A switching theory for bilateral nets of threshold elements

DIGITAL COMPUTER LABORATORY UNLVERSITY OF ILLINOIS lTFBANA,

ILLINOIS

REPORT NO

153

A SWITCHING THEORY FOR BILATERAL NETS OF THRESHOLD ELEMENTS by William Donald Hrazer

October

( This

11, 1963

work is being submitted in partial fulfillment of the requirements for

Degree of Doctor of Philosophy in Electrical Engineering,

August,

1963.)

ACKNOWLEOOMENTS

The author is indebted to several people for various forms of assistance given during the preparation of this thesis. particular,

The following,

in

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; former for suggesting the problem,

and the latter for valuable suggestions

given during preliminary investigations in the summe r of Professors F. o f Illinois,

and Dr.

preliminary results,

S.

Eo

Hohn,

Ro

the

Narasimhan,

and S.

1961.

Seshu of the University

Amarel of RCA Laboratories for critical reading of the

and suggestions for extension thereof.

Members o f the Computer Theory Research Group at RCA Laboratories and the Staff o f the Digital Computer Laboratory of the University of Illinois for many helpful discussions" Professor To

A.

Murrell for his services as chairman of the final

examination committeeo Mrs.

Phyllis Olson whose exceptional typing skill,

good sense,

and

experience will be obvious in the format of the final draft. Elizabeth B. the past two years,

Frazer,

the author's wife,

for her encouragement over

and her sharp proofreader's eye over the past two weeks.

TABLE OF CONTENTS Page

1.

INTRODUCTION

1.1 1.2 1.3 1.4 2.

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0

0

.,

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0

0

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Networks of Two-Terminal Bistable Devices

•

Threshold Switching Functions

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Rationale

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The Model

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State Behavior of the Model

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6

The Balance Condition

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Nets with Inputs

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Analysis of Synchronous Nets

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Analysis of Asynchronous Nets

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6 6 8 9 10 11 12 13

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Speed Independence and Semi-Modularity

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15

Introduction

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15 15 16 19 24

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Graph-Theoretic Properties

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The Cycling Theorem A Canonical Form

1 2 2 4

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Equilibrium States

PROPERTIES OF A FAMILY OF FUNCTIONS DESCRIBING BILATERAL

4.1 4.2 4. 3

•

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Introduction

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Conventional Threshold Logic

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0

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Pairwise Monotonicity

0 •

0 .

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Extensions of Pairwise Monotonicity

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Introduction

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Clocked Automata and Combinational Logic Realizability of Idempotent Automata The

( p72 1

Realization Scheme

Summary

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Suggestions for Future Work

BIBLIOGRAPHY

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Generai Synthesis Techniques

AFTERWORD

6.1 6.2

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27 27 27 28 29 31 39

SNYTHE S IS OF BILATERAL THRESHOLD SW ITCHING NETS

5.1 5.2 5.3 5.4 5.5

. . . .

A Family of Functions CharacteriZing Bilateral Threshold Nets

4.4 4.5

6.

•

An Outline of Bistable Device Characteristics

THRESHOLD NETS

5.

0

1

SOME GENERA L PROPERTIES

3.1 3.2 3. 3 3.4 3.5 4.

•

A MATHEMATICAL MODEL FOR BILATERAL THRESHOLD NETS

2.1 2.2 2. 3 2.4 2.5 2.6 2·7 2.8 3.

Preface

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• •

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39 39 40 47 55 57

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57 58 59

•

-iv-

TABLE OF CONTENTS ( CONT INUED ) Page APPENDIX A

61

•

APPENDIX B

65

B.l B.2

65 67

Proof of Theorem 3.10 . Another Characterization of Equilibrium States

APPENDIX C

•

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68

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-

v-

A SWITCHING THEORY FOR

BILATERAL

NETS OF THRESHOLD ELEMENTS

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 aLwhich are basically nondirective with respect to information flow--has created new problems associated with the artificial imposition of directivity, and thus raised the question of whether one. might ptofitably revise 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 A

direction along a connection between devices. for a very general class of such "nets":

To

is assigned an ordered pair (z.,k.), where z �

vertex i and k

i

1

i

a

ij

connection between vertices i and j. i is defined by a function F[ =

-1 (x < A

each vertex (i) of a linear graph =

+ -

1 is called the "state" of

is an integer called the "threshold" of the vertex.

tion matrix, A, is also defined:

F[x)

linear graph model is proposed

0).

A

connec-

represents the weighted bilateral The "next" or irdesired" state of vertex

� aijz j - ki]

where F[x)

=

+1 (x

� 0)

and

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

proceed from any on

the

number of

very general assumpt i ons,

initial state

t o an equilibri um

vertex state changes which

can

any

state,

net and

of

a

take place in

Results are presented which describe the character

of

this

kind must

b o und

is

g iven

such a transitiono

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 and of the inputs.

of

the states of the other vertices

This family of functions is shown in this case to have a

structural property called "pairwise monotonicity," a generalization usual concept of Boolean monotonicity to families of functions.

of

the

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 iiidempotent

automata"--automata incapable of distinguishing the single input "ail from the input sequence ua followed by aVi--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.

10

INTRODUCTION

101 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 threeterminal deviceso The growing number of two-terminal bistable devices-- networks of which are basically nondirective with respect to information flow--has, as observed by several authors,

[12 , 15 " 17 18Jt

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 question thus arises:

The

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 IInets ll 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 b r a ckets refer to references listed in the bibliographyo -1-

-2-

properties

of

a set of Boolean functions associated with such nets and

demonstrate the realizability of a large class of sequential machines two schemes for achieving such realizations.

by

giving

We shall discuss also the problem

of realizing sequential machines us ing techniques which will lead to more general net topologies than the ones resulting from the two schemes mentioned above.

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 .112,16,21] O t1C O O shown 1n F 19o 1 aD ch aracter1S 0

Th e var1a bl e x 1n thO1S f O19ure 1S a 0

0

generalized coordinate representing the quantity which is switched.

0

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 systemj and is shown in Fig. lb.

The ordinate y thus shows the

external force necessary to maintain a given value of x; y is therefore the iiswitchingBi variable.

The quiescent values of the switched variable are seen

to be � xo' and the system is constrained to move along the curve. any attempt to increase

y

Consequently,

beyond +yo when the system is in state -x will result O

in a transition to the +x state, and similarly for -yo and +x . O O

1.3

Networks of Two�Terminal Bistable Devices The most Vlnatural" 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

y

for a given element

-3-

v

(a)

Potential. fo r

Energy Ctlrve

a Bistable Device

t

(b)

"Force· vs. Distance" Curve a Bistable Device

tor

J'1gure 1

�-

will ideally be a function at any given time with the

neighboring

with its

own

of the

values of

x

elementsJ and independent of the value of

element, and of timeo

associated

x

associated

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 sum

of the

neighboring

x's.

The choice of such a function is

prompted

primarily

" or " 1 inearly separable " behavior is common by the fact that suc_h " threshId 0 to most known two-terminal bis table devices, including all of previously, and only secondaril y by its simplicity.

those mentioned

We allow both positive

and negative weights to attach to the variables in the sum because physical networks�m

to justify this also; that iS i it seems feasible

in

at least same

cases to make connections of a bilaterally inverse nature between devices. parametron or core nets such a connection might be achieved by rev ersing winding,

or alternativelYi in the former case, by adding

wavelength of wire between devices.

an

In

a

extra one-half

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 disallowing 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 w ide ly

fram author to author.

F or

clar ity� therefore, we pause at this point

to make some definitions. A switching function, as used here, will be a mapping

Definltion� from ( _ l ,l )

n

to (-l,lJi

that is, a

sw

itching function maps n- tuples

of elements from the set ( - l,lJ onto that set.

-

Defin it ion :

5

-

A s �itching func tion f is a threshol d s �itchin g func tion

�hen the re e xis ts a s e t of pos iti ve intege rs { k, a , l n fo r all X € ( -l, l } :

+1

•

•

, a } such that n

n

if

f ( X ) ::

-1

•

L a 1. x.1 - k > 0 i=l n

if

L a.x . - k < 0 . 1 1 1 1=

We no � proceed to the development of a model fo r bilateral threshold s � itching nets .

(101)

2. 20 1

A MATHEMATICAL MODEL FOR Bll.ATERAL THRESHOLD NETS

Rationale The advantages to be gained from the development of a simple and

accurate mathematical model for bilateral threshold swi tching nets are manifold. One is enabledj

through use of such a model�

to consider the general properties

of network behavior without reference to physical parameters such as voltage level or current flow in each analysis

0

The results are a Ish6:cten;i..ng�·ofltthe

arguments associated with various demonstrations and a concise method of analysis, as well as some new insights.

In additiony our choice of a graph-theoretfumodel

will enable us to use results from the literature of that course,

field.

It is,

of

of the greatest importance that we refer frequently to the characteristics

of physical devices to avoid making any unwarranted assumptions.

An

important characteristic of the nets un der consideration is that,

for a given set of elements,

network topology completely determines network

This fact leads us to draw upon linear graph theory for a model,

behavior.

and

the additional fact that the connections between elements are nondirective leads us to propose a nonoriented linear graph as a model.

2.2

The Model Let I be the set of integers,

set

{ �l, l)

"-

I the set of nonzero integers,

and B the

o

Definition� weighted,

A

bilateral threshold switching net,

linear graph consisting of a f inite set,

which we shall number numbers--and a set}

I

Rj

.

N, V,

is a nonoriented, of vertices--

through n and thenceforth denote by their of weighted branches satisfying�

'..

-6-

-

1) R

C

7

-

'"

(I x V x V) E

R, where q

2)

for no i is (q, i, i)

3)

( q, j, i)

4)

for at most one q is ( q, i, j)

Defi nit ion :

E

R

-

( q, i, j)

E

€

'"

I

R €

R

The "degree, " p . , of ve rtex i will be e qual to the sum 1.

of the magnitude s of the we ight s of the bra nches incident ; that is , p. = L 1.

Iql

such that, for some j, ( q, i , j)

Def init ion :

R.

To each vertex i , of degree p . , we will ass ign a n ordered 1.

pa ir (z.,k.), where z. 1.

E

1.

1.

E

called the " s tate " and k

B and k.

1.

E

I and -p 1.o < k. < p 1.1. .

" i the !'threshold of vertex

•

z. 1.

will be

1.

From this orde red pair, and from the topology of the l inear graph, we can comput e

a th ird quant it y, d . , the " des ired state " of ve rtex i, in the 1.

follo wing wa y: Def in it ion :

Let F[x ] be a fu nct ion mapp ing I to B defined as follo ws :

F[x 1

=

{

l

if

x > 0

-1

if

x < 0

Exte nd ing this def init ion to ve ctors , �, the ith.

compo nent of F[�]

is defined : F [X]. = F[x . ] -

Definit ion :

1.

For a n autonomous b ilateral t hreshold sw itchin g net of

n vert ice s , let defined b y:

1.

AI

be a s ymmetr ic " connection matr ix " of dime ns ion n

-8-

( q, i, j)

if

€

R

=

a� .

