the addition and multipli-
cation processes differing only in the realization of each block. Element
a
e
IY
3
t
/i.	 1s
re .t 'ham	 i i:.	 ..
awjlGis1 AL tlti" .y
() Inc)()E3 QUALITY
representations may be chosen such that they are the equivalent binary re-
presentations. Given the decomposition and representations, the ring struc-
ture may be realized in hardware using standard procedures.
Polynomial arithmetic can also be decomposed, as in the proceeding ex-
ample, if a ring structure is evident.
Example 2. Ring of Polynomials Modulo x 2 + 2 over
Ground Field Modulo 3
The addition and multiplication tables are:
+ I	 0	 l	 2	 x	 2x x+l	 x+2	 2x+2	 2x+1
0	 1
1
2
x
2x
x+l
x+2
2x+2
2x+1
and
0	 1	 2	 x	 2x	 x+1 x+2
	 2x+2	 2x+1
0 1 2 x 2x x+l x+2 2x+2 2s+1
1 2 0 x+1 2x+1 x+2 x 2x 2x+2
2 0 1 x+2 2x+2 x x+1 2x+1 2x
x x+l x+2 2x 0 2x+1 2x+2 2 1
2x 2V+1 2x+2 0 x 1 2 x+2 X+l
x+l x+2 x 2x+1 1 2x+2 2x 0 2
x+2 x x+l 2x+2 2 2x 2x+1 1 0
2x+2 2x 2x+1 2 x+2 0 1 x+l x
2x+1 2x+2 2x 1 x+1 2 0 x x+2
0
1
2
X
2X
x+l
0 0 0 0 0 0 0 0 0
0 1 2 x 2x x+l x+2 2x+2 2x+1
0 2 1 2x x 2x+2 2x+1 x+l x+2
0 x 2x 1 2 x+ 1 2x+1 2x+2 x+2
0 2x x 2 1 2x+2 x+2 x+l 2x+1
0 x+l 2x+2 x+l 2x+2 2x+2 0 x+1 0
	
'
..
°I	 x+2	 0	 x+2 2x+1 2x+1	 x+2	 0	 x+2	 0	 2x+1
K	 ^:	2x+2	 0	 2x+2. x+l 2x+2 x+l	 x+l	 0	 2x+2	 0
	
4	
2x+1	 0	 2x+1	 x+2 x+2 2x+1	 0 2x+1
	
0	 x+2
4	
i°
7
The id 4ls are:
1) The ring itself
2)	 + 0 x+l 2x+2
0 0 x+l 2x+2
x+l x+l 2x+2 0
2x+2 2x+2 0 x+l
3)	 + 0 x+2 2x+1
0 0 x+2 2x+1
x+2 x+2 2x+1 0
2x+1 2x+1 0 x+2
4) +	 0
	
0	 0
• ! 0	 x+ l	 2x+2
0 0 0 0
x+l 0 2x+2 x+ 1
2x+2 0 x+l 2x+2
0 0 x+2 2 x+l
0 0 0 0
x+2 0 x+2 2x+1
2x+l 0 2x+1 x+2
0
0	 0
Note that the two non-trivial ideals are generated by the linear factors
of the modulus, x2 + 2.
The partitions and associated lattice corresponding to the ideals are:
.l) to, 1, 2, x, 2x, x+1, x+2, 2x+1, 2x+21 = III
2) 10, x+l, 2x+2; 1, 2x, x+2; 2, x, 2x+11 = 111
3) 10, x+2, 2x+1; 1, x, 2x+2; 2, 2x, x+11	 = 112
4) {0; 1; 2; 2x; x+1; x+2; UT-1; 2x+2; '2x+1; 2x+21 = rid
n•
5
11Z
H2
ORIGINAL F'A€ " "
OF POOR QUAL1 N
This lattice indicates a parallel decomposition. The element representa-
tions may be made as follows:
0	 1	 2
111 = 1 0, x+l, 2%c{2; 1, 2x, x+2; 2, x, 2x+1 }
0 1_	 2
112
 = f0, x+2, 2x+1; 1, x, 2x*2; 2, 2x, x+l}
It is important to note the implication of this representation. Poly-
nomial arithmetic can be performed using simple integer arithmetic. As an
example, consider 1(2x)* (2x+2) 
I 
x +2"
1(2x) . (2x+2) I x2+2 =	 • 2x2x + 2x • 2 I x2+2
_ 4x2 + 4x I x2 +2
x2 + x I x2 +2
x + 1.
This result may also be obtained by using the representations
I (2x) .	 (2x+2) I x2+2) I + (1,2)- (0,1) = (1 . 0, 2 . 1) = (0,2)
(0,2) + x+l.
A,
^r
Fr.
6
In other terms, multiplication of polynomials modulo another polynomial can
be considered a cyclic convolution. Interpreting the results of the lattice
analysis, a cyclic convolution may be performed by a term-by-term multipli-
cation of two integer sequences representing the polynomials. This is
equivalent to performing a cyclic convolution of two sequences by transform-
ing, multiplying and inverse transforming the sequences using a method such
as the finite Fourier transform. It is interesting to note that this result
was obtained quite systematically using lattice analysis.
SECTION II. UNIVERSAL. ALGEBRAS
It has been shown that Group [1] and -,ring decompositions may be obtain-
ed in a systematic fashion using lattice analysis. This method may be gen-
eralized to any algebra belonging to a class following the rules of univer-
sal algebra [2 ] .
An algebra may be defined as a pair {S,F}, where S is a nonempty set of
elements and F is a set of operations which map a Cartesian power of S into
S, that is,
Fa : Sn(a)+ S, for all FaeF.
Groups, rings, and lattices are all examples of algebras. A field is not an
algebra under this definition since the inverse function is not defined for
zero.
A subal.gebra may be defined as a pair {T,F,}, where T4S and
Fa : ' 
1(a)
+ T for all FaeF. That is, the set T is closed under all
Fa	and T is a subset of S. The importance of sub algebras to decompos.i-_.
Lion is evident by the following theorem and corollary from Birkhoff [2].
Theorem 1: Any union of subalgebrad of an algebra is a subalgebra, and an
algebra is a sub algebra of itself.
7
Corollary 1: The subalgebras of an algebra form a complete lattice.
Corollary 1 provides the vehicle by which arithmetic systems may be
decomposed in a systematic fashion. The structure of the system is first
examined to determine if it forms an algebrrs., All subalgebras are then
found and a lattice of the subalgebras is constructed. By augmenting each
subalgebra with its cosets a lattice of partitions is formed from which the
possible decompositions are obtained.
SECTION III. UNARY FUNCTIONS
Unary function3 are of particular interest, especially with the avail-
ability of new memory technologies which allow the implementation of unary
functions by look-up tables stored in read only memories. If the range of a
unary function is a subset of its domain, then the function forms an algebra
and readily lends itself to possible decompositions. The value of decompos-
ing a unary function, in terms of look-up tables, is a reduction of the to-
tal memory required to implement the table.
Example:
	 x+l Modulo 12
N
x x+l	 x
1	 6
x+l
0 7
1 2	 7 8
2 3	 8 9
3 4	 9 10
4 5	 10 11
5
1	
6	 11
1	
0
This table forms an algebra, but there are no apparent subalgebra, but there
are no apparent subalgebras for the function x+l. If the table is treated
as a sequential machine, the partitions may be readily obtained.
8
4	 I
R^
i
tr
GPartitions:
	 OF POOR QUALITY
1<	 { 0, 1, 2, 3, 4, 5, 6, 7 1 8, 9 1 10, Ill
I(1	 { 0, 2 1 4, 6, 8,10; 1, 3, 5, 7 0 9, 11}
112	 { 0, 4, 8; 1, 5, 9; 2, 6, 10; 3, 71 Ill
113 = { 0, 31 6, 9; 1, 4, 7, 10;	 2, 5, 8, 11}
H4 = { 0, 6; T,_7; 2, .8; 3, 9; 4, 10; 5, 11
The 9^ _ IT; 1	 2 . 3	 4; 5	 6, 7	 8	 9 10 11f	 f	 f	 f	 f	 f	 9	 !	 f	 )
The Lattice of Partitions:
nz
IIi	 R3
II^	 .n4
9 3 ' 82 =11
The element r
11 2 = { 0,
11 3 = { 0,
suggesting a
epresertation,
0	 1
4) 8;
	
