A switching theory for bilateral nets of threshold elements

75 Pages·1963·4.056 MB·English

by  W. D. Frazer

DIGITAL COMPUTER LABORATORY
UNLVEFSITY OF ILLINOIS
lTFBANA, ILLINOIS
REPORT NO 153
A SWITCHING THEORY FOR BILATERAL NETS OF THRESHOLD ELEMENTS
by
William Donald Hrazer
11, 1963
October
(This work is being submitted in partial fulfillment of the requirements for
1963.)
Degree of Doctor of Philosophy in Electrical Engineering, August,
ACKNOWLEOOMENTS
The author is indebted to several people for various forms of
assistance given during the preparation of this thesis. The following, in
particular, deserve special mention�
Professor David E. Muller; advisor, mentor, and oracle; with whom ·
it has been the author's privilege to be associated these past years, and to
whom the author owes his continuing interest in switching theory.
Drs. James Gibson and Robert O. Winder of RCA Laboratories; the
form.er for suggesting the problem, and the latter for valuable suggestions
given during preliminary investigations in t e summer of 1961.
h
Professors F. E. Hohn, R. Narasimhan, and S. Seshu of the University
of Illinois, and Dr. S. Arnarel of RCA Laboratories for critical reading of the
preliminary results, and suggestions for extension thereof.
Members of the Computer Theory Research Group at RCA Laboratories and
the Staff of the Digital Computer Laboratory of the University of Illinois for
many helpful discussions.
Professor T. A. Murrell for his services as chairman of the final
examination committee.
Mrs. Phyllis Olson whose exceptional typing skill, good sense, and
experience will be obvious in the fonnat of the final draft.
Elizabeth B. Frazer, the author's wife, for her encouragement over
the past two years, and her sharp proofreader's eye over the past two weeks.
TABLE OF CONTENTS
Page
1. INTRODUCTION 1
• 0 D
1.1 Preface . 1
. . o o • o o • • • • • • • o • • •
1.2 An Outline of Bistable Device Characteristics 2
1.3 Networks of Two-Terminal Bistable Devices . 2
1.4 Threshold Switching Functions 4
• • • • • • • • •
2. A MATHEMATICAL MODEL FOR BILATERAL THRESHOLD NETS 6
2.1 Rationale 6
• • • •
2.2 The Model . 6
• • • . • • . • • . • • • . •
2.3 State Behavior of the Model . . 8
. . .
2.4 Speed Independence and Semi-Modularity . . . . 9
2.5 The Balance Condition . 10
2.6 Nets with Inputs . . . 11
• • . . • . . • .
2.7 Analysis of Synchronous Nets 12
. . • .
2.8 Analysis of Asynchronous Nets 13
. . . • •
3. SOME GENERAL PROPERTIES 15
3.1 Introduction . 15
• . • • • . .
3.2 Graph-Theoretic Properties 15
. . • • .
3.3 The Cycling Theorem 16
. • • • . . . • • .
3.4 A Canonical Form 19
. • . . . . . • • . . • . . . . • .
3.5 Equilibrium States 24
. • • . • • . . • . . . . .
4. PROPERTIES OF A FAMILY OF FUNCTIONS DESCRIBING BILATERAL
THRESHOLD NETS 27
. • . . . . . . • •
4.1 Introduction . . 27
. . . . . • . . . • . • •
4.2 Conventional Threshold Logic . . 27
• . • • . . . .
4.3 A Family of Functions Characterizing Bilateral Threshold
Nets . 28
. o o o o o • o o • • • • •
4.4 Pairwise Monotonicity 29
• . . . . • .
4.5 Extensions of Pairwise Monotonicity 31
5. SNYTHESIS OF BILATERAL THRESHOLD SWITCHING NETS 39
5.1 Introduction . . 39
• . . • . . . • • • • . .
5.2 Clocked Automata and Combinational Logic 39
• • • .