1.J

otherwise

Note again that the diagonal entries of A are zero,

Definition: n verti ces,

1 et Zi be a co 1umn

" state

vector

=

1.

2.3

where�

where:

For an autonomous bilateral threshold switching net of let

D

be a column

Then for any vertex,

iJ

"

desired state vector

Vertex i wi

"

ZS

i

"

defined by:

(� .1)

D

=

d.

is the desired state.

11

be said to be

1.

Defini tion � if

d·l.mension nJ

threshold of vertex i

�]

"

0f

" " threshold vector of dimension n,

k.

relaxed

"

For a bilateral threshold switching net of n vertices,

let K be a column

n vertices,

1s unique.

state of vertex i

1.

Definition:

a�j

For an au�onomous bilateral threshold switching net of

z!

Definition:

and

91

' exci ted PI if z !

1.

r

d.

1.

and

d i·

State Behavior of the Model Information is transmitted through the network by the changing of

state of excited vertices.

No vertex which is not excited may change state.

It is easily verified that the

" " changing of state behavior modeled here corres-

ponds in all relevant respects to the switching of a threshold element from one stable state to the other in the physical network.

Implicit in our model also

-9-

is the fac il it y for mak ing o ne a djacent vertex have greater influe nce tha n others on the behav ior of a given vertex through appropr iate cho ice of branch we ights . One influence s the behav ior of the net work throu gh inputs ; the se may be treate d as a ddit ional vert ices which, once set , ma inta in the ir state In

value s , in depen dent of the behavior of the vert ices a djace nt to them .

some

cases it become s convenient, for cons istency, to as s ign a th re shol d of +p . to 1

each input , i, in the ( +1 ) state and a threshold of - p o to each input in the (-1) 1

s tate , where p . is agai n the de gree of the ( input ) ve rtex . 1

be treat e d in the same fash ion as other vert ices .

Then the inputs can

Th is requ ires chang ing the

threshol ds of input vert ices fo r chang ing input state, however .

2. 4

S pee d In dependence and Semi-Mo dula rit y Two a ddit ional concepts wh ich bea r intro duct ion at this t ime are those

[22] · depen dence " an d II seml -mo du1a rlt y . " of " spee d ln •

Def init ion :

.

An " internal state " of a net with n internal ve rt ices and

q inputs is an n- tuple of valuat ions the net .

z.

1

over the internal vert ices of

An " input state " is l ikew ise a q-tuple of valuat ions

over the input vert ices of the net .

z.

1

A " total state, " o r merely a

" state " of the net is an ( n+q ) -tuple compose d of the Cartes ian pro duct of an inte rnal state and an input state . Def in it ion :

A b ilateral thre shol d sw it ching net will be said to be

" spe e d- independent with re spect to a g iven ( total) state " if, under constant input , the f inal ( total ) state is in dependent of the orde r of -I

chan ge of exc it e d vert ices at all stages of the trans it ion .

-10Definition�

A bilateral thre shold switching net will be said to be

vVsemi-modular with respect to a given (total ) state" if, under constant input, all possible sequences of changes of excited vertices have the property that no vertex passes from excitation to equilibrium without changingu It has been shown [

22

] that �emi-modularity is

a

sufficient, but not

a

necessary, condition for speed independence.

2v5

The

Balance Condition In

the course of tracing the behavior of a net, we allow the excited

vertices to change state, one by one.

We do not allow simultaneous changes to

take place at adjacent vertices for the following reason:

The sole case in

which allowing simultaneous changes would yield ambiguous results is that in which two adjacent vertices are excited and a change in the state of either alone would also relax the other.

The vertices are thus Vlmutually exciting"

one another. If

we return to consider the physical net and recall again the device

characteristics shown in Fig. IJ we note that if we assume a positive branch joining the two vertices, 1 and 2, then vertex

1

is in the state

�

corresponding

to the point lettered 8vd�' in Figo Ib, and vertex 2 is in the state g corres­ 2 "goil

ponding to the point lettered is unlikely that tre points e the behavior of devices 1 and be symmetrico

Now because of manufacturing tolerances, it

and e J or f and f on the two curves governing l 2 2

l 2

will coincideJ or even that either curve will

As device 1 begins to move along its curve from d to e , device l l

2 begins to move from g2 to f2; thus values of Y are opposite in reference 2 with respect to Y v l

N010T

a ssume l Y 1 e 1

<

I Yf I; then when device 1 reaches el 2

and turns to approach f } dEvice 2 will also reverse direction and head back l

- 11> I Y I, both dev ices will come to r es t in s tate do f l 2 s imilar arg ume nt, beg i nning with like state s and re sult ing in u nl ike states , Co nversely , if

A

I ye I

holds for the case i n wh ich a negat ive branch jO ins the t wo vert ices . This "bala nce condition" s ituat ion c learly doe s not oc cur in semimodular nets, although it may occur in speed- i ndepe nde nt nets , a nd must occur in speed-depe nde nt nets ( see theorem 3.7).

S i nce semi-modular c ircuit s are the

largest subclas s of speed -independe nt log ical c ircu it s which are curre ntly well understood, the s ituat io n would not , in practice, be e ncountered ofte n.

It is

worthy of co ns ide ratio n, ho wever, because it represe nts the s ole pos s ible except ion to the cycling theorem (3.4).

Although s ome b istable t wo-termi nal

devices may, under certain c ircumsta nces, exhibit behav ior which seemingly is i n conflict with the s imulta neous change condit io n, such behavior whe n it does occur depe nds not on any e s sent ial characteri s t ics of the devices themselve s, but rather on such experime ntally introduced parameters a3 inter-eleme nt s ignal tra nsm is s io n delay .

The re i s nothing in the phys ical characte ris t ics of the

dev ices whi ch cont radicts the hypothe s i s that another co nfigurat i o n, for example depos ited elements or hybrid net works, might not make such behav ior i mpos sible .

2. 6

Nets with Input s We have already noted that the de s ired state of a n auto nomous net

may be expre s sed i n terms of the prese nt state us ing equation (2 . 1 ) 0

We have

also noted previous ly that inputs to a net may be treated as addit io nal vert ices whose state s do not change , regardles s of the behavior of adjace nt internal ve rtice s .

Th is last propert y make s it unne ce s sary to add ro ws to AI to corre s -

po nd to i nput vertices . Defi nition :

Thus we have :

For a b ilate ral threshold s witching net wit h n internal

vert ices a nd m inputs , let A be a " connect io n mat r ix " :

-12-

A

(n

=

n

x

a ..

=

a� . l

A

+

m)

where:

1) for all

and 0< j < n�

13

2) for all i) and n < j

<

n

IJ

+ m:

joining vertex i and input j

aij

J =

weight of branch

0

Note that AU is the principal square submatrix of Aj and that the inputs numbered beginning at n Definition�

�

1.

In

are

like manner, we have:

For a bilateral threshold switching net with n internal

" vertices and m infuts, let � be a column 'state vectorPf of dimension (n

+

m) where:

1) for 0< i < - nJ 2)

for n < i < n

z. 1

=

+ ffi,

z�

1

Z. 1

state of input i.

We can now write., for an arbitrary network�

F[A� Furthermare5

WE

Definition:

�]

can characterize the "steady-state " behavior of nets: An "equilibrium state" of a bilateral threshold switching

net is a total state in which no vertex is excitedc equilibrium state,

Z

-e

ClearlYj for any

�

F'[AZ --e

2.7

(202 )

D

K]

... -

z'

-e

Analysis of Synchronous Nets The mat�ix equation (2.2) provides a convenient description of the

state behavior of a bilateral threshold switching net subject to synchronous

-13ope ratio n

If

0

one impresses a n e xternal "clock" signal on all of the vertices

o f a ne t i n suc h a fas hion that e xci te d vertice s can change o nl y at re gular, spec ified in tervals and mus t c hange i n any s uc h inte rva l, the n � be c omes the ne xt in ternal s ta te of the ne t.

We thus ge t a who le sequenc e of e q ua tio ns o f

the form ( 2.2 ) as the net pas ses throug h a s tate s equence F [ AZ.

-1

where in eac h case, D .

-1

In

=

Z�

-1 + I

-

_K]

=

i

D.

-1

=

I, 2,

•

•

�1'�2'o

• .

:

0

( 2. 4 )

"

this waY J one can eas il y trace the s ta te be havior o f the ne tworka

pro vided he a ls o c he cks to see that t he "balance condi ti on Vi desc ribed in sec tion 2. 5 does no t occur .

In s ync hronous o perationj t his condit ion wi ll

be indicated b y the exc i tation of a pair of vertices in bo th a given s tate and its succes sor .

2.8

Anal ys is of As ynchronous Ne ts In

the case of as ync hronous ne ts , the s i tuation i s more c omplex .

One

o f the principal ques tions to whi c h one seeks an ans we r t hrough analys i s is that o f spe �d i ndependence .

Since no s ystemat ic procedure ex is ts for c hecking t his l

o ne no rmall y c he cks for semi -modulari ty ins tead l 1 and even in this c ase the only currentl y well -de fined method for c he cking this c ons ists of trac ing all poss ible s e quence s of exc i t ed ve rtex c banges . [

2

]

Even in the general c ase , thi s is no t

alwa ys neces sary, bu t no reall y good met hod ex is ts for eliminat ing unnece s sar y c hecks in a s ys tematic fashion; t �is is true in convent ional log ic net works as well .

t

We thus proceed in the following manner :

In these ne ts , there i s an alte rna ti ve ; see remarks follo wing theorem 3.7.

We begin with the ini.t ial state of the net3 the corresponding desired (internal) state,

fOo

�3

and from it determine

From this pair3 we determine

the excited vertices and allow these to changej one by one, checking all possible � sequences, in the manner of Bartky and MullerJ [ ] to insure that the same equilibrium state results in all caseso

The procedure is somewhat simplified

for bilateral nets over those of the directional variety in that the latter can cycle, while the former cannot,

as

will be shown in the next chapter.

example of the matrix techniquej see Appendix A o

For an

3. 3.1

SOME

GENERAL

PROPERTIES

Introduc tion In thi s c hapter, we s hall demons tr ate s e veral pro pe r ties whic h are

c har ac teris tic of bil ate r al thre s ho ld swi tc hing ne ts .

Some s imple one s deri ve

direc tly from the pro per ties of the model de ve lo ped i n th e l as t c hapter; o thers require slightl y more e l abor ate deri vations .

Toge ther, howe ver, the y gi ve

addi tional ins ight into the be havior o f suc h ne ts .

3.2

Gr aph-Theoretic Pro pertie s We beg in b y c alling u pon the l i ter ature of l inear gra ph theory for a

few c las s ic result s : Theorem 3.1:

An y fini te bil ater al thre shold swi tc hing ne t mu st h ave

an e ven numbe r of ver tices of odd degree . Defini tion �

Two gr aphs will b e s aid to be " homeomorphic " i f the y c an

be converted into the s ame gr aph b y an arbitr ary number of re peti tions of the following o peration : Gi ven a br anch PQ of a gr aph, insert a vertex R into PQ to form two new branche s PR and RQ. Theorem 302:

(Kur atowski )

A ne ce s s ar y and sufficient condi tion th at

a ne twork be pl an ar is that i t cont ain no subgr aph homeomorphic to the gr aph of Fi g . 2a or th at of Fig . 2b .