1, 5,
0
3, 6, 9;
	
1,
parallel decomposition.
s may be made:
2	 3
9;	 2, 6, 10;	 3, 7, 11)
1	 3
4, 7, 10;	 2, 5, 8, 11}.
9
F POOR^ I.I
'thus the original function (table) may be decomposed into two functions
(tables):
Y	 P(Y)	 Z	 P(Z)
0	 1	 0	 1
1	 2	 1	 2
2	 3	 2	 0
3	 1	 0	 ,
Notice the total nuuber of memory locations needed is 4 + 3 - 7 as opposed
to 12. In this decomposition, the transformations are systematic:
Y	 I x I k and Z- I x 13.
If the transformations are not systematic the decomposition may be more dif-
ficult to implement than the original function, unless it is a part of a
hirer s ystem i±gi. ng the de composed representat ions. lhi so is the biggest
disadvantage to using this method for unary functions.
SECTION IV, CMENT TOPICS
Polynomial evaluations may be represented in various decompositions in-
cluding serial and parallel. A simple example of a parallel decomposition
is the representation of the polynomial as a product of its linear factors.
That is, a polynomials may be factored:
an-1 xn-1 + ... + a2x2 + a lx + a  =
( ... (an-1 x + an-2) x +, ..a 2 ) x + ,a o .
10
V 
0000•
s e m
ORIOM'AL PAC"'-' Kq
OF POOR QUALITY
The structures of the two decompositions may be represented in block form,
Parallel
x —^^^ .,-----
Serial
F (x)
As of yet, a correspondence between these decompositions and a lattice from
which they may be obtained has not been found. Such a correspondence will
help generalize the lattice method.
lI
i
.	 K
Decompositions of matrix products are very desirable from the stand-
point of decreasing computational complexity. There exist decompositions of
matrix products which reduce the number of element multiplications. For
example, the product of two 2 x 2 matrices can be performed using six
multiplications rather than the usual eight. Matrix operations are highly
algebtaic in structure and should lend themselves to many possible decompo-
sitions.
Current vesearch involves using lattice analysis to obtain beneficial
decompositions. There is a major problem encountered using these tech-
piques. Match algebras are very large, For example, a 2 x 2 matrix over
a field of sixteen elements has 65536 representations. This size problem
may be overcome by generating the technique to large problems by examining
smaller problems in detail and using computer assistance in handling the
large information sets.
4
n:
G•
i
12
r ,	 ^
REFERENCES
1. Livingston, David and Varanasi, Murka, " Investigation of Analytical
Methods for Efficient Functional Partitions of On-board Processing
Functions," Final Report for N.A.S . A. Under Task Order NAS1 -15648-29,
Sept. 1980.
2. Birkhoff, Garrett, Lattice Theory, American Mathematical Society Collo-
quium Publications Vol. 25., American Mathematical Society, Provi-
dence, Rhode Island, 1967.
S
N^
k^
13


Investigation of analytical methods for efficient partitioning of on-board processing functions for remote sensing applications

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