The computer-aided design of nano-scaled digital circuits

Rochester Institute of Technology

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6-2004

The computer-aided design of nano-scaled digital

circuits

Frank Alva Krueger

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The Computer-aided Design of

N ano-scaled Digital Circuits

by

Frank Alva Krueger

A Thesis Submitted

III Partial Fulfillment

of the

Requirements for the Degree of

MASTER OF SCIENCE

in

Electrical Engineering

Approved by:

PROF. _ _

S_e_r-=-9_

e-"....-y_L-"...-y_sh_e_v_s_k_i _ _

_

Sergey E. Lyshevski (Advisor)

Vincent Amuso

PROF. ____________________________________ _

Vincent Amuso

PROF. _ _ _ _ _

D_a_n_ie_I_P_

h

_

i

p'---S

_ _ _ _

_

Daniel B. Phillips

Robert Bowman

PROF. ____________________________________ _

Robert J. Bowman (Department Head)

DEPARTMENT OF ELECTRICAL ENGINEERING COLLEGE OF ENGINEERING

ROCHESTER INSTITUTE OF TECHNOLOGY ROCHESTER, NEW YORK

Thesis/Dissertation Author Permission Statement

Title of thesis: The Computer-aided Design of Nano-scaled Digital Circuits

Name of author: Frank Alva Krueger Degree: Master of Science

Program: Electrical Engineering College: College of Engineering

I understand that I must submit a print copy of my thesis or dissertation to the RIT Archives, per current RIT guidelines for the completion of my degree. I hereby grant to the Rochester Institute of Technology and its agents the non-exclusive license to archive and make accessible my thesis or dissertation in whole or in part in all forms of media in perpetuity. I retain all other ownership rights to the copyright of the thesis or dissertation. I also retain the right to use in future works (such as articles or books) all or part of this thesis or dissertation.

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~~

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Abstract

Theuse ofCMOS-based transistorsto implementdigital logicisthe prevalent

means of modern computation. It _is, however, not _{the only}means. Advances innano-scienceand_engineeringdemonstratethatnano-scaleintegratedcircuits

are in fact a viable_technology for computation. The dominant means for in formationpropagation in thesedevicesisquantum_tunneling

-a phenomenon

that isnot _whollycompatible with currentdesigntechniques. Thispaperisan

explanation of one process usedtoboth designand simulatedigitallogiccircuits

utilizing thetopologyofthehypercube. Theaim ofthepaperis todemonstrate theease of_designingand_implementing astreamlined designenvironment and

to demonstrate _{the utility that}such an environment affordsthe designer. The

hypercubetopology is usedasthe dominantexample forconstructing 3D cir cuits. In this topology, each device isrequired to operate as a _doubly gated

switch and computationis performed _utilizing a concept similarto pass-gate technology. Thepaperdetailsthesoftware requiredtogeneratethelogiccircuit

and themeans of simulation. Each deviceofthe structure ismodeled _usinga

non-linearstate-space representation. Thepaper concludes with twoexamples

ofimplementabletechnologies: single-electrontransistors(wrap-gatestructures

Table

Contents

Abstract ... ii

Table ofContents . . . . iii

ListofFigures . v

1. Introduction 1

1.1. Nano-scale digital circuits 1

1.2. Computer-aided design ... 2

1.3. Overview . . . 2

1.4. Practicalconsiderations . . . 3

2. Representations oflogic functions 6

2.1. Circuits asdirectedgraphs. . 6

2.2. Scheme_programminglanguage . ... 9

2.3. _Binarydecision diagramsandtrees . 12

2.4. Hypercubes . . . , . 14

2.4.1. BDDtohypercubetransformation algorithm 14

2.4.2. Methods of optimization .... 15

3. Simulationof3D logic circuits 17

3.1. Cellularnon-linear networks . 17

3.1.1. Cellularautomata . 18

3.1.2. Cellular nonlinear networks . 20

3.1.3. Higher-orderconfigurations ... 22

3.1.4. Higher meaning . . 26

3.2. Extensionofthemodeltothehypercube. 28

3.2.1. Logiccircuitdesigner 30

3.2.2. Dynamic modelintegrator 38

3.2.3. Matlabsimulation 45

3.3. Sample hypercubedesign and simulation . 61

3.4. Conclusion . .66

4. Technology-specific integration 70

4.1. Integrationof single-electrontransistors ... .70

4.1.1. Physicalmodeling 71

4.1.2. Example integration . . . . 78

4.2. Integrationof molecular components . ... . . 84

4.2.1. Physicalmodelof_N@C6o 85

4.2.2. _N@C6o asalogic device . 92

5- _{Conclusion 96}

A.

Displaying

List

Figures

1 Example feed-forward logiccircuit . 7

2 Example feddback logiccircuit 7

3 Examplebinary decision diagram . . ... 12

4 Example3Dhypercube . 15

l Output function for CNNnode ... . 21

2 Exampleprogression ofCNNstates overtime . . 24

3 Exampleprogression ofCNNstates . 25

4 Simulationof reaction-diffusion equation 26

5 Reaction-diffusionsimulation with randominitialconditions 27

6 Multiple_input, single output node 29

7 Singlenode simulation . . . ... 30

8 Hypercubeof exclusive-or 37

9 4Dhypercubeexample 38

10 5D Hypercubeexample ... 39

n Comparisonoflinear andhermitian outputfunctions 41

12 Anexample oftwo_{automatically}generated inputfunctions 59

13 Sample6Dhypercube . . 64

14 Outputof6Dhypercube . . ... .66

15 Input parametersto6D hypercube . . 67

16 Samplemodel script 68

17 Samplesimulation script . . 69

1 Single-electrontransistor 71

2 Single-electrontransistorequivalent circuit . 75

3 SETmodel script . . 83

4 C60cage . ... 84

5 PotentialmapofC+N>C electron path. 88

6 Currentvs. voltagefor fullerene. ... . . 93

7 Differentialconductance vs. voltage for fullerene . 93

i.

Introduction

ThisTHESISisthedetailed descriptionof a process usedto_developa

computer-aideddesignenvironmentforthecreation ofthree-dimensionalnano-scale _dig ital circuits. Its goal is to demonstrate how the concepts of _(i) logic design,

(ii) the transformation (or _compilation) ofthose designs to physical _devices,

(iii) methods forsimulation, and _(iv) the derivationof physical models canbe integrated into one cohesive environment and used tosolve specific problems. Thatspecific problemis the dynamic analysis of nano-scaledigitalcircuits.