-15-

-16-

(b )

('a) Figure

Theorem 3.3�

A

2

necessary condition that a network which contains no

self loops or branches be planar is that r

r

=

v

�

3(v - 2) where

number of branches

number of vertices

The proofs of these theorems have been ornittedi in several books. Seshu and Reed,

[27]

Theorems

3.1

and theorem

and

� 3.2

for they may be found

are treated in Konig

3.3�· is treated in Ringel.

[26]

[13]

and in

The applications

of these theorems to printed circuit synthesis and other topology- dependent considerations are clear.

303

The Cycling Theorem We now come to a very powerful theorem which demonstrates an essential

difference between the behavior of bilateral threshold switching nets and that of conventional directed logical networks.

t

t

The method of proof is a generalization of an idea suggested by R.

in connection with a weaker theorem.

O. Winder

-

17

-

Two ve rtices , i and j, o f a net wil l be said to be

Definit ion :

"pos itively jo ined " if a

ij

> 0, an d "negat ively jo ined " if a

ij

< O.

The ver tice s will be said to be " ad jacent " if the y are e ither pos itive ly or negat ive ly jO ined . A branch of a bilateral t hre shold switching net will be

Definit ion :

said to be "ex cited " if 1)

it negat ive ly jo ins ve rtices of like state

2)

it pos it ively jOints vert ices of u nlike state .

or

Theorem 3.4:

An y state sequence of a bilate ral threshold switching

net under constant . input is nonrepetit ive ; that is , no state appears twice . Proof :

In

order that s ome state of a net repeat , it is ne c es sar y that

s ome se t of vert ices unde rgo an e ven number of changes .

Recall from

sect ion 2 5 that we can de te rmine the behavior of a network b y allowing .•

the ve rt ices to change one at a t ime .

Now c ons ider a vertex with degree

p and threshold k which changes state twi ce : In

order for the vertex to change from the state ( - 1 ) to the

s tate ( +1 ) , the sum of the magnitude s of the weights of the inc ident exc ited branches , (p - r ) , mus t " outwe igh " the sum of the magnitude s of the we ights of the inc ident relaxed branches , r, b y at least k :

exc ited branche s I we ight l sum = (p -r )

{

� �

-f

---

I-----

Figure 3

}

relaxed branches I we ight l sum r =

-18Thus: (p - r)

r

-

> k

or 2r ::: p

-

k

Similarly� when the vertex changes from (+1) r - (p - r) 2r

<

+-

k

to

!-1}� we mUst have

< k

P

Now in each case the net gain in weighted magnitude sum of excited branches is given by� r

(p - r)

-

2r - p

Combining (303) and (301) we g€t� 2r - p :::: -k Combining (303) and (302) we get: 2r

Thus� the total net gain) excited branches for the pair

-

T; of

p

<

k

in the wEighted magnitude sum of transitions is bounded above by the

sum of the right-hand sidES of (3.4) and (3.5)� T

<

k - k

Therefore any pair of changes

in

0 the s�ate of a particular vertex

decreases �he weighted magni�ude sum of excited branches.

Since this

-19-

sum is unique for a given total state,

no internal state in a sequence

can ever repeat under constant inputo

Corollary 1:

No bilateral threshold switching net can cycle--that is,

pass repeatedly through a sequence of internal states--under constant input

0

Corollary

2:

Any bilateral threshold switching net has at least one

equilibrium state corresponding to each input state.

It was mentioned in section

2.5

that the

Vl

balance condition

there constituted the only possible exception to this theorem; be seen,

"

discussed

this can easily

for if a pair of vertices in such condition were both to change simul­

taneousl� the sum of the excited branch weights might remain undiminished for a pair of changes at each vertex.

3. 4

A Canonical Form We shall now demonstrate a useful canonical form for bilateral

threshold switching nets.

Definition:

A majority vertex is a vertex of odd degree and threshold

zero.

Definition:

A regular net is one in which all vertices have the same

degree and threshold.

Theorem

3.5:

To any bilateral threshold switching net,

vertices which are not majority vertices, net,

M,

N,

containing

there corresponds a regular

which is composed excl usively of majority vertices,

state behavior indistinguishable from that of

N

and has

in all cases.

-20-

Because of' the symmetry of' the vertex state representation

Proof':

and the asymmetry of the discrimination function, there exists

(E ai z

- k lo

When

i

j j

z

=

eroo

F[L a

value

of

Furthermore�

under

these conditions:

z - (k - l)]� since ij j i

function is always F[La

Pi' and

degree,

both the

the

vertex i are evenj or

ConsIder the expression

any elemento

ambiguity in the threshold of

Z - k ] i ij j

=

F[L ai Z

j j

thr esho ld,

the

ki, of

this expression will be an even n�ber

argument,

the

if both

SimilarlYj

oddo

an

- (k

of the

-P

i

a

Z Ij J

l�t"te

k1] r

land kr ..6re cxk1�

1)]0

4

i

F[E

for

the

simulating vertex and connecting it to sources of (+1) andJor ( - l ) j

one

We now demonstrate that,

by

choosing a large enough degree

can simulate the behavior of a vertex of threshold x and degree Yo Consider a majority vertex of degree (2q - 1) with a connection of weight b to

an

input of (+1) and one of weight

c

to an input of

For such an elementi we have:

(-1) 0

F[La"Z lJ j

-

k.1 ]

d.

=

1

As far as the rest of the net is concernedJ the degree of this vertex

is (2q -

b -

c - 1)9 and it behaves as if it had threshold {c - b)o

Accordingly� we may try to choose: x y Observe, however} that (c even

or oddo

ambiguity

If

x and

y

=

2q

�

(b

+ c) - 1) cannot both be

add,i< we.·taJte

..advan:t.i:Js€ of the

in threshold assignment and set:

J

.

- (b + c) - 1

are both evenj or

6."

_

(c - b)

and (2q -

- b)

.6

•

=

.. I..

(

•

,

, -:

J

C

(

JO '.

, \

)

-2 1x

y

=

=

(c - b ) - I

2a - ( c

+

(308a )

b) - I

It is easily verified that one of the systems (308) always has a solut ion.

One thus chooses t he minimum value of q for each vertex of Na

then makes all of the others in t he net M equal

to

the maximum of these minima .

The advantages of the use of the canonical form will become clear as we proceed to demonstrate further prope rties of these nets , but the following theorems should give some indi cation of its power . Theorem 3,6:

An upper bound on the number o f ve rtex changes which can

take place in a bilateral threshold sw itching net in proceed ing from any pre s ent state to e quilibrium, under constant input , i s g iven by the

SUIT.

of the magnitudes of thE we ights of the ex ci ted branche s in the canonical network . Proof :

The canonical network is cOffip osed exclus ively of maj ority vert ice s

and these have the prope rty that each c hange of state reduces the magnitude we ight sum of ex cited branche s by at least one . '

T hus the total number

of changes canno t exce ed the magnitude of the original sum . Pursuing this approach st ill fur ther , we can define a quant ity whi c h i s like entropy in that it always in creases . Definition :

For a canonical b ilateral threshold sw it ching net, N, with

internal state �v} input state �j and connect ion mat rix n

n

L z�{ L a . . z� +

i =l

1

Cf . , equat ion (3.3).

t1

The supers cript T denote s transpo s e .

1

j=l

lJ J

A,

n+m I.

j=n+l

let a. .s . ) lJ J

Indee d , this provides an alternate proof of theorem 3040

-22The quantity U{�!�J

is

difference between the sum of the magnitudes of the If y'

weights of the relaxed branches and that of the excited branchesg

is a

successor state not equal to �Vj then

Indeed, this format may be used to specify the possible outcomes to a balance

and only if, for

z.

1

all

Z·

if

Z'

to

y' :

y'

is a

excited vertices,

i,

which change in going from

specifying

n+m

n

( L. 1-) i'Z� + L. j=n+l j =l J J

We now come to Theorem 307�

state to

that

condition situation by

If

a

a.. s.) < lJ J

legal

successor

n+m

n

L. a. . s. ) y. ( L. n.. . y� + j=n+l lJ J 1 j=l lJ J

very interesting

result,

suggested

by

Do EQ

Muller:

a bilateral threshold net is speed-dependent with

respect to a given

initial

state) then there must be some sequence

of

vertex state changes which can lead to the balance conditiono If

Proof: sible

the net is speed�dependent, there must be

at lea st

two

pos-

equilibrium states to which it can proceed from the initial state.

Call these states A and B and the inital state

10

Now since no state

can repeat, we may partially order the states which occur in proceeding from I

the

to A

states

and

B; allowing only one vertex at a time to change state;

may be ordered chronologically.

partial ordering resembles Fig. 4. and

I

to

Bj

The

and the unidirectionality

that the net may

t

For

justification of

go from T

this

to U or

V,

finitene ss of

of

must exist a group of states such as Tj

The Hasse diagram for the chains

the transitions insure

U)

this I

to A

that

there

and v,1 having the property

but that there exists no state W

cl a im, see Appendix C.

-23acce ssible from both U and Vo ma y be Vo

Note that T may be I, U

may

be

A

and B

We now c laim that the ba lance condition e xi st s in state T .

�I

Figure 4.

State s of a Speed-Dependent Net

To see that this is so, conside r the two ve rtices , i and j , which change in going from state T to U and V, respect ively .

Now i and j

are both exc ited in state T ; suppose that the net proceeds to state U . Then i is no longe r ex cited, and j may or may not b e excited . it i s .

Suppos e

Then there is a state W ac ce ss ible from U in which i and j are

both relaxed and both have changed .

By hypothes is , however, W i s not

acces s ible from V, and so i and j are both re laxed in V also , when only j has changed .

We now have a cont radict ion, howeve r, for states V and

W differ only in the state of ve rtex i , and it i s relaxed in both cases . Thus W does not ex i st and both i and j must be relaxed in both U and V. But this is exactly the balance condition . The proof may be extended to cases in which more than two equi librium states are acce ss ible by grouping the equ il ibrium states in pai rs .

-24It

is fairly easy to construct examples of non-semi-modular

speed-independent nets in which the balance condition occurs or fails to occurg Thus the class of nets in which the balance condition fails to occur is speedindependent, and properly contains the class of all semi-modular nets.

An

analysis of a net for speed independence might therefore be conducted by looking to see that no sequence of excited vertex state changes can lead to the balance conditiono More use will be made later of the canonical formg

3.5

Equilibrium States We will now demonstrate some properties of GistableU or equilibrium

states of bilateral threshold switching nets. Theorem 308: Whenever two stable states differ in the state of the jth (internal) vertex, they must differ also in the state of one or more of the vertices adjacent to jo Proof�

This is simply a restatement of the uniqueness of the state of

any vertex as determined from the states of those vertices adjacent to ito An autonomous net or a net with constant input can thus have at most

2 n-l stable states if it has n verti ces.

Even this bound is too high, however,

as will be seen latero Theorem 3D9�

In

any bilateral majority switching net, the complement of

a stable state is stable. Proof:

We note

that Z

=

-Z.