5.3 Realizabi) lity of Idempotent Automata 40
•
5.4 The ( 7 Realization Scheme 47
p 2 • • . •
5.5 Genera Synthesis Techniques 55
i . • • •
6. AFTERWORD 57
• • . • . • • . •
6 . 1 Summary 57
• . • • • • . • • • •
6.2 Suggestions for Future Work . 58
BIBLIOGRAPHY . 59
• • •
-iv-
TABLE OF CONTENTS (CONTINUED)
Page
APPENDIX A 61
•
APPENDIX B 65
B.l Proof of Theorem 3.10 . 65
B.2 Another Characterization of Equilibriwn States 67
APPENDIX C 68
. . •
-v-
A SWITCHING THEORY FOR NETS OF THRESHOLD ELEMENTS
BILATERAL
William Donald Frazer, Ph.D.
Department of Electrical Engineering
University of Illinois, 1963
Extant theories of logical design conventionally make several
assumptions regarding the characteristics of the logical elements used in
forming networks; among the most fundamental of these is that of directivity
of information flow. This assumption has historical basis in the fact that
most of the devices used heretofore in the construction of logical elements
have been three-terminal devices. The growing number of two-terminal bistable
devices--networks of._which are basicalyl nondirective with respect to information
flow--has created new problems associated with the artificial imposition of
directivity, and thus raised the question of one.might ptofitably revise
whether
one's concepts of logical design to conform to the characteristics of such
devices.
It is the object of this thesis to initiate a theory of logical
design for networks in which information is not constrained to flow in only one
direction along a connection between devices. A linear graph model is proposed
" "
for a very general class of such nets : each vertex (i) of a linear graph
To
" "
is assigned an ordered pair (z.,k. ), where z 1 is called the state of
1. 1 i = -+
" "
vertex i and k is an integer called the threshold of the vertex. A connec-
i
tion matrix, A, is also defined: a represents the weighted bilateral
ij
" " " "
connection between vertices i and j. The next or desired state of vertex
�
i is defined by a function F[ a z - k ] where F[x) +l (x � 0) and
ij j i =
F[x) -1 (x o).
=
<
A method of analysis based on this model is proposed for learning the
state behavior of nets, and a canonical form for any net proposed. It is
demonstrated that, under very general assumptions, any net of this kind must
proceed from any initial state to an equilibrium state, and a bound is given
on the number of vertex state changes which can take place in such a transitiono
Results are presented which describe the character of the stable states of such
nets and place bounds on the number of such states.
Approaching the problem from a more conventional point of view, one
can define a family of functions which expresses the state behavior of any
vertex in the net as a Boolean function of the states of the other vertices
and of the inputs. This family of functions is shown in this case to have a
structural property called "pairwise monotonicity," a generalization of the
usual concept of Boolean monotonicity to families of functions. It is demon­
strated that this property and its extensions play a major role in determining
logical design techniques for nets of this kind; they make it impossible, for
example, to simulate any directed logical net having feedback.
The presentation concludes with a discussion of the synthesis of
"
bilateral threshold switching nets. It is demonstrated that all idempotent
automata"--automata incapable of distinguishing the single input "an from the
input sequence ua followed by an--are realizable as nets of this type, and two
schemes are given for achieving such realizations. The possibility of develop­
ment of synthesis techniques capable of achieving more general net topologies
than those resulting from the two schemes mentioned above is also discussed, and
shown not to be feasible with present knowledge of the theory of threshold logic.
lo INTRODUCTION
lol Preface
Extant theories of logical design conventionally make several
assumptions regarding the characteristics of the logical elements used in
forming networks; among the most fundamental of these is that of directivity
of information flowo The convention which leads one, when building logical
networks, to think in terms of logical elements with separate sets of input
and output terminals, has practical origin in the fact that most of the devices
used heretofore in the construction of logical networks have been three-
terminal deviceso
The growing number of two-terminal bistable devices--networks of which
are basically nondirective with respect to information flow--has, as observed
[12 ' 15 ' 17 ' 18Jt
by several authors, created new problems associated with the
artificial imposition of directivityo These problems are in some cases so severe
as to counterbalance other desirable features which a device may possesso The
question thus arises: Instead of forcing basically bilateral networks to meet
conventional logical design requirements, might one profitably revise one's
concepts of logical design to conform to the characteristics of such devices?