Tothatend thedetailed descriptionofthesoftwaredesign processisgiven in addition to all code usedto implement the designenvironment. Although thiscodeimplementsonly theminimum of what shouldbe includedinadesign environment, it is, nonethe_less,a sophisticated applicationthat can adaptto and beadaptedtoa varietyof circumstances.

An introduction to the concept ofthe design environment is presented in thischapterinadditiontoadescriptionandjustificationofthegeneraldirection taken throughout theremainder ofthethesis.

1.1. Nano-scale digital circuits

The choice of problem _area, nano-scale digital circuits, stems from a present needtoperform_exploratoryresearch. There isageneral_{understanding in the} design community that CMOS technology, the present ubiquitous solution to large-scale logic _design, will _somedayhaveto besupplantedinorder _{to satisfy} desires for more complex devices. Ofcourse, there is neither consensus nor general _{understanding} of what form this new _technology will take. Present studies _{are,therefore,in their exploratory} phase.

Thestudyof physical sciences atthenano-scalehasbeenenhanced_{greatly in} thelastyearsthrough theuse of computer simulation. _Technologyis_beginning toreach the point whereit has become feasibleto simulate _{(or model)} multi-electronsystemsinorderto determinetheirproperties [4]. While thephysical and mathematical_{understanding}ofthese systemsismore than a_half-century old, the computationalcost of _{analyzing them}in a meaningful _way had been toogreattoundertake.

Thecouplingoftherecent (withinthelast_decade) availabilityofthis com putational power with other advances in _fields, such as _{biotechnology,} offers scientists and engineers an amplefieldtoinvestigate foralternatives toCMOS technology. _Today, structures that act upon the influence ofsingle electrons and other particles canbeconsideredforuse as computational devicesthrough theuse of simulations.

1.2.Computer-aideddesign 2

absolute_necessityfor _{thoroughlydeveloping}thesenew computational_devices, simulations are requiredtoexamine_{the earlyfeasibility}of each ofthesedevices. Which of _{the many} possible physical combinations of nano-scale elements is worth_{investigating}can_onlybe determinedthroughaccurate simulation. There is, therefore, a needto create simulation tools and environmentsthat canac curatelymodelthesenovel physical systems.

Thesetoolsmust also providethescientistandengineer(heretoforereferred toasthe_designer) withthefreedomtoexperiment withdifferentconfigurations ofdeviceswith an eye alwaystowards thedesignof computationaldevices. That isthegoal ofthegoal ofthecomputer-aided designenvironment.

1.2. Computer-aided design

Thisthesis definesthe maintool of explorationfor designersasthe computer-aided design _(CAD) environment with ideastowards rapid prototyping. This CADenvironment, asdefinedin thisthesis, iscomprised ofthreeparts:

1. Logic Synthesizerthat is ableto transformhigher-level descriptionsofa computation (perhapsan entire_program)and synthesize the digitallogic functions_bywhich it canbeimplemented.

2. Logic Designerthatisabletotransformthoselogic functionsintoahard ware_(physical) representation.

3. Simulatorthatutilizesthe physicalproperties ofthe devicesusedto im plementthelogic functionstogeneratea reliable view ofhow thephysical deviceswould operate shouldthey bebuilt and assembled as perthe re quirementsofthelogicdesigner.

Thisthesisconcentrates onthesecondandthirdelements oftheCADenvi ronment, thelogicdesignerandthesimulator. _However,foraysinto thepossible implementationofthe logicsynthesizer are madefromtime totime.

1.3. Overview

Theremainderofthisthesisissplitintothreechapters_{that cover, (i)}justifica tionsforthedesignoftheCADenvironment, (ii)thedesignof saidenvironment, and _(iii) theuse oftheenvironment.

Chapter 2 provides anintroduction topossible notations_{that may}be used to specify logic functions. Logic circuits, Scheme functions, binary-decision

1.4-Practicalconsiderations 3

Chapter 3 details the design of the CAD environment based around the

hypercube_topologyand a state-spacemodel ofdevices. Theuse ofthatenvi

ronmentis demonstrated inan examplethatmakes useofnearly-idealdevices. Chapter 4 extendsthe discussion oftheuse ofthe developed design envi

ronment_{byconsidering}theintegrationof non-ideal devicesinto thehypercube

topology. Two examples are given based upon _{two promising} technologies:

single-electrontransistors and endohedralfullerenes.

1.4. Practical considerations

Throughoutthis_thesis, practicalimplementationsof alldiscussedideaswillbe shownintermixedwiththeideasthemselves. That _is,this thesisnot_onlyseeks

to reveal the concepts involved withthe computer-aided design of nano-scale

digital circuits, it also serves as a reference or an example ofhow thoseideas canbepractically implementedand used.

Thetwodominant languagesusedthroughout thediscussionareSchemeand Matlabm-scripts. Eachisusedin theareas where_theyarethemost powerful

- Scheme

as an evaluator andinterpreteroflanguages _{(includingmathematics)}

andm-scripts asnumerical workhorses.

Schemeisused_{predominantly through the discussion}becauseit servesdis

tinctly as the most clear and _thoroughly unambiguous description of ideas.

Although the reader_may notbe familiarwith _{Scheme, they} should findthat its syntax _(or, lack there_of) willbe readily comprehended. When in need of

reference, theuser is directedto _[1] and [10].

Scheme statements in this thesis (and in _general) are distinctive in their abundant use of parenthesis andindentation toshow program structure. _They are also_readilyrecognizable dueto theirreliance upon a prefix notationfor all functionsand syntax. Asanexample,aScheme functiontocalculatethevalue ofex

throughn iterationsisgiven_by,

(define (my-fact n) (define (iter i val)

(if (= i 0)

val

(iter (- _{i 1)} (* val i)))) (iter n 1))

(define _(my-exp x n) (define (iter i val)

(if (>= _{i n)}

val

(iter (+ i 1)

i-4- Practicalconsiderations 4

(my-fact i))))))

(iter 0 0))

Whentheexpression _(my-exp 1 30) isevaluated (withthe intentionof calcu

lating thevalueof_e),aScheme interpreterwill_give,

2403440095914245030058787948979

2.7182818284590452353602874.