Therefore

- 2 5Now i f

Z

is a stable state of a majorit y net , since

K

=

0:

But then

and Z i s a stable state . Note that it i s the total state ( and not just the input state ) which i s complemented .

In

a canonical network, there fore , the auxil iary source s mu st

al so be c omplemented in order to apply thi s theorem .

Complementat ion of the

source s re sult s, of course , in lo ss of the significance of the canonical network . That not all networks have more than one stable state i s shown by the following example , t for which the only stable state is ( +1 , +1 ) :

Figure 5

An open question at thi s po int i s that of a good bound on the number of stable state s which a network may po sse s s.

By mean s of an involved geometric

ar gument, pre sented in Appendix B, one can obtain the bound of theorem 3 . 10 . . Thi s, however , i s still muc h greater than the large st number of stable state s po sse ssed by any network so far d i scovered .

The current record for number of

t Throughout thi s paper , number s on vertice s will repre sent thre sholds , and numbers in parenthe se s will indicate branc h weight s.

-26stable states is held by the net of Figo have the common state

�l) or {- ll

6g1

Since each pair of vertices can

the total number of stable states is 2

/

n 2

reader is invited to consider the problem of finding a good upper bound

•

� pairs

..

•

of vertices

Figure 6

Theorem 3ol0�

No bilateral threshold switching net of n internal

vertices may have more than 2

Proof:

T

See Appendix B.

Suggested by Do Eo Muller

0

n-2

stable states» when n > 3.

•

o

The

4. 4g1

PROPERTIES OF A FAMILY OF FUNCTIONS DESCRIBING BILATERAL THRESHOLD NETS Introduct i on The matrix des cript ion of state behavior proposed in chapter 2 i s

well su ited to the nets be ing cons idered here .

Additiona l ins ights may be

gained, however , from cons ideration of another des cription- -one in co mmon use i n the treatment of conv entional dire cted logical networks .

Informat ion obtained

from cons ideration of this latter set of funct ions will therefore have the addit i o nal advantage that it will lend itself eas ily to compari s on with known re sults in the other area .

We begin with a s ummary of some result s conce rning threshold

logi c ; our treatment resembles most closely that of Winde r . [30]

4.2 Convent ional Threshold Logic Let f be a threshold switching funct ion ; then by a "valuat ion,et

Definit ion :

V, on some subset of the argument s of

fj

(-1) or (1) to each member of the subset .

we mean an as s ig nment of value If

V

i s a valuat ion ) then

V

is

a valuation over the s ame subset, with ( -1 ) and ( 1 ) interchanged in all case s .

We denote by

g � f if :

2)

g

3)

g

Definit i on :

the corresponding restr icted function .

If f and g are switching functions , we say that

Definit ion : 1)

fv

a)

f and g have the s ame domain

b)

g has va lue (+1) whe neve r f doe s

=>

f if �

g

=>

f but f

t:.

g

=

f if :

g

=>

f and f

=>

g

A switching function, f ) is lit -monotonic " whe n, for every

pair of valuat ions

V

and W , on s ome c o mmon subset of t or less variables

we have :

- 2 7-

-28-

or

Clear ly ,

if Y

Theorem

t� t-monotonicity implies y-monotonic ity .

<

4.l�t

monotonic

or both.

A

threshold switching function of n variables i s n-

( c ompletely

monot onic ) .

For reade rs more familiar with the terminology of

A switching funct ion is

Def init ion� ment

"

"

Vi O' unate functi ons:

posit ive

( negat ive )

in a given argu-

when the corresponding var iable appears everywhere in some dis j unct ive

normal Boolean ex�res s ion without

( with )

negation.

In e ither case, the

funct ion is !lunate il in that argument. Theorem

4.2:

A sw itching funct ion is one-monot onic if and only if it is

unate. Coro l lary :

Theorem

A threshold function is unate.

403�

junct ive form

A unate funct ion has exactly one irredundant normal dis-

( ion.dofJ »

this form cons i sts of the dis j unct ion of all

prime implicants of the funcL ion.

Further, this express ion is unate,

and s o demonstrates the unateness of the funct ion

403

0

A Family of Functi ons Characterizing Bilateral Threshold Nets As

an

alternative to the treatment proposed in Chapter 2, one can

describe a bilateral threshold sw itching net of n internal vert ices and p input s

.

by a set of n Boolean equat10ns:

1

Proofs of theorems

4.1J

[22]

4.2; 4.3

may be found in Winder,

[30]

among other places .

-29d. 1

=

f . (s ' 1 l

• • •

,s ,z ' P l

• • •

,Z ) n

i

=

1, 2,

• • 0,

The s . are inputs and the z . internal vertex s tate s , as before . 1 1

n

E ach of these

equat ions des cribes the behavior of a s ingle vertex in terms of the state s of the other vert ices in the net and the input s , and all are threshold functions. In the follo wing treatment we will cons ider additional properties possessed b y We begin with a central property result ing

the members of this s et of funct ions .

from the b ilateral nature of the conne ct ions bet ween vert ice s.

4.4 Pairwise Monotoni city Definition :

Let f . 1

=

f . (s ' 1 l

constant , we denote by ( f . ) 1

( f 1. ) w.

w. J

.

•

's 'Z l ' P

.

•

•

'zn ) .

.

and s imilarly for w

=

Then if w E B is a

the function :

f . ( s ' . .. , s , z '·.·' z . l l 1 J P

=

J

Definition :

.

-

l' Vi,

Z . 1' J+

••.

'Z ) n

-We

Let {f.) be a family of funct ions , as above . 1

Then if , for

all i, j :

( f 1. ) w.

J

�

( f 1. ) -w .

J

-+

( f . ) w. J

1

�

( f J. t-:w.

(4.2 )

1

the family of funct ions will be said to be "pairwise monotonic . " Le nrrna : t

If f . is a threshold s witching function and 1

on a c o mmon subset of the argument s of f . , then 1

1

Used by several authors, s ee [30], pp . 9, 10 .

V

and U t wo valuations

-30-

1) L.

a. . v . >

lJ J -

L.

a . .u .

lJ J

.

(f.1 ) V

�

( f.1 ) U

::>

-

common set of argum.ents and V

where the sum is over the

and u are the value s assigned j

j

by V and U, respect ively, to the j th argum.ent of

2)

If

( f )v i

E

( f i ) U'

then, with the s ame interpretat ion:

L.

The two funct ions

Proof :

a. .V lJ j

( fi ) V

L.

<

a. U . 1j J

( fi ) U

and

are thems elves threshold functions.

They differ only in the value of the threshold term k

(1.1)

curs ory examinat ion of the inequal ities

w ith the smaller threshold contains the other . negation of

Theorem

404:

Proof :

The family of functi ons

(401)

e q.

1.1).

A

shows that the function

2)

merely s tate s the

is pairwise monotonic .

.

it is one-monoton ic by theorem

( f. ) 1 w

.

J

Now suppose

( see

1).

Consider a member of this family, f 1

functi on,

( f1. ) .

::>

(f.1 )w .

J

( f1. )w.

::>

4.1.

S ince f . is a threshold 1 Thus , we know that

( f1. )W

or

J

•

or both

-

j

(r.)� ; 1

2)

by

of the lemma , thi s means that

j

W

a.

.

lJ

•

> -w

•

a. .

lJ

But then

W

Now cons ider f . . J

•

a.

.

lJ

(4.3)

> 0

Aga in:

or

(r.)::> ( fJ. )w . J w. 1

1

or both.

-

But no w suppose (f . ) J

w i

� ( f . }J

w i

31

-

Then, again by 2) of the lemma :

0

&-

w

0

a. . < -w Jl

0

a. .

Jl

But then w which, s ince a . .

lJ

=

0

a. . < 0 Jl

a . . contradict s the hypothes i s ( f . ) Jl

1

w.

J

::>

( f1. )-w.

J

0

There is every indicat ion that this generalization of the concept of monotonic ity to families of funct ions is a central property of the family of functions (4 . 1 ) for bilateral threshold s witching nets .

4.5

Extens ions of Pairwise Monotonic ity We no w come to some iwportant re sults which shed s ome light on the

question of why one cannot, for example , bui ld a logical os c i llator using bilateral thre shold s witching net s . in Chapter 3 to be impos s ible .

The reader wi ll recall that this was sho wn

It has been po inted out that one can eas ily make

one ne ighbor have more influence than another on the behavior of a given vertex . One might speculate therefore about construction of a logical osc illator taking the form of Fig . 7 . In

this figure , the arro wheads indicate s chemat ically the direct ivity

of informat ion flo w which one might try to achieve by appropriate we ight ing of the connections .

It turns out that such a patte rn of dire ct i vity i s not

poss ible because of a loss of s orrlething analogous to logical " fan-out . "

This ,

in a sens� is a generalizat ion of the property of pair wise monotonicity . Before we inquire into the pos s ibility of achieving an informat ion flo w topology s imilar to that of Fig . 7, it is necessary to e stabl ish what is

- 32 -

Figure 7

meant by tffe Arrowheads there in terms of the fam ily of funct ions

f 2

_ ; �

In terms of these, the pair of funct ions

de s cribing the behavior of the net . fl and

(4 . 1 )

have, for exampl� the following propert ies :

2) That the configuration of Fig o 7 may not be achieved is demonstrated by the following :

Theorem

4. 5 �

If a bilateral thre shold switching net , N, contains

three ve rt ices,

then :

1, 2 J

l and connect ions such that :

::>

(f2 >:-: , W

but

( f 3 )w

::>

(f )3 w2

but

(r1 )w 3

::>

1)

( f2 )W

3) 5)

I

2

I

( fl �

3

,

;

2)

(

f

) 2

=

l w

4) ( f2 )w 3

. ( f 1 )w ' 2

=

(f � 2

3

;

- 33For some valuation, �, over input s, the conditions 1 ) and 4) yield

Proof :

the strict inequal ity :

(4.4 ) or

( Note that condit ion 4 allows one t o choose the sign of the a term at 23 will i n (4.4) 0 ) tion,

a,

In

a s imilar fashion, 2 ) and 5 ) yield, for s ome valua-

over the inputs , the strict inequal ity :

or

(4 0 6) Now s ince f

3

i s a threshold funct ion, and hence one-monotonic , we know

that (f ) ::> ( f )3 w 3 wI Since f

3

and f

l

(f ) 3 w

or

-

I

-

I

::> -

(f ) 3 w

or both . I

must be pairwise monoton ic, we know from 5 ) that

::>

( f � . Therefore, it remains only to check whether ( f )- ::> ( f ) 3 w 3 w 3 w I I I I ( f )w 0 Suppose s O o Then, combining or, equivalently whether ( f ) 3 3 w 1 I this hypothes is with 3 ), for s ome valuat ion, 5, over the inputs , we have : (f ) 3 w

-

=

or

- 34-

Combining

(4 . 5 ) and (4 . 6 ) � (4 . 8 )

which contradicts

(4.7 ).

Therefore

Extending this method of proof, we obtain the following corollary .

Coroll ary : n

If a bilateral thre shold switching net,

ve r t ices

1, 2,

. 0 0 ,

N, contain s

n and connect i ons such that :

.'.

and

then :

This theorem and its corollary demonstrate what happens if one forces direct ivity of informat ion flow to occur in a net of thi s kind .