It is the purpose of this dissertation to initiate a theory of logical
design for networks in which information is not constrained to flow in only
one direction along a connection between deviceso A model for a general class
of such "nets" will be proposed, and an analysis procedure derived for learning
their state behavioro We shall prove several theorems demonstrating properties
of these nets, including some which demonstrate essential differences between
them and conventional directive logical networkso In addition, we shall show
i Superscripts in brackets refer to references listed in the bibliographyo
-1-
-2-
properties of a set of Boolean functions associated with such nets and
realizability of a large class of sequential machin s giving
demonstrate the e by
two schemes for achieving such realizations. We shall discuss also the problem
of realizing sequential machines using techniques which will lead to more
general net topologies than the ones resulting from the two schemes mentioned
aboveo
102
An Outline of Bistable Device Characteristics
One can associate with most known bistable devices a quantity,
corresponding to potential energy, which is similar in form to the idealized
ch arac te ri. s ti· c sh o wn i. n Fi" go 1a o.1 12116121] Th e var.i a bl e x .i n thi" s fi" gure i. s a
-
generalized coordinate representing the quantity which is switched. Corres
pondingly, one can plot the derivative of the potential energy curve of
( )
Figo la3 this is proportional but opposite in sign to the restoring force
exerted by the system, and is shown in Figo lb. The ordinate thus shows the
y
external force necessary to maintain a given value of x; y is therefore the
11switchingn variable. The quiescent values of the switched variable are seen
to be .:!: x0, and the system is constrained to move along the curve. Consequently,
any attempt to increase beyond +y when the system is in state -x will result
y 0 0
in a transition to the +x state, and similarly for �y and +x o
0 0 0
1.3
Networks of Two�Terminal Bistable Devices
n "
The most natural configuration for a net of two terminal .devices
is one in which one terminal of each device is used for reference, while the
other is available for connection to other elements. This requires the
sensing of x and y at a common terminal, and precludes their being the same
kind of variable. In a logical network, the value of for a given element
y
-3-
v
(a)
Potential.En ergyC llrve
a Bistable
tor Device
e
d
t
(b)
"Force·Divsst.an ce"Curv e
a
for BistableDe vice
1'1.gur• 1
-4-
will ideally be a function at any given time of the values of x associated
with the neighboring elementsi and independent of the value of x associated
with its own element, and of timeo This idealized behavior is not always
strictly adhered to by actual devices.
In the networks which we will consider, y will be a weighted linear
of the neighboring x's. The choice of such a function is prompted pr aril
sum im y
by the fact th a t sueh_ " th res ho ld" or " 1i n early separable " behavior is common
to most known two-terminal bistable devices, including all of those mentioned
previously, and only secondarily by its simplicity. We allow both positive
and negative weights to attach to the variables in the sum because physical
networks seem to justify this also; that isi it seems feasible in at least same
cases to make connections of a bilaterally inverse nature between devices.
In
parametron or core nets such a connection might be achieved by reversing a
winding, or alternativelyi in the forrr.er case, by adding an extra one-half
wavelength of wire between devices. In the case of tunnel-diode networks, there
may be some question as to the present feasibility of such connections, but the
art and technology of these devices is enough in its infancy to preclude dis-
allowing such connections on this ground alone, particularly when such excellent
physical models exist in the other cases.
1.4 Threshold Switching Functions
The notation used in the treatment of threshold functions varies
widely from author to author. For clarity� therefore, we pause at this point
to make some definitions.
Definition� A switching function, as used here, will be a mapping
n
from (-1,l) to (-131); that is, a switching function maps n-tuples
of elements from the set (-1,1) onto that set.