884176199373970195454361600000

Becauseexact numbers (1and₃₀₎were providedto the_interpreter, itreturned an exact value forthefirst_thirtytermsoftheTaylorexpansion ofe1

Matlabm-scripts are used whenthevast librariesand_{plotting functional} ityoftheMatlabenvironment are needed. Giventheextent ofthese_libraries, thereare_many operationsthatcan bewrittenin Matlabmuch more_quickly

than inScheme. Thiswillbeseen_veryplainlywhenMatlabisused asthesim ulationenginefor_performingdynamicanalysis of nanoscaleintegrated-circuits.

Asimilar exampleto that givenfor Schemecanbewrittenasan m-script:

function val =

my_exp(x, n) val = _0;

for i = ₀

: (n-1) val =

val + x"i / my_fact(i); end;

function val =

my_fact(n)

val =

1;

for i = ₁ : n val =

val * i; end;

Whentheexpressionmy_exp(l, 30) is evaluated, Matlab returnsthevalue,

2.71828182845905.

Here,Matlabinterpretsall numbers_{using the}64-bitIEEEfloating-pointstan dardandthereforedoes not attempt an exact solution.

Asafinalnotefor assistance when_readinglanguagesused inthisthesis, it

should benotedthatSchemeprograms areusuallywritteninafunctionalstyle of programming. That _{is, they} are written without _using explicit variables. This is the traditional way ofcoding in Scheme. Matlab m-scripts, on the

other_hand,are writteninan imperativestyle

-itstraditionalwayof coding.

This thesis will demonstrate the comfortable _relationship established be tween Matlab andSchemecode. In truth, allofthecodeinthis thesiscould have beenwritten in one language orthe other. _However, neither language is

particularly well suited to the full scope ofthis thesis. For example, Scheme hasno standardfacilities forsolving differentialequations orfor_plottingresults

manipulat-1.4- Practicalconsiderations 5

ing expressions. _Therefore, each language (and its accompanying libraries) is usedinthe areaswhere it isthemost useful.

You thinkyouknowwhen you_learn, aremore sure when youcan write, even more when you can teach, butcertain when you can program.

2.

Representations

logic functions

This chapter _briefly discusses four different notations for representing logic

functions. The firstnotation, the use oflogic_circuits, isapredominant means

for _specifying simple circuits. _However, due to notational difficulties when

those circuits becomecomplex, it must be abandonedin favor of more robust methods. The high-level (in terms of abstraction from _hardware) means of specifying logic throughout this thesis is then presented. It isshownthrough

example that thisnotation is capable of_specifying all manner oflogic circuits

- from

combinational logic to _memory circuits. The high-level approach is

then abandoned fortwo notations that have simple and direct expressions in

hardware. The last of_{these notations,} the_hypercube, is used throughout the

remainderofthis thesisasthedominantexampleinlogic-circuit topology.

2.1. Circuits as directed graphs

Thetypicalrepresentation for logic functions isthelogiccircuit. Thesecircuits

graphically providesthe datapathsthrough which information may propagate

alongwith gatesordevices_{that may modify}a singlesignal or a_groupof signals.

Such gates include logic functions such as and and or or ways ofcombining

or _{extracting information} from the signals _{(multiplexers,} switches, &c.) To

continue withtheelectrical circuit_analogy,thedatapaths are often referredto

simplyas wires.

Logiccircuits_{may employ}multipleinputsand multiple outputs. Inthecase

ofthe latter,thecircuit _trulyrepresents amulti-_{valuedfunction.}

Furthermore, logic circuits_typically utilize feed-forward datapropagation. That is, each device and each data path is used once and only once _during the computation of the output or outputs. If a component of the circuit is

reused, thereexists some sort ofintrinsicstate

-theoutput ofthecircuit will

not _necessarilytakeonthesame valuesfor alltimegiven constant inputs.

Theterm "feed-forward" isused forthissimpler typeof computationsince

logic circuits are often drawn as directed graphs. Each device is represented

as a node onthe graph and its outputs are directedto new nodes forwardin thegraph. When componentsare_reused, thenthedirectionofthegraph must

necessarily go

"backwards"

torelocatedused nodes. _Thus,computationsthat

require Illimitable!!! state are saidtoutilizefeedbackor backpropagation.

Asexamples ofthese two typesof_{circuits, Fig.} 2.1 and Fig. 2.2are given. Fig. 2.1 demonstrates a simplefeed-forwardcircuit. It isusedtorepresentthe

logic function,

2.1.Circuits asdirectedgraphs

Figure2.1. Anexample of alogiccircuitthatuses_onlyfeed-forward datapaths.

Figure 2.2. Anexample ofalogic circuitthatuses_onlyfeed-forwardandfeed back datapaths. Thisis thebasic D-type latch.

whereA andVarethebinary-operatornotationforthelogic functionsandand or_respectively andthenotation₂₇isusedtorepresentthecomplement ofthe input (or function _parameter)x.

In ordertodeterminethe output value ofthe circuit given _{in Fig. 2.1,} one must _simply substitute the inputs into the circuit and use the definitions of thecircuitelements_{to simplify}or computeintermediatevalues ofthefunction. These substitutions and simplifications _may happen at _{any time} and _{in any} order

-so _long as _they are done correctly, the output of the circuit will be consistent.

Forexample,one_maybeobligedtocomputethevalueofthecircuit withthe inputsx\ =_0,X2=_1,

x3=_0, wherethesymbols0 and 1are usedto represent thelogicvalues

"false"

and "true" Thiscanbe done_byfirst_{substituting these} symbols into the definition ofthelogic function (which is readily determined fromthe logicdiagram):

/(0, 1,0) =_0V(0A1) V(0A1). _(2.2)

Next,thedefinitions forAand_{V along}withcomplement_maybeusedto_simplify the expression_to,

2.1. Circuitsasdirectedgraphs 8

and

/(0,1,0) =_0V0V1.

(2.4)

One lastsimplificationstep (actually,twobecausetherearetwomore operators

to _satisfy) are usedtodetermine/(0,1,0) = 1.

This is thesubstitution method for_determining thevalue offunctions. It

workswonderfullyforall functionsthat contain nointrinsicstate

-those that

can be drawnas circuits_{using only}feed-forward datapaths.