One can build

a chain of element s in which this takes place J but thi s chain cannot c�ose :on

- 35 itself becaus e o f the atten u at ion in the we ights o f the connect ing branches indicated by (4 . 8 ) for the three -vertex net .

One thus loses what amount s to

logical fan-out by c reating such a s ituat ion . That it is pos s ible to construct a net with t wo vert ices which satisfy

( f 1. ) w .

J

::>

( f 1. )-w.

and

J

i s sho wn by the net of Fig . 8 .

Here w

( f 1. ) w.

1

:J

( f . )-w. J

1

+1 .

Figure 8

The remaining question is :

What happens if one chooses to make

vertex 3 in Fig. 7 independent of vertex 1 1 t wo vertices affe cted then? Theorem 4 . 6 :

Ho w is the relation bet ween these

The ans wer is found in the follo wing theorem .

If a bilate ral threshold s witching net , N, contains

three verti ce� 1, 2 , � and connections such that : 1) 3)

( f3 ) w

2

::::>

( f 3 )--w , but 4 ) 2

( f 3 )-w

1

then :

:I

.

- 3 6-

Proof :

As in theorem

4 . 5,

we still have the relation

(4 0 5 ) :

Proceeding in the s ame fashion as before j we have , combining and

(5 )}

for some input valuati on�

(3 )

�:

or

Now, by the re sult of t heorem

we know that

( fl )w P ( f ):-: w l

.

There -

3 3 Then, combining this hypothes i s with

fore , suppose

(2)

4 . 4,

above , we have , for some input valuation, u,

or

( 4 . 10 )

But

(4 0 5 )

and

(4 0 9 )

combine to yi eld

( 4 . 11 )

( 4 . 10 ) .

The rEfore

( fl )P ( fl )w w

But s ince f i s l 3 3 a thres hold swit ching func t ion, and hence one -monotonic, we must have

which contradicts

•

or both;

- 37 so, s ince ( f ) l w

3

-:P ( f >:-: , and ( f >:-: -:P ( f ) ' it must be the case that l w l w l w 3

3

3

Again we have an obvious corollary : Corollary :

If a b ilateral thre shold sw it ching net , N, contains

n�·'-

ve rtices and connect ions such that (f ) 2 w

::>

l

( f � , but ( f ) 2 l w 2 1

and

Then : ( f )l w n Theorems '. 4 . 5 .' and ramifications of bilaterality.

4 .6

togethe r give an idea of the more subtle

That something like thes e relations exists

become s apparent all too quickly when one sets out to des ign bilateral threshold net s, but the nature of the diffi culty is not clear .

It may at first s eem

surpri s ing that no more direct extens ions of the concept of pairwise monotonic ity are pre sented .

An effort to discover such relat ions was made , but met with no

succe s s , largely because of the difficulty in defining a functional relat ion between elements not dire ctly connected .

-

38

-

It would at first seem the case that the se theorems impose very stringent limitat ions on the capab ilit ies of such nets , but this turns out not to be the case .

In

fact, a surpris ingly large clas s of automata are real izable ,

as we shall demons trate in the next chapte r . Before we devote attention to the quest ion of real izabil itYj however, we ment ion one last theorem concerning the family of functions (4 . 1 ) . Theorem 4 0 7 :

Let

N

be a bilateral maj ority switching net j and let the

family of functions (4 . 1 ) describ ing

N

be :

Then the net described by the set of equations

is real ized by the same topology as replaced by one with we ight Proof :

-

g

N

with each branch of weight g

o

For a maj ority net , we may write

Complementing all branche s corre sponds to replac ing ( A ) by ( -A ); thus : n+p L.

j =n+l

a . s ] = -F[ L. aij z iJ j j j =l

d 1. = f 1. ( s l �

n+p

n

•

0 0 '

S

P

,z ' l

•

•

•

+

,z ) n

L. a . o s ] j =n+l l J j

5. 5.1

SYNTHESIS OF BILATERAL THRESHOLD SWITCHING NErS

Introduct ion No theory of swit ching for any class of logic networks would be in

any sense complete without a discus s ion of the problem of synthe s i s of a net­ work to meet a de sired need . sion are two :

(1)

The central questions ar is ing in such a dis cus­

What class of automata is reali zable ?

membe rs of thi s class be reali zed?

and (2 )

How may

In sp ite of the impre ss ion of severe limita­

tion which may have been conveyed by theorems ( 3 . 4 ), ( 4 . 5 ) and ( 4 . 6 ) - -and which will probably be re inforced by any attempt to des ign such nets - - it turns out that it is pos s ible to realize a large class of automata us ing bilateral threshold switching net s, albe it in somewhat devious fashion .

5 .2

Clocked Automata and Comb inat ional Logic We begin our discuss ion of the synthe s i s of net s with a limitat ion :

Note that by theorem 3 . 4, we cannot reali ze any automaton which requi res a clock, for we cannot build the clock . It was pOinted out to the author some t ime ago by R . O. Winder that b ilateral threshold swit ching nets are capable of p erforming all combinational logic operations . 4.5

and

The scheme for demonstrat ing this i s suggested by theorems

4.6 ;

Theorem 5 . 1 :

Bilateral threshold switching nets can be used to pe rform

any combi nat ional logic operat ion . Proof :

Cons ider the element of Fig . 9 .

If k

=

2, the state of the

element and the s ignal appearing on line c will be the Boolean "or " of the state s of the vertices at a and b, and will be independent of the state of the vertex at c .

Similarly, for k

- 39-

=

4 , the state of the

- 40-

vertex will be the Boolean VIand" of a and b . directive .

The element i s thus

Negat ion is available through the use of inverting branches .

The extens ion of this technique t o multiple input elements is obvious 1 and one can build another element to "drive " this one by merely doubling the value of the threshold and all branch we ights .

Thus one

can s imulate, element for element, any directed combinat ional net .

a

I---..a.-","-- C

b

Figure 9

We have so far demonstrated the unreal izability of clocked automata and the realizability of all c ombinat ional logic .

We now c ome to cons ider what

class of sequential machines are realizable .

5.3

Realizability

of

Idempotent Automata

We begin this sect ion w ith a definition . Definit i on �

An

II

8 idempotent " automaton [ ] is one wh ich cannot di stinguish

the input sequence

t

81

a

•

a if ( ila followed by a " ) from the input sequence "a . "

Numbers in parentheses indicate branch we ights .

-41Idempotent automata have been called by other names , most fre quently " asynchronous " automat a .

It will b e demonstrated now that all su ch automata

can be realized using bi lateral threshold switching nets exclus ively .

Two

scheme s will be given for achieving such reali zations , the first most straight forward, the second often more _effic ient . For the first real ization scheme ] we require specification for the [ 20 ] . automaton i n the form of a Moore state dlagram .

We beg in with the following

observations concerning the propert ies of a state i of such a diagram for an idempotent automaton ( see Fig . 10 ) .

In this diagram, S . represents the s et of 1

all input combinat ions which can caus e the automaton to enter state i from s ome M . is the set of all input "memory " comb inations - - input c omb ina-

other state .

1

t ions which will cause the automaton, once in state i, to remain there .

W.

1

repre sent s the set of all input comb inati ons which can cause the net to leave s tate i and go to some other state . Now if Fig . 10 represents a state of an idempotent automaton as adverti sed, then we must have : M . ::> S .

1

1

and M.

1

where ::> and

n

n

W.

1

=

¢

denote set inclus ion and intersect ion, re spe ct ively, and

¢

is the

empty set . We realize the state diagram topologi cally, inserting in the net for each state of the state diagram a configurat ion such as that shown in Fig . 11 . this figure, the branches with arrowheads are connected to inputs . f

s

In

The inputs

and f ( vertex C in the figure ) are defi ned as follows ( cf . Fig . 10 ) : w

- 42 -

.";

�

�

�----------�

r

�----------�

"

..

.. . l�

____-......

.."J

_____

y-

branches ,

each

t i on shown at top

typ i c al conf igura­

of we ieht

�

st ate

preceding

(3)

to aux iliary

vertex for

(3)

Su

Figure

li .

each

(2 )

branche s,

of weight

�

w

"l'

(r)

I I I

{_ Realization of State 1 of

(1 )

( -1 )

Fig .

&

�

10

(2r

,

1\ 2 t3 )

=

+

I

:

-

3d

(2 )

-

2 �)

. �

each

(3 )

b ranche s , of we ight

'] 0

1

each of we ight

thre s hold

a branche s ,

I

I

� W

-44-

By ( 5 . 2 ) : f w The i nputs g

il

v

f

s

- 1

( po int A in the figure ) w ill be explained later .

We shall now

outline the operat ion of the net : To beg in with, assume that all vertices in the figure are in the state ( - 1 ) , and that the automaton is in some state whi ch can lead into state i . Now as sume that an input s . .

lJ

E

S . occurs ; the automaton should pro ceed into 1

This state change will be manifested in the net by a change in the

state i .

state of vertex C and its aux iliary vertex,

D.

As sume� for the moment , that one of the vertices of the type A assumes the state ( +1 ) .

The we ighted sum of the states observed by vertex Now s . .

(3 - 2� ) , so it be come s ( +1 ) also . that

ff

w

=

+1 by (503 )0

lJ

Since vertex

E

E

S.

1

implies that f

S

=

B

is now

+1 and als o

i s in the state ( -1 ) and the branch

j oining it to vertex C has negat ive we ight , the weighted sum of the states observed by ve rtex C is equal to ( +1 ) and it, being a maj ority element, as sume s the state ( +1 ) also .

It will be noted that the state of vertex

determined by that of vertex C .

D

is completely

Its only funct ion i s to provide a hysteretic

behavior of the ir common state with respect to changes in the states of other ve rti ces adj acent to C, and to provide a means , through the other branche s inc ident to it, of transmitting information about the state of C to other vertices without affe cting the state of C .

t

VlV"

Therefore, the state of vertex

denotes Boolean dis j unction and

u--- "

denotes negat ion .

D

becomes ( +1 )

-45als o .

We have now reached the po int where vertices B, C, and D, as well as

some vertex such as A, are all in the state ( +1 ) . Now the branches at F go to vert ices of the type A for each of the pos s ible suc ce ssor states to i .

It was by means of s imilar branches that the

vertex of type A which init iated the pre sent chain react ion was set to ( +1 ) . Next, assume that the latter vertex of type A, that to the left of vertex C in the figure , as sume s the state ( -1 ) .

The me chan ism through which thi s change

take s place will become apparent in a moment .

This causes the ve rtex B to

revert to ( -1 ) . As long as the input is not a member of the set W . , the automaton will 1

remain in state i .

This is clear because , unless the input i s a member of W . ,

the sum of the two inputs f

1

s

and f w will be at least zero .

The we ighted sum of

the state s observed by vertex C under this cond it ion is thus at least (2r + I ) . We have thus far shown that s . . E S . will set vertices C and D and any input lJ

1

not in the set W . will not re set C ( or D ) to ( - I ) . 1

Next, suppose that the input as sumes value w . . E W . . lJ

1

It will be noted

that w . . E S for some state x which can follow i in the state diagram x lJ

By

( 5 . 3 ) the sum of the inputs to ve rtex C is now equal to ( +1 ) again, the conne ct ions from D and E be ing the only one s having pos itive we ight .

The vertices

C and D thus remain in the state ( +1 ) until vertex E changes to the same state . Thi s will come about when the succe ssor state vert i ces corresponding to C and D have assumed the state ( +1 ) .

The 0 branche s of we ight ( +2 ) inc ident to vertex E

come from the points corresponding to G on each of the pos s ible next vertex pairs ; thus when one of these pairs assumes the state ( +1 ) , exactly one of these o

branche s will be j o ining vertex E to a ve rtex in state ( +1 ) .

Thus the weighted

sum of state s observed by E will be ( 3 - 20 ), and it will assume the state ( +1 ) also .

This " turns off " C, and subs eque ntly D, and st ill more subsequently the

vertex of type A which e nabled the successor state to be set .

-46We have thus reached the condition in which the succes sor state vertex and its auxiliary have been set and the pres ent state vertex has been reset

( to ( -1 ) ), all in a semi-modular fashion . t We now come to the much-pos tponed explanation of the functions g

iJ

o

Cons ide r the s ta te diagram segment shown in Fig . 12;:

1

Figure 12 .

Illus trat ion of the Need for the Functions g

iJ

Now if we were to connect the vertex B assoc iated with s tate d in a reali zation of the Fig . 11 type directly to the vertex s ince

�

E

S ' an input of d

�

as soc iated with state a, then,

when the net was in state a would cause the setting

of the vertices B, and subsequently of the both s tate f and state d .

D

C-D

vertex pairs, corresponding to

This could be prevented by forming at pOint 1 the

logical " and il of ( 1 ) the s ignal indicat ing that the net is in state a and (2 ) the logi cal " or " of the s ignals which can caus e a transit ion from state a to state d .

1

The latter function (2 ) is the g . . for this particular pOint , and lJ

Veri ficat ion is left to the reader .

-47-

. the ind icate d 1oglca1 " an d FJ 1. S .f onne d at a vertex Fig . 11 .

0

. f the t ype 1 ab e 11 e d A ln

One final remark is that it is be cause more than one input may well

be capable of caus i ng t.he same trans ition t,hat the nwnbe rs are not necessa rily as large as

t

a

and t3 of Fig . 11

and q� respectively "

The re remains one problem t o be cons ide red i n this real i zation .

Note

that if a state di agram ha s any loops such as that shown in Fig . 13aj then one of the branches

F

G

and one of t he branche s

of Fig . 11 wi ll ul timate ly i nfluence

the same vertex of type C of the same figure .

This could result in a failure

of the net to behave in the spe c i f i ed manne r and thus result in a nOD--e erri 'I'Ee s olut i on t o th is prob lem

modular net .

diagram of Fig . 13a, that of Fig . 13b,)l in

i E-

to realize ins tead of

wb i ch

U£

state

2

s tates al and a and b and b 2 l

are equ ivalent . Thi s di scus s i on prove s the important theorem � Theorem

5. 2 :

It is pos s ible t o c ons tru� t any idempotent automaton

us ing bilateral threshold nets exclus ive ly . As an example o f the applica t i on o f the technique j a reali zat ion o f the state diagram o f .Fig . 1 3b i s sho wn in Fig o 14 .

In

this state d iagram,)'

there are no verticES correspond ing to tbose of the type A} B,)l or s ince none ar e nece ssary .

Although thi s

l imi t s

E

in Fig . 11 ,

the resemblance of the example

to Fig . 11, it none theless make s it much easier to vi sual i ze the operat i on of the net .

To examine the operation of the ne tj assume that one of the four

state ve rtices and i t s auxil iary are i n the s � ate ( +1 ) and the others are all in state ( - 1 ) ; operat i on from such a poirt on depend s upon the inpu t sequence .

5.4

The

(P/2)

Feal i za t i on Scheme

Now that we have sho wn that it is pos s ible to cons truct a net reali z ing any idempotent automatonj

it

cecomes meaningful to att empt to iJnprove the

-48-

10 � o 0 o 1 1 1

A.2! o 0

o 1 1 0 11

(a )

State Diagram of ifF" Element

0 0 0 1 1. 0

o 0

q 4 1 �

11

10

10

o 0 o 1 1 0

11

Equivalent State Diagram Realized . by Method of Section 5 . 3

Figure 13

0 0 o 1 1 1

(1)

&

&

Figure 14 .

(- 1 )

(1 )

(-1 )

Realizat i on of State Diagram of

1

Fig . 13b

&

(1 )

f

a

I

I

+:­ \0

-5 0 technique employed in such a real ization .

Criteria for evaluating the relative

" goodness " of various reali zation s cheme s are normally expressed in terms of minimality- -or more accurately, economy- -although in some cases , reliab il ity has be come the primary cons ideration .

One should also give some cons iderat ion

to que st ions of what might be termed " des ignability " ; there should be a systematic method for achieving a "good ll realization .

We need a good definition of what

is

meant by a " realization . " Suppose that we have an automaton spec ification given in terms of a flow-table or state diagram . Definit ion �

A " reali zat ion " of a given automaton spec ificat ion by

means of a bilateral threshold switching net i s any net which has state behavior described by a specification equivalent to that above ; that is, we will at the pre sent stage disregard cons iderations of outputs . A reasonable apprai sal of the merit of a parti cular reali zation might be defined as follows : Definit ion � and

r

Let

�

be the spec ification of an idempotent automaton

the set of all net s which realize

for a reali zation

1

number of branches .

1

E

r

�

in the above sense .

let n . be the number of vertices and 1

Then the " cost " of

1.

1

Then �.

1

the

will be approximated by : (5.4)

and C2 are constant s called the " cost of the vertex " and the " cost of a branch " respect ively .

where C

l

- 51Under any definition of thi s kind , the reali zation schemes of s ect ion 5 . 3 will be costly, for it requi re s two informat ion storage vertices for each state .

Also a large number of inputs are re quired, and thus � will

be large als o .

The s cheme is of inte re st, however, in that it demonstrates i n

a clear and conclus ive manner the realizability of all idempotent automata .

In

thi s section) we present an alternative re ali zat ion techni que --one which conforms more closely to clas s ical methods of directed log i � des ign .

This method can

s ometimes be more economi cal than the previous one in the sense of ( 5 . 4 ) , but uses the same hyste res i s -pair configurat i on for memo ry . The alternat ive s cheme is dep i cted in Fig o 15 0

The central memory

of the net is contained in the hyste re s i s '-pairs C-D through

B-J,

p

in number .

Assume for the moment that p i s even; then the bas ic idea of operat i on of the net is that each of the stable states will htve an equal number of (+1) and ( - 1 ) states among the memory pairs

0

The number of such states available for ass ign­

ment to the vertices of the state diagram is thus

(P72)

restriction in the as signments of the s e states , however :

.t

There is a further the state as s ignment

made to each state must differ from that made to each of its ne ighbors in the states of exactly two memory elements o uti lize all

�72) states ,

Thi s means that one cannot always

and a dis cuss ion of this que s t ion later in thi s

chapter will culmi nate in the proof of theorem

5 0 4 , furthe r detailing the

re qu irements for realization o To understand the operat ion of the ne t of Fig . 15 , as sume first that the net i s in a stable s tate o

Then ex actly half of the memory vertex pairs

are in the state ( +1), and half are in the state ( - I ) .

Note that each of the

auxiliary vertices - -thos e of type D and T - - i s conne cted by means of a branch

t

( p7�

denotes the number of combinat ions of p items taken p/

2 at a time . U N I V ERSITY OF

(+1 )

-

typical at bottom

(+1 )

p-l branches , ea.ch{ of we ight

typical at top

of we ight

eacb.{

- '--

p-l branche s ,

eleme nt

aux iliary

fran

.....

I

h:t, (p-1 ) 1)

(1 )

! ---P"""'!-

-

Pigure 15 .

each of we ight

��

(2r

=

+

1 -

2 (-2 )

4(p - 1 ) - 3)

branches , . to a type A element for each

(p-1 )

to a type B

- 4p

each of we ight

Realization Scheme

&

other pair .

{,., ,

&

b ranches,

2r

=

eleme nt for each other pair; also,

(p-1 )

(2 {p - 1 ) ( -2 (p - 1 )

threshold

I

I

\Jl I'\)

-

53

-

of we ight ( -2 ) to a vertex of type A as sociated with each of the ( p

-

1)

remaining elements , and s imilarly by means of a branch of we ight ( +2 ) to a ve rtex of type B as sociated with ��ch of the (p - 1) remaining elements . inputs f

M

and f

N

s erve �uch the same funct ion as the input s f

S

and f in W

. . of Fig . II . . . are s i�ilar in funct ion to g lJ Fig . II} and g . . and h lJ lJ

idea of operation is th is :

The

The bas ic

When an input change oc curs which requi re s that the

ne t change state jl the state change take s place in two phases .

In the first

phase, the appropriate C-D vertex pair changes from state (+1) to (-1); in the se cond phase} the othe r appropr iate C-D vertex pair changes from s tate ( -1) to (+1), complet ing the trans ition .

Spe c ifically, in e quilibrium, the s tate of

each C- type vertex i s susta ined by its aux iliary} for , although the range of values whi ch the inputs may as sume is ( -2r ) to ( +2r ), the vert ices of type E and F assoc iated with each C-D pair normally disagree in s tate .

The E-type

vertices are normally i n the state (-1) and the F- type (+1) . When an input occurs in conj unction with the proper combinat ion of ( -1) state s to require a state change, however, the F vertex as soc iated with the C-D pair whi ch must change from (+1) to (-1) assumes the s tate ( - I ) . corre spond ing E vertex remains ( -1) and the i nputs f

Mi

thus the we ighted sum of states obse rved by the

C

The

and f go to (-1); Ni

VE r tex is ( -1) and it change s

s tate, causing the as soc iated D vertex to follow sui t .

This marks the end of

the first phase of the trans it ion; at th i s poi nt, (p/ .? + 1) elements are in the state (-1), and the remaining l P /2

-

1) element s in the state (+1) .

When the proper comb i nation of (p/2 + 1) elements in s tate (-1) occurs in conj unction with the above input combinat ion , the appropriate

E

vertex

associated with that C-D pai r which mus t change from (-1) to (+1) as sume s the s tate (+1) 0

The corresponding

F

-

vertex remains �+l ) and the input s f and f Ni Mi

go to (+1) ; thus the we ighted sum of s tate s obse rved by tLe C vertex is (+1)

- 54and it changes state, caus ing the as s oc iated D vertex to follow suit, and complet ing the trans it ion . To s ummari z e , the F vert ices remain in the

(+1 )

state unles s the

required input occurs in conjunct ion with the proper set of

(-1 )

S imilarly, the E vert ices remain in the

( p/2 )

states

( -1 ) .

s tate unle s s the required input

( p/2 + 1 ) states ( -1 ) . Trans itions from ( p/ 2 ) states ( -1 ) to ( p/2 + 1 )

occurs in conj unct ion with the proper set of always take place in the s ame order, going s tates

( -1 ),

then back to

( p/2 )

s t ate s

( -1 ) .

This real i zation se·heme can be

altered to i nclude the cas e in wh ich p is not even by appropr iate adjustment of thres holds and we ights . As mentioned previous ly,

it may not be pos s ible to real ize an

arb itrary state diagram of n states with a number of memory element pairs p

(/)

whi ch is minimal in the sense that

is the smalles t integer greate r than

A bound on the number of memory elements required i s specif ied

or e qual to n . by Theorem

P 2

5.3. For a dire cted linear graph G wi thout parallel branches ,

Definition : define the

"

image graph

"

to be the nondirected graph, G I , derived

from G by appli cat ion of the following operations :

1)

Replace each

( dire cted )

element of G with a nondirected

element . 2

)

3)

Eliminate all S l i ngS .

1

Replace any pair of parallel branche s resulting from with a s ingle branch .

t

Branches which begin and terminate on the same vertex .

1)

- 55 Let. S be the state diagram of an idempotent automaton

Theorem 5 . 3 :

and let S ' be the image graph of S . of the

c;/�

(1 )

Y

O

Then S is real izable by means

reali zat ion scheme with p � max (x ' Y ) where o O =

smallest integer y such that

�7� �

number of

nodes of S ( or S ' ) .

(2 )

X

o

=

smalle st integer x such that

(� )2 � degree

of

each vertex in S ' . Proof :

The ne ces s ity of the re qui rement ( 1 ) is clear; we have thus

only to worry about the si tuation which might ar ise if the degree of one or more vert ices were too great to allow the number Y determine p .

o

to

We observe that what we really require is that the

graph S ' be imbeddable in the subgraph of the x- cube cons i sting of pre cisely those vert ices wh ich have x/

2

will be true when x is chosen such that

nonzero components .

(2 )

Th is

is satisfied, for

( � )2

is the degree of each vertex in the subcube . We make me nt ion at this pOint of a practi cal note . -

In the case of

-

both of these reali zation schemes , we have made use of vert ices of arbitrarily large degree .

The assumpt ion that such ex ists i s i n one sense unrealist ic , in

that circuit element tolerances will normally restrict the degree of ve rtices . This doe s not make such real i zat ions impo s s ible , howeve r, for one might construct nets of vert ices of lesser degree which s imulate the behavior of a vertex or group of verti ces of larger degree , and replace the latter by the e quivalent net in a reali zation.

5.5

General Synthe sis Techniques The observant reader will have not iced that both of the real ization

techni ques outl ined so far are bas ically of a direct ive nature .

The trick of

- 56 -

forming hysteret ic pai rs has enabled us to achieve memo ry in nets , as well as l imited directivity, and we have relied heavily on the se in both s chemes .

One

might well ask why no scheme is presented which rel ies more on the inherent lack of directivity of bi lateral thres hold switching net s - -one wh ich would go from state to state in a completely deterministic but nonobvious fashion .

This

problem has been considered at length, but no sat i s facto ry solut ion has been found, nor,

it is felt ,

is one l ikely to be found in the immediate future .

In order to achieve a systemat ic synthe s i s procedure based on,

for

example , the matr ix analys i s technique of Chapter 2, we might require that D of e quat i on

(3.2 )

be the next s tate for any state and input, and that the network

be semi -modular .

Thus for each s tate of the s tate diagram, we would have a set

of relations like

(3.2 ).

II

The problem at f i rst seems the refore to resemble the

1 . . c aSS l caI " secondary var labl e as slgnment pro blem .

0f

1

async hronous sequent la ·

swi tching c i rcuit theo ry .

In the latte r, however, one i s two main concern s are

e conomy and eliminat i on of

" c riti cal races . "

In the present instance ,

one must

f ind a solut ion to a very large sY·3tem of s imultaneous linear inequalit ies as well .

The re are some state as s ignment s wh ich are impo s s ible in the sense that

they result in inconsi s tant ine qualities .

In other words , the threshold character

of the network functi ons limits the po ss ible state assignment s in a manne r as yet undete rmined .

The problem i s in ess ence a general i zat ion of the problem of

characteri z i ng the set of all funct i ons reali zable by a s i ngle threshold element - -a problem which its elf is as yet incompletely unde rstood .

For these

reasons, the search for such general techniques was , for the pre sent, abandoned after consi de rable effort .

60 6.1

AFTERWORD

Summary An effort has been made in this paper to preE ent a theory of swi �chi ng

for networks of two- terminal b i stable devices .

A model has been propos ed in

Chapter 2 for what seems the mos t reas onable fonm for suc h ne tworks � and an analysi s procedure based on this model developed - -also in Chapter

20

In

Chapter 3, several propert ies of such ne� s were demons trated - - includ ing the critical property of proceeding uniformly t o e quilibrium- -and a canoni cal form The ques t i on of tee number of s�able state s w� ich might be pos s e s s ed

developed .

by such a ne t was also cons idered) and an atterr.pt was made to find a bound on this number . Next, we cons idered characteri z ing the behavior of such net s in a manne r more familiar to de s igners of convent ional logi c, and we re able thus to begin to charac te rize the elus i ve tf fan-out " pror,erty wh ich plagues from the very start any attempt to use convent ional logical des ign te chni ques on net s of this type .

It was he re also that, through development of the property of

II

•

•

palrwlse

monotoni c ity, 1I some ins ight was gained i nto the relat ionship between tee bilateral nature of the connecti ons between elements and the mono tonic nature of the s tate functions .

The properties demonstrated up to th is point did not bode well for

any hope s of meaningful logi c capabil ity, and the reali zat i on that one cannot realize any automaton re quiring a c lock d id no tting to disr,el the gloom section 5 . 3, howeve r, it was demonstrated that

the

0

In

set of all asyncr ronous

machines are real izable , and in sec t ion 5 4, a firs t attempt was made to find a reali zat ion which would be i n s ome sense economi cal o To summar ize the summary} we bave a model J an analys i s I rocedure, a canoni cal form} a handful of important propert ies , and two synthes i s s chemes .

- 57 -

- 58 -

6.2

Sugge st ions for Future Work To attempt to detail areas for fruitful future i nvest igat ion is in

one sense ludicrous , because one can proceed in virtually any d irect ion he de s ire s .

Some attempt should be ' made to build nets of this kind, to see how

well the theory will predi ct re sults .

Further inve st igat ions into t he propert ies

of the family of funct i ons d i s cussed in Chapter

4

could well shed more light on

logical de sign te chni ques , and pe rhaps provide a bas is for a technique of the type d i s cussed in s ection 5 . 5 .

Another approach which appears promis ing but

has s o far yielded few re sult s i s that of cons idering trans format ion of the n- cube . state s has yet to be found .

( A�

-

�)

as an affine

F inally, a good bound on the number of stab�e

BIBLIOGRAPHY 1.

Arden, D . N . , " Delayed Logic and Finite State Machine s " ; Proc . First and Se cond Ann . Symp . SCTLD; AIEE Pub . S- 134; Sept . , 1961; pp . 133-151 .

2.

Bartky, W . S . , and Muller, D . E o ; " An Illiac Program for Simulat ing the Behavior of Asynchronous Circuits and Det ect ing Undes irable Race Conditions " ; pres ented at ACM Nat ional Mtg o ; June , 1957 0

3.

Berge , C . , Theorie des Graphes et ses Applications ; Paris : English Ed . - -New York : John Wi ley and Sons, Inc . , 1962 .

4.

Cadden, W . J . , IIEquivalent Se quent ial Circuits If ; IRE : pp . 30-34; ( March, 1959 ) .

Dunod, 1958 .

PGCT; CT-6, 1,

5.

Caldwell, S . H . , Switching Circuits and Logical Des ign; New York : Wiley and Sons, Inc . , 1958 .

John

6.

Cameron, S . H . , "An Est imate of the Complex ity Re qu i s ite in a Universal De c is ion Networkll ; WADD Tech . Rep . 60- 600; March, 1961; pp . 197-212 .

7.

Crane , H . D . , " On the Complete Log ic Capability and Realizability of Trigger- Coupled Neuristors " ; Stanford Re s . Inst 0 , Proj ect No . 3286 Interim Rep . No . 4; July, 1961 .

8.

Elgot, C . C . , and Rutledge, J . D . , " Operat ions on Finite Automata " ( Extended Summary ) ; Proc . First and Second Ann . Symp . SCTLD; AlEE Pub . S- 134 ; Sept . , 1961 ; pp . 129-132 .

9.

Fan, K . , " On Systems of Linear Ine qualities Vl; in Linear Inequalit ies and Related Systems ; Kuhn, H . W . , and Tucker} A o W . , eds . ; Princeton : Princ eton Univ . Pres s , Ann . Math . Studies , No . 38, 1956 .

I

10 .

Frazer, W . D . , " A Preliminary Inve stigat ion of the Propert ies of Switching Networks of Bilatera l Elements " ; ( Unpublished ) RCA Laboratories , Internal­ Memorandum, 1961 .

11 .

Ginsburg, S . , "Examples of Abstrac t Machines Wi ; IRE : pp . 1)2- 135; ( April, 1962 ) .

12 .

Goto, E . , et al . , " Esaki Diode High-Speed Logical CirCUits " ; IRE : EC-9, 1, pp . 25 -29 ; ( March, 1960 ) .

13 .

�onig, D . , Theorie der Endl ichen und Unendlichen Graphen; New York : 1950 .

14 .

Kuhn, H . W . , " Solvab ility and Consistency for Linear Equat ions, and Ine qualities " ; Amer . Math . Monthly, 63, 4, pp . 217-232, ( April, 1956 ) .

15 .

Kunihiro, T . , " Applications of Tunnel Diodes in Switching CirCUits " ; Univ . of Illinois , Digital Computer Lab . Rep . No . 102 ; Oct . , 1960 .

- 59-

PGEC; EC-ll, 2 , PGEC, Chelsea,

-6016 .

Landauer, R . , " Irrevers ib ility and Heat Generation in the Comput ing Proces s " ; IBM J . Res . and Dev . , 2., 3, pp . 183- 191, ( July, 1961 ) .

17 .

Lewin, M. H . , "Negat ive Res i stance Elements as Digital Computer Component s "; Proc . EJCC, pp . 15 -27, De c . , 1959 .

18 .

Lo, A . W . , "Some Thoughts on Digital Components and Circuit Te chniques " ; IRE : PGEC, EC-IO, 3, pp . 416-425, ( Sept . , 1961 ) 0

19 .

Minsky, M . L . , " Some Universal Elements for Finite Automata"; in Automata Studies ; Shannon, C o E . , and McCarthy, J . , eds o ; Princeton : Pri nceton Univ. Pres s , Ann . Math Studies , No . 34, 1956 . 0

20 .

Moore, E . F . , "Gedanken Experiments on Se quential Machine s " ; in Automata Studies ; Shannon, C . E . , and Mc Carthy, J . , eds . ; Princeton : Princeton Vniv . Press, Ann . Math . Studies , No . 34, 1956 .

21 .

Moser, J . K . , "Bistable Systems of Differential Equat ions with Applications to Tunnel Diode Circuits "; . IBM Journ . Res . and Dev . , 5 , 3 , pp . 226-240, ( July, 1961 ) . -

22 .

Muller, D . E . , and Bartky, W . S . , " A Theory of Asynchronous CirCUits " ; Proc . Int . Symp . on Theory of Switching; Cambridge : Harvard Univ . Press, 1959; pp . 204-243 .

23.

Muroga, S o , "Functional Forms of Dual- Comparable Functions and a Necessary and Sufficient Condition for Realizability of a Maj ority Function " ( Revisedl Revis ion of paper in Proc . First and Second Ann . Symp . SCTLDj AIEE Pub . S-134; Sept . , 1961, pp . 39-46 .

24 .

Onyshkevych, L . S . , et al . , "Parametric Phase-Locked Os cillator- ­ Characteristics and Applications to Digital Systems " ; IRE : PGEC, EC-8, 3 , pp . 277-286; ( Sept . , 1959 ) .

25 .

Paull, M. C . , "Review of 4 Pape rs in Threshold Logic "; IRE : 3 , pp . 541- 542 ; ( Sept . , 1961 ) .

26 .

Ringel, G . , Farbungsprob1eme auf Flachen und Graphen; Berlin : Verlag Gewissens chaften, 1960 .

27 .

Seshu, S . and Reed, M. B . , Linear Graphs and Electrical Networks ; Reading, Mas s . : Addison Wes ley Pub . Co . , Inc . , 1961 .

28 .

Unger, S . H . , "Hazards and Delays in Asynchronous Sequent ial Switching CirCUits "; IRE : PGCT, CT-6, 1, pp . 12 -25; (March, 1959 ) .

29 .

Whitney, H . , "On the Abstract Propert ies of Linear Dependence "; Amer . J . Math . , LVII, pp o 5 09 -5 33 , ( 1933 ) .

30 .

Winder, R . 0 . , "Threshold Logic " ; Ph o D . TheS iS , Princeton Univers ity, 1962 .

31 .

Winder, R . 0 . , " Characteriz ing Threshold Funct ions " (Review of 6 papers in threshold logic ) ; IRE : PGEC, EC- l1, 5, pp . 717-718; (Oct . , 1962 ) .

PGEC, EC-IO, Deutsche

APPENDIX A

AN EXAMPLE OF THE MATRIX ANALYSIS TECHNIQUE Cons ider the net shown in Fig . 16 :

-1

Input +1

Notes :

1)

All thresholds zero .

2)

All branche s of we ight + 1 unle ss othe rwise indi cated 0

3)

Numbers bes ide vertices indicate mat rix order .

4)

Shaded vertices have init ial s tate -1; others , +1 . Figure 16

The matrix A for this net is :

-61-

-62-

A

The vector � i s ,

=

0

1

0

0

1

0

0

0

1

0

1

1

0

0

0

0

2

0

1

0

1

0

0

0

0

1

0

0

1

1

0

-1

0

0

0

0

0

1

0

0

-1

0

-1

0

0

1

1

0

0

0

0

-1

0

1

1

0

0

0

0

0

0

0

1

0

1

0

1

0

0

0

0

0

1

1

0

0

1

init ially :

1 1 1 -1 Z

=

1 1 -1 -1 1 -1

U s ing the relat ion

F[�]

=

�, we find that :

1 1 1 D

=

1 1 -1 -1 -1

1

0 0

-63From this , we can see that vert ices It may also be verified that

Q

4

and

6

are excited in the init ial state .

is an equilibrium state .

Thus as a synchronous

machine with an external clock s ignal, the net make s but one state trans ition . As an asynchronous machine , on the other hand, the net can enter e ither of two addit ional states from the init ial state; vertex

4

alone , or vertex

6

alone may

change . In these cases, the corresponding states would be :

Zl -4

=

1

1

1

1

1

1

1

1 1

1

1

1 1 1

-1

1

-1

-1

-1

-1

-1

1

Zi =u

=

� =

1

-1

-1

-1

1

1 -1

-1 -1

-1

Note that the latter case leads again to the same equi librium state , for only vertex

4

remains excited .

In the former case, vert ices

5

and

6

are excited .

Allowing these verti ces t o change individually, we find that the corresponding states are :

1

1 1

1

1

1

�46

=

1 1 1 1 -1 1

Q45 = 1 -1 -1

1 1

1

-1

1 1 -1

-1

-1

-1

-1

-1

-1

-1

-1

1 1

D 6= -4

V -Z45

==

-64Now we have a violat ion of semi-modularity, for vertex 5 has pas sed from excitation to equilibrium without changing when vertex 6 was allowed to change alone ( compare � and � ) o 4 46 Vertex 6 is still exc ited in the latter case; so we allow it to change and f ind that the result ing s tates are :

.

:

�4 56

=

1 1 1 1 -1 -1 -1 -1

124 56

=

1 1 1 1 1 -1 -1 -1

Therefore this change sequence leads also to the same equilibrium state . This net is an example of a speed- independent but not semi-modular net in which the balance condit ion never oc curs .

The fi nal state will be that

given by � regardless of the order of change impos ed upon the excited vertices , but one sequence of changes --notably 4, 6 - - causes a violation of the semi ­ modularity condit ion .

APPENDIX E E.l

Proof of Theorem 3 . 10 We cons ider only autonomous canonical nets , for the que stion of stable states has meaning only with re spect to constant input, and the inputs may then be regarded as part of the constant s ources The rows of A for such a net may

as soc iated with the canonical net .

be looked on as a set of vectors , a . , in Euclidean n+2 space having -1

the following properties � 1)

The set of n vectors is not necessarily linearly independent .

.!: )

2)

The first ve ctor must lie in the x the se cond in the x 2

3)

=

l

=

0 hyperplane ,

0 hyperplane , etc .

For all i f j , i, j � n, the x . component of the ith J vector must equal the x . component of the j th vecto r . 1

4)

The lengths of all vectors are the same .

The fourth constraint , stemming from the regularity of the canonical net, implies that the loc i of poss ible cho ices for the ve ctors , a . , -1

are hypers quares in each of the zero-c oordinate planes . In like manner, the 2

n

pos s ible states of the network may be

looked on as vectors from the origin in n-space to each of the vert ices of the cube of s ide 2 centered at the origin . one of the se s tates having z

l

=

Now suppose we choos e

+1 and try to determine how then to

choose A so that 1)

The net will have the chos en state as an equilibrium state

2)

The net will have a maximum number of additional equilibrium

and

I,

states . -6 5 -

-66We note that for a canonical net , equat ion

F[ A�]

=

(303)

becomes

Z·

The proces s of matr ix mult ipl icat ion thus corresponds to forming the dot product of the total state vector t he constant sources

)

with eac h,

( includ ing

pos it ions for

in tur� of t he row vectors of A .

A stable state i s one for wh ich, for each of these mult ipl icat i ons , the cos i ne of the inc luded angle has the same s ign as the corre s ponding component o f the state vector .

Now t o go back t o our chosen

e quil ibrium state . In the x

=

l

0 hyperplane l ie the proj ecti ons of the 2

state s ; there are 2

n- l

-1 -1 -1

-1 -1 -1

-1 -1 1

-1 1 -1

o

1

1

1

1

2

n-l

ve ctors

We now des i re to p ick an � whi ch will max imize cond it ion l

s tate

( +1 )

(1 ) .

Now vertex

1

(2 )

above ,

w ill be in equilibrium in the

whenever the re st of the ve rtices of the net are in any

state who se proj ection makes an angle of 90 t ion of � i n the x l l ve ctors .

pos s ible

of these and we can list them :

o o o

subj ect to condition

n

=

0 hyperplane .

0

or le s s w ith the proj e c ­

The re are at most 2

Be cause of theorem ( 3 . 8 ) however, at mos t 2

can be equilibrium state s .

By t.he o.r.efn.

ponds t o two equilibr ium state s . states i s bounded above by 2

. 3 . 9,

n- 3

n-2

such

of these

however, each corre s-

Therefore the number of e quilibrium

n-2 .

•

-6 7 The interested reader is invited to try to extend this argument and thus improve the bound .

B.2

Another Charac teri zation of Equil ibrium States Another approach to �he quest ion of counting the number of pos s ible

equilibrium states is this .

Recall the equation for an equilibrium state :

( Here A has been augmented to be s quare ) .

This is equivalent to ;

where Q . is a diagonal matr ix with finite pos it ive entr ies . 1

Now Q . is non1

s ingular , so we have

We the refore des ire to know how many unit e igenvalues we can have as Q. ranges over all pos s ible Q o 1

The s et o f suc h Q i s finite because each row

entry of Q . is bounded by the maximum degree of a ve rtex o 1

APPENDIX C

SUPPLEMENT TO PROOF OF THEOREM

We wish to show that ex is t o

To s ee that this i s

SOi

�

3Q7

group of states s uch as T I U1 and V mu st always cons ider the follow ing argument x

Suppose we have a speed-depende nt net N which can proceed from initial s tate I to any one of s eve ral equi libr1um state s i E l l know that each of E l l

Q Q Q; E o t

E is ac cess ible from II and that none of E l l t

. Q .i

i s ac ces s ible from any other, s ince all are equ ilibri um states .

an

Now we Q "

E t

�

As indicated in

the main body of the proof, the s ingle -vertex- change s e quence diagram must be a Has se diagram s ince no state can repeat . Now cons ider an arb itrary equilibrium state E acces s ible from s ome set of states a

( ai l ,

=

•

•

•

,a

ik

)Q

.

1

•

Such a state i s

Cons ider a member, a

of this set; either some othe r equilibrium state is ac ces s ible from a If

SOl

iq

t o an equilibrium state other than E

If no equilibrium state o ther than E

members of al then cons ider the set � of a are ac ces s ible from I .

or not .

1

then the appropriate � .

U

=

( �il "

corresponds t o V of F1S .

•

I

�

i

P)

of state s via which members

is ac cess ible from same member of � or not o

corre sponds to T I the appropriate a '

1j

If

to Uj and"

to another equilibr1um asain, the first state i n any sequence leading from � ix state, to V .

If not , then we consider the set of s tates from whi ch members of

� are ac cess ible .

1

t

It is worthwhile to not e that while the intersection of the sets a and � need not be emptYI any � . I as above , may not be a member of a . lX

-68-

4.

i s acces s ible from any of t he

i •

i

Again we apply the s ame argument; e ither s ame

equilibr ium state other than E . SOl

�

then a . corresponds to T, E to UI and the first state i n any sequence i lq

leading from a .

iq

iq

-69Now the state sequences are e quilibrium

all finite and nonrepetit lve,

states are acces s ible from I .

Therefore, we must,

if

and

all

we proceed i n

this manne r, eventually find some first set, � , of states in this sequence with the property that more than one equilibrium state is access ible from same member �

iz

and

of thi s set .

In this case, �

via which the net can reach

iz

Ei

corre sponds to

T,

any state adj acent to �

iZ

corresponds t o U, and any state adj acent to

� . and via which the net can reac h any other equilibrium state corresponds to 1Z Since E . was chosen arb itrar ily, this argument applies to each of 1

V.