Onthe other _hand, thecircuit given in Fig. 2.2 does haveintrinsic state

-itsoutputisnot _readilycomputable given itsinputs. The feedback datapaths

requireustousesomedefinitionoftime inordertodeterminetheoutput ofthe

circuit. _{Specifically,} additional steps must beperformed in order to calculate

the output ofthe function. _First, because the feedback datapaths cannot be determined_immediately, it isrequiredtoutilizeinitialconditionsforthosegates

that are driven_by feedback datapaths. Oncethis isdone, the output ofthe circuit, inaddition to thetrue values ofall thedata_{paths, may}be determined using thesame substitution methoddescribeearlier.

However,one complete substitution and simplification_{iteration may}not be adequate to represent the output ofthe circuit since it represents _{merely the} output ofthecircuit at one instantintime. Instantaneousvaluesare_typically

not of interest. _Usually, knowledge ofthe steady-state value ofthe output is

required

-thosevaluesthatact asif_theyare nottime dependent. To determine

the steady-stateoutput, the circuit must be _continually solved (through the

substitution_method) untiltheoutputhas stabilizedto a set of valuesthatcan not change (determinable _{byobserving the internal}states ofthecircuit).

Returningattentionbackto Fig. 2.2, it isfirstnotedthatit is impossibleto

write aclosed-_{formdefinition}ofthefunction_usingbasicmathematical notation.

Therefore, theoutput ofthe circuit mustbesolved for_by utilizing the circuit

directly. This is best done_bylabelingstates in thecircuit asthevalues ofthe datapaths. _Fortunately, the only states in the circuit are _already labeled as

thetwo outputs.

Theinitial conditions shallbechosen somewhat mischievously. Sincethere

aretwo feedback paths, two initialconditionsmustbespecified: _g(0),and <j(0).

Since q and_q are complementary, there is an_opportunity to forcethe circuit

intoan illogical initialstate. Itwill then beinformative tosee how thecircuit respondstosuchapredicament. Theinitialconditions arebothsetto thesame value, 0.

It isnow possibletosolvethecircuit_{using the}substitutionmethod. This is donein fourtime incrementswhile thetwostates (q, and_q)are tracked. This is solutionisgiven in Table2.1.

Attime step2,thecircuit reached itssteady-statevalue; however, thesim

ulation was continued for another _{time step to test that this} was in fact the

2.2. Scheme_programminglanguage

t d elk _{q q}

0 110 0

11111

2 1110

3 1110

Table2.1. Sample timeprogression ofoutputsofFig. 2.2.

was expected. Ifhowever, the circuit were _arbitrary and at all complex, the

simulationwouldhave had tocontinue foralonger periodto demonstratethe factthat ithadreached steady-state.

Theuse oflogiccircuitdiagrams havenowbeen demonstrated for_describing two types of logic functions: those with and thosewithout state. While the logic circuit isuseful for simple _logic, it becomes somewhat less useful when the number of_{input,outputs,} andinternalstagesincreases. Atthis point, the

designer becomesencumbered_{by following}wires(data_paths) andisdistracted from his true goal of_developinga computationalfunction.

Let us therefore consider a more compact and efficient representation of

logic functions: theuse of a computer _programminglanguage.

2.2.

Scheme

A directmeans for specifyinga logic functionis to use a notation _specifically

designedto express mathematicalfunctions. Such a notation was used in the

previous sectionto define thelogic function _/ in (2.1). That notation isvery

convenient because it is similar to the notation used to define the values of

functions used throughoutmathematics and is therefore familiar to scientists

and engineers. _However,such notations are not_typically used when_interfacing with a computer. _Instead, oneof_{many programming}languages _maybe used

to define the values offunctions. Forreasonsthat will be discussed here and re-enforcedthroughout thisthesis, theScheme programming language is used asthemeansfor_defininglogic functions.

The function given in _(2.1) can be written, slightly more _verbosely, in Schemeas,

(define f

(lambda (xl x2 x3)

(or x3

(and xl (not x2))

(and (not xl) x2))))

Thelambdafunctioncreatesafunctionthatacceptsthe threeargumentsxl,x2,

2.2.Scheme_{programming language} 10

theoutputofthelastor gate.

TheuseoftheScheme keyworddefine,here,simplyassociatesthefunction created_bylambdawiththesymbolf.

When Schemeisusedtodefine functionssuch asthis, thegeneral shape of thefunction (asrevealedinits typographic_conventions)revealsthe distribution

of_complexity ofthat function. Forinstance, in theabove example it is easily surmised at a glance that the functionultimately has to compute some three quantitiesbeforeitcan performthefinalor operation and computethevalueof thefunction. It isalso seenthat thecomputation ofthose threepartsare rather simplistic. This convenient structureismaintained so _longasthe function has no intrinsic state.

Let us now turnour attention to _developing a Scheme function forofthe

second example logic function: the D-type latch. This logic function differs fromtheprevious intwo importantways: it hasmultiple output values andit contains state. For those reasons, the Scheme definitionofthefunction willbe slightlymore complex.

InordertomodeltheD-type latchgivenin theprevioussection,a function withlocal state must be created. _Before, the lambda functionwas utilized to create thestateless _function; now, adifferent _(custom) constructor is needed. Thatconstructor andthefunction thatitconstructs are givenintheirentirety:

(define make-d-latch (lambda (qO nqO)

(let _((q qO) _(nq nqO))

(lambda (d elk)

(let _((new-q (nand (nand d elk) nq))

(new-nq (nand q

(nand (nand d elk)

elk))))

(set! _q new-q)

(set ! _nq new-nq)

(list q nq))))))

There are a lot of_interesting concepts usedto create this _function, and some moments will be spent to discuss them to contrast _{the simplicity}of thefirst functionf.

2.2. Scheme_programminglanguage 11

The constructing functionthen_trulycreatesthelogic functionutilizing the lambdafunction. Becausethe lambdaexpressionis deeper inlexicalscopethan

the let expression that creates_q and _{nq, that}created function has access to

thosestates. The logic function itself hastwo _inputs, dand elk andthose are given_explicitlyasthefirst parameterto the lambda function.

Similarto thesimplerfunctionfdefinedearlier, thisnewfunctionnow goes onto computethevalue ofitsoutput; however,in afunctionwith state, those statesmust_{additionally be}calculated. This isalldonein alet statementthat

creates the two new values of_q and nq. Once that is _done, the states ofthe

functionare updated_usingset! (theexclamationisusedtomarkthefactthat

this _{function has side-effects)} andtheoutput values are returned as alist. The natural lexical structure ofScheme functions may again be employed toseethat the computation ofq (as seeninthe definition of_{new-nq) is}more involved _(by oneadditionalnand_gate) than thecomputation of q. In discrete hardware implementationsoflogic functionsthathavesuch unbalanced output datapaths, there exists possibilities for mismatched outputs. _Typically some sort of_auxiliary synchronizationcircuit must be usedto compensatefor this.

Coincidentally,suchtimingcircuits are often comprised oflatchessuchasthese.

Thisthesiswill not considerthe synchronization problem asit is well covered inbasictextsondigital design.

Nowthattheconstructorhas beencreated,a newfunctiondmaybe defined thatrepresentstheD-type latchwith aspecific set ofinitialconditions. Inorder

to_keepparitywiththeexamplesimulation oftheprevious_section,letusdefine d_thusly,

(define d (make-d-latch #f #f))

Theconstants#t and#f are usedtorepresentsthelogicalvaluestrueand_false, respectively.

As_before, dmaybetestedagainstboth inputs_beingsetto#t. Onthefirst

computation of_d, it isseen thattheoutputsareboth #t:

(d #t #t) -> (#t #t)

Once, however, the second andthird iterations arecomputed, the output sta bilizes to theproper values:

(d #t #t) -> (#t #f)

(d #t #t) -> (#t #f)

It has therefore been demonstrated that Scheme is a sufficient and even convenient _{way to define} and simulate digital logic functions. The notations usedfor logic functions can behigh-level (easierto comprehend, and generate

2.3. _Binarydecision diagramsandtrees 12

Figure 2.3. An exampleof a_binarydecision diagram for the logicfunctionof

(2.1).

previous section.

Attentionwill nowbeturnedback tomore low-level structures asit isdif ficult tolocate hardware that implements theScheme _programminglanguage. However, in the next _{chapter, the} transformation (or _compilation) ofScheme definitions for functionsinto theseother low-level notations willbe considered

in depth.

2.3.

Binary

The binary decision diagram _(BDD) is yet another notation for _representing

digital logic functions. Itsredeeming attribute is itssimplicity: thereare_only

three elements that comprise the notation as opposed _{to the nearly} infinite

variety of components used in the previous two notations (logic circuits _may

haveany typesofgates,andScheme functionsmaydefinearbitraryfunctions).

Thosethreeelementsaredatapaths,functionalnodes, andterminalnodes.

The BDD for the example logic function _(2.1) isgiven as Fig. 2.3. It will be

usedasareference asthegeneralstructure oftheBDD is discussed.

For eachinput to the function, there exists a decision layer. Such a layer iscomprised of a set offunctional nodes (circles in thereferencefigure). The

2.3- _Binarydecision diagramsandtrees 13

number offunctionalnodesfor a p-inputfunction_is,

Nf(p)= 2"- _1.

(2.5)

Each diagramortreeis rooted withtheoutput functionalnode. Thisnode branchestotwonodesthatrepresentthenextinputto thefunctionand, equiv alent_ly,thenextdecisionlayer. One branch designatesthepathofinformation flow inthecasethattheinput is_logicallytrue(shownasl'sinthe_figure)while theotherbranch designatesthepathfortheinput_beingfalse (0). Eachofthese nodesthenbranchesto twonodesthatrepresentthenextinputparameter. The totalset of nodes associated withthis lastinput formsthenext decisionlayer. Each functionalnode is essentially adecision node: if the inputassociated withit is_logicallytrue,then theappropriatepathis takenuntilanother nodeis reached. Thevalue oftheinput associated withthat nodeisthenexamined to determinewhichofitspaths shouldbe followed. Thiscontinues until aterminal node isreached.

Onceaterminalnode(markedasboxesin the_figure)is reached, theconstant value that it represents becomes the value ofthe function itself. The tree is constructed suchthatfora set ofinput values,one and_onlyoneterminalnode canbereached.

To use Fig. 2.3 to determinethevalue ofthe function for X\ = _{0, x2} = _1,

and_x3= _0,a walk ofthe treeis begunatthe topmostnode. SinceXi is_0,the left branch istakenfromthisnode. Anx2nodeisnextencounteredthatdirects thewalkto take theright path (1-path). Atthe x3 node, theleftpathis taken

and the walk completes at a 1-terminal node. The value ofthe function for thoseinputvalues isthen 1,justashas been determinedinprevious sections.

The BDD is a_very direct representation ofstateless logic functions

-no computation facilities are needed to determine the values of functions aside from _{the ability to}switch the path ofinformation propagation. The BDD is thereforeaveryconvenient representation of alogicfunction for implementation in hardware. The circuit _topology is completely regular for _any computation (it dependsonly on thenumber of_parameters), and all parts ofthehardware are identical

-theyare switches.

These switchingnodes canbemodeled_{simply as,}

n(g,u0,ui) =

(5AU0)V(jAui). _(2.6)

This logic function is the fundamental building block required to implement any state-less logic function using the BDD. It could be _{implemented using} discrete logic components (one not gate, two and _gates, and one or gate). Alternatively,more efficient solutionsmay be devisedsuch aspass-gate CMOS technology. Chapter4 discusses two alternatetechnologies.

However, before _movingon to those _{technologies,} let us consider the final logiccircuit notation

2-4-Hypercubes 14

as its_primary examplefor _developingtheCAD software.

2.4. Hypercubes

The so-called hypercube notation _[18] is a _functionally equivalent topological realization ofthe BDD. Its main contribution to the realm of logic function notations is the development of a specific 3Dstructure that is ubiquitous for alllogic functionsirregardlessofthelogicthat _theydescribe.

Traditionally, a hypercube that represents an m-inputlogic function isre ferred to as an m-dimensional hypercube or an m-hypercube. This is merely an odd nomenclaturefor all_hypercubes, irregardless ofthe numberof_inputs, arestructuresthatexist _in, and are_completely defined_in, 3Dspace.

2.4.1. BDD to hypercube transformation algorithm

Themethod of_{constructing the}hypercube isasimplerecursive process whose understandingis basedupon a select few definitions.

A nodeis a_connectingpoint ofdatapaths. Anodehas twocomplemen

taryinput datapaths and one outputdatapath.

A datapathisa connectionbetweennodesthatallowsfor_{(unidirectional)} informationpropagation betweenthenodes.

A data path is gated when it can control whether information is propa gated or not.

A terminalnodeisa source of constantinformation. In logic_designs,this informationisone ofthe Booleanvalues trueor false.

Withthese_definitions, thehypercube constructionprocess_is,

1. Beginwitha singlenode,n0,aO-dimensional hypercubethat musteven

tually represent an m-dimensional hypercube. This node is the output node ofthe fully-realized hypercube.

2. Extend that node to be a ID hypercube _{by connecting two} nodes to it through complementary datapaths. These two data paths extend in oppositedirections from the node (ifthenode isvisualized as a sphere, thenthedatapaths extend from theopposite poles). Eachofthesenew nodes must eventuallyrepresent _(p l)-dimensional hypercubes.

3. Connecttwo nodes toeachoftheselast nodestoextend themto be ID

2-4- Hypercubes 15

5>-^>-w)

Figure2.4. Hypercubeofthelogic functionf(xx,x2,x3) =

x3V(i1 A12)V(xiA

x2).

4. Continuetoincreasethe_{dimensionality}of nodes_byappendingnew nodes throughorthogonal_{complementary}datapaths. This processterminates when_{uq is,} in _fact,an m-dimensionalhypercube.

5. The last nodesto be added, those thatremain tobe O-dimensional _hy percubes, formtheset ofterminalnodes.

As an example of a3D hypercube, Fig. 2.4presentsthe hypercube for the example logic function (2.1). Because the logic function has three input pa rameters, Fig. 2.4takeson a cubicalform. Ifthefunction hadconsisted oftwo input parameters, itwouldhavetakenon a planarform.

The method usedto determinethe output value ofthe hypercubeis iden tical to that used with binary-decisiondiagrams. As such, no example ofits evaluation willbepresented.

2.4.2. Methods of optimization

All hypercubestructures oflogicfunctionswiththesamenumber ofinputsare identical

-theironlydifferencecomesfrom theconstant values associated with the terminalnodes. Therefore anymethods formutating thestructure of the hypercube(forthesake of some optimizationcriterion,forexample)mustbegin

byconsideringthe terminalnodes.

Giventhehypercubeconstruction_process,these terminalnodeswill always exist on theperipheryofthehypercubestructure. This isan importantchar acteristicifone considersthefeasibilityof_{actually manufacturing}logiccircuits based upon the hypercubestructure as itsimplifies themeans _by which con stants (be _they voltages or electron _injections) are _physically interfaced with thecircuit.

2.4. Hypercubes 16

This grouping is due directlyto theorder inwhichdatapaths are encountered in thedesign. From a_{manufacturability standpoint,} thehypercube structures

maybeoptimized_{byattempting to physically group} (as in_proximity)terminal nodes ofidenticalvalues.

Later in this thesis, it will be seen that the order ofthe data paths will simplybe specified _by a lexical sort ofthe names ofthe function inputs that control them. This ordering is therefore arbitrary. The hypercube structure can be optimized from a _{manufacturing} standpoint _{by examining the many} combinations ofordersofdatapaths inan attempttodesign hypercubeswith

groupedterminal nodes.

Othermethods of optimizationbeyond_{manufacturability may}considerthe

size ofthe hypercube. There is an obvious case where constant _folding may

beappliedto thestructure. That _{is, any}nodes thatare guaranteedtoreceive inputs of identical values for all _times, under all _{conditions, may simply} be

replaced withterminalnodes ofthosevalues. Thisspaceoptimization,however, canpotentially (andlikely) destroy theregularstructureand_symmetryofthe

hypercube. Becausethis regular structure isa majorbenefit ofthe hypercube

3.

Simulation

3D

logic

The Hypercube that was _lastly presented in the previous chapter is used

throughout the remainder ofthis paper as the dominant means to describe

and structure3D logiccircuits. The dynamic behaviorofthesecircuits canbe

determined in both technology-independent and technology-dependent ways.

Thischapterisconcerned with theexact means_by whichthis is done.

3.1. Cellular non-linear networks

Nano-scale digitalcircuits_verywell_maypresentthecircuitdesignerwithade

sign process_{contrary to}what_theyare accustomed: thedevicesthat comprise

thesystemsofthis_technology may come inpredetermined_{topological config}

urations. This _{is contrary to the} typical use of_topology to define a circuit's

behavior. Theseconfigurations of nano-scaledevicescould comein the formof

regular grids or lattices. Much more complexforms such as the structures of foldedprotein molecules as circuits _mayalsobeutilized.

In thesecases, the designermust implement thedesign without _{the ability}

to redefine the structure ofthe final system. It is therefore sensible to con

sider whether regular structures of similar components can perform _anysort of

computation; and, if so,howthat computation canbeanalyzed.

A methodology must be determined for reasoning about the control and

optimization ofmany devicesto achieve some system-wide goal. Thiscontrol

must accommodate the concept that a particular device's reference state (or

required _{state, target,} or _goal) is merely related to the reference state ofthe

entire_system,andthat thepresent stateof a particulardevicecanbeinfluenced

not onlyby itsown actions but_bytheactionsofitsneighbors.

Some _{[5], [7]} feelthat _theyhave found such a suitable methodology. Their

formulation is basedupontheuse ofCellular Nonlinear Networks (CNNs).1

The analogue of CNNs is the organization of cells in complex organisms.

Althoughsuch organisms_may possess some sort of central nervous system or

central controlnetwork, on a smallerscale, individualcells performtheir func

tion basedonlyontheirownstate, theirenvironment's_state,andthestates of

their neighboringcells. Whilethe exact operation ofthesecomplexorganisms is not _thoroughly understood, the premise of distributed coordinated control

seems_worthyofinvestigation.

Thissectionisconcerned with theconceptof_CNNs, andtheirapplications

1Atone pointintime, [5],theacronym "CNN"stoodfor Cellular Neural Networks. The word "neural" hassince been replaced with theword "nonlinear" to convey the ideathat these networks ofdevices are not limited to thoseapplications forwhich neural networks seemforeverassociated.

3-i.Cellularnon-linear networks 18

to computation and to the solution of partial differential equations (PDEs). The two topics seem at first unrelated, but it will be shown that the latter is anatural consequenceofthestructure ofCNNs. It is not readily clear how

effectivecomputationduetoCNNsisas a general_{means, and,}as a consequence

ofthis lackof_knowledge,a_varietyof perspectives ontheir designand analysis shall be presented. This section is a search for ideas for the simulation of nano-scaledigital circuits. In_{the proceeding}sections ofthischapter, theCNN concept willbemodifiedtosuit ourlogic designneeds. _However,forthepresent time, letus considerCNNsintheirpurest form.

3.1.1. Cellular automata

The CNNmethodologyisaccreditedtoLeon 0.ChuaandLin_Yangat_Berkeley, Californiain 1988 [5]. Theydescribeitas a generalization of another methodol

ogy: cellular automata. This _is,infact, thebestmeanstobegina presentation of_{the underlying} concepts of CNNs. As such, some moments shall be spent

examiningcellular automata.

Cellular automata is an exploration of emergent computational behavior.

It is a _study ofhow naturalevolution (in the context of genetic _algorithms)

could produce coordinated global _{information processing through} the action and interactionof simple components_(cells) [15].

A classical cellular automation system is a lattice ofindividual cells each

witha _binarystate one of either on or_{off, 1} or0. Each cellsis aware ofits neighborhood andits "next"

stateispurelyafunctionofitsown state andthat ofits neighbors.

A simple example is a one-dimensional lattice consisting ofN cells. Each

cell is indexablefrom 1 to N andisnoted as C. For example, C(l) is avalid cell;however,C(N+₁₎isnot. Eachcellhasa state x and an outputdenotedas y. Atransitionrule_table,analogoustoalogicaltruthtable,isusedtodescribe

thenext state of each ofthesecells.

There exists a neighborhood (or ^-neighborhood), Na, about each cell i

representedasa set of cells. One definition of such a neighborhood_is,

Na(i)= {C(r):\r-i\<a) (3.1)

where a isreferred toastheradius oftheneighborhood. If, for example, aisset to _1, then _Nx(i) = {C{i

-l),C(i),C(i + 1)}. This

definition of a neighborhood presses the immediate need to define _boundary conditions. _Classically, theseone-dimensionalstrips of cells are interpretedas ringsandthe _boundaryconditionsfor d>0 are given as:

C(l-d)=C(N+_{l-d) (3.2)}

3-1.Cellularnon-linear networks 19

Withtheserulesandaset ofinitialconditions,a complete view ofthesystem canbepresentedthrough time. Forexample,giventherule table (fora= 1):

neighborhood state: 000 001 010 011 100 101 110 111

output (y): 0 10 10 10 1

the _followingnetwork withN= 11 wouldchangefrom

1 0 1 0 0 1 1 1 0 1 0

0 1 0 0 1 1 1 0 1 0 1

to

By definingvaryingsizesof a and differenttransition functions (or_tables), oneissaidtoprogramthesystem. It is easy toimagine a program (rule_table) toperformoperations suchasleftand rightlogicalshifts. Melanie Mitchelland her colleagues developed somewhat more _interesting programs _{by employing} genetic algorithmstosearchfor_{(or evolve)}programsthatwould performcertain tasks[15]. One example of such programsisthe _"majorityrule"

program that determineswhethertheinitial stateofthe latticeconsisted of more ones than zeroes. Iftherewas a _majorityof_ones, all cells would _eventuallyturn_{to one;} otherwise, theoutput wouldturntoall zeroes.

Althoughthesecellular automata programs are not_{easilydevised,}theentire system possessessomevery redeemingattributes:

ParallelProcessing. Althoughtheprocessingoccurs_{incrementally,}it isdone in a _completely parallel fashion

-that is, each cell is _doing itsbest to solve the problem on each increment oftime.

This fact is demonstrated nicely by the majority rule program mentioned

above. The serialform ofthis algorithm would process each cell _individually to accumulate thenumberofonesin theentirelattice. Itwould thenhave to comparethatnumbertothenumber of cellsinthelattice. Thus,the lengthof executiontimeforthisalgorithmis _linearlydependentonthesize ofthelattice (N).

Inthe case ofthe cellularautomata_program, thelength ofexecution time isdependent on theneighborhood size and therandomnessofthe initialdata [15].

Furthermore, theserial algorithm requires priorknowledgeofthe totalsize

ofthe system. Thesystem of cellular automatadoes not needthis knowledge becauseofthenext important attribute:

Integrated Communication. Thisadvantageisone ofthemore_interestingtraits of cellular automata. Sincethesize of a neighborhoodis_finite,andtherequired

ac-3-i. Cellularnon-linear networks 20

complished _{by transferring} state from one cell to another in a _relay fashion untilithaspropagatedtothe cellthatneedsit. Ofcoursethisprocess requires timeand_{the efficiency}ofthecomputationis_directlyrelatedto_{the efficiency}of

thecommunications scheme. Itis, nonetheless, a robust system inthat itcan tolerateintermittent failures.

Efficient Implementation. The cellular automata network is both a compu tational and storage medium. _Furthermore, the implementation of each cell is identical. Implementationsofdifferent programs _rely on _{changing only the}

transition rules. Thispromises a _{cheap way} of_{manufacturing these}computa

tional devicesifone is able to devise a simple_way of_changing the ruletable

while _maintaininga consistent structure of cells.

However, one must notforget the disadvantages. It is verydifficult to de

vise correct programs fora given _task, and some tasks_may not be able tobe

programmed atall. Onecan imaginethat thedesignof a programtoperform binary arithmetic_maybesuch a_sufficientlydifficult task.

3.1.2. Cellular nonlinear networks

The advantage of a _cheap and robust parallel _processing architecture is too

enticingtoignore. LeonChuaandLin_Yangtookituponthemselvestoextend theideasof cellularautomata. _Theyextendedtheidea in three importantways.

Continuous states Instead of each state _being a _binary state, Chua and Lin

extendedthestate of each celltobe_anyreal number.

Continuous time Whereas cellular automatadescribes state transitions as in

stantaneousstep-_wise

changes, aCNN's states are continuousthroughtime.

Addition ofInput and Bias The state of a cell is now _(optionally) dependent

on inputs toitsneighborhood and onbiasconditions.

Thestate equation of a cell_C(i) in thea standard CNN isgiven as [5]:

xt =

-xi+

^A(i,r)yr[C(r)

J2B(i,r)ur[C(r)

r r

wherexr, yr, ur,and zTare _{the state, output,input,} andbiasofthecell C(r).

Thefunctions_A(i,r) and_B(i,r)givetheweight orinfluencebetween_C(i) and

its neighboringcells _(includingitself).

Furthermore, thesummation notationisthat ofDonald E. Knuthgiven in

"TwoNoteson

Notation"

(http://www-cs-faculty.stanford.edu/~knuth/

preprints.html). Thisnotationis basedontheoperators_[and_] whose values is 1 if thelogicstatement withinthem istrue,and0otherwise. Inthis context,

3.1. Cellularnon-linear networks 21

Figure3.1. Exampleoutput function defined_bytheequation_y(x) = \\x+_1|

\\v _l| =i(|x+_l|-|x-l|

Thisstate equation canbecomparedtothestateequation of an equivalent cellularautomata system:

Xi(t+1)

-Xi(t) =

Y,

wheret isan integerindicator oftime.

Generally speaking, the Aand B functions can _vary with respect to time and the specific cells upon which _theyare applied. In practice, however, they

remain constantbetween cell groups. A cell _{group is the simply}a cell and all thecellsin itsneighborhood. Whenthefunctionsareindependentofindividual cell groups and_time,the are referredtoastheCNN's template(ortemplates). As the inputand bias values are free tochange, attention mustbe turned toward the important definition ofthe output function y. _{Fortunately, (3.5)} gives some hintthat theoutputs shouldbestronglyrelatedto thestate. Chua

andLin defined theoutput _as,

y=_h(x) = -\x+1\

1, , 1

2l*-H

= _{H II (3-6)}

This functiontakes the formgiven in Fig. 3.1.

Therefore,forstatesintherange_{(1, 1)}theoutputisequaltothestate. For largeror smaller state_{values, the} cell issaid tobesaturated, andthenoutput takesonthevalue sgn(x). Thisfunction isessentiallyahold-over fromtheori gins ofCNNswhen_theywereusedtomodel neuralnetworks. _{However, nearly}

all literatureon thesubject continues to use this function or adifferentiable formofit.

Theuse ofthe saturatingoutputfunctioncanalsobetracedbacktoimple

mentationdetails. Aswillbepresentedshortly, thesimplesthardwareelement to implement the output function is an operational amplifier configured for

unity-feedback with built-in or predetermined saturation points. It _{is recog}

nized then, that the output _{function is simply} a convenient function that is

representative ofthe state andeasytoimplement.

3-i. Cellularnon-_{linearnetworks 22}

Ofcourseitwouldbepossibletodefiney=x andimplementthatrelationship.

In _fact, the _saturating output function exists for the important purpose of

stability

-it lim-itsthepossibility ofstatesever _increasingwithoutbound.

3.1.3. Higher-order configurations

The ideas ofCellular Nonlinear Networks _{is easily} extended beyond that of a

stringofcells to thatof_arbitrarygeometriesin _{arbitrary dimensions. In fact,}

most in-depth studies of CNNs have been performed on 2D regular grids of

dimensions N x M [5].

The extensionto higher dimensions is accomplished _{by slightly modifying}

thestate equation and_redefiningwhat constitutesa neighborhood. Forexam

ple, the state equation and neighborhood definition for a cell in a3D regular

grid couldbegiven as:

Xi,j,k=

-xt,j,k+

Y/A(i,j,k,r,s,t)yrtSit

r,s,t

^2B(i,j,k,r,s,t)uriStt[C(r,s,t)

and

Na(i,j,k) =

[C(r,s,t)

a)

wherean alternativeformoftheneighborhoodcouldbegiven as

Na{i,j,k)= _{C(r,

8,t) : maxflr -i\,\s

-j\,\t

-k\)<_{a} (3.9)}

The firstoftheseneighborhooddefinitions disallows diagonalneighborswhereas

the second_specifically allows them. Such definitions dictate the exact sizeof

theneighborhood andthusdictate thesize ofthe Aand B templates.

Two-layertwo-dimensionallattices

One ofthe first publicized uses ofCNNs beyond imagemanipulation was the

initiation and control of self organization or patternformation. The idea is to

initiateand sustain adesiredcomplex pattern(asrevealedthrough theoutput

of allthe_{cells) by} asimple set ofinputs.

It has beenrecognizedforsometime thatnatural patterns canbegenerated

by mathematical partial differential equations called reaction diffusion _(RD) equations. Thesegenerated patterns are referredtoas_Turing patterns [16].

However, reaction diffusion equations are much more _{than simply} pattern

generators. _They are a model ofthekinetic distributionofsome elemental in

someenvironment. The reaction diffusionequations are a macroscopic view of

3-i.Cellularnon-linear networks 23

tomodel. The most commonformofthereactiondiffusionequation_is,

dc

=f(c) +Z)V2c (3.10)

wherec isthe vector of elemental_{concentrations,} f represents thereactionor

generationprocess, D isa matrix ofdiffusioncoefficients, andV2 isthespatial Laplacian operator. In3D Cartesianspace, theLaplacianisdefined as

n2

d2 d2 d2

.

,

v _=dx~2+W2 + dT2' (3ai)

where,here,x _yand z representthethreespatial coordinates. FromtheLapla cian andthepartialderivativewith respectto_time,itisobviousthat thisequa tionisspatial-temporal elemental concentrations canvary in bothspace and

time.

Turning back, one can generate patterns under certain circumstances _by

utilizing the reactiondiffusion equations oftwo elementals. _Sometimes, these elementalsarereferredtoasactivatorAandinhibitor_{/; however,}thischoicein

vocabulary iscompletelyarbitrary. What_{is actually}desired isa set of_opposing

elementals. Whenthese twosides areleftto_interact,then theirprogress canbe observed_{bysimply differentiating}betweenthe twodifferentelementaltypes.

Thetwo-layer two-dimensionalCNNwasdevelopedtomodel_{these opposing}

elementals. Oneofthelayersrepresentstheactivators andtheother represents

the inhibitors. Thetwo lattices aretwo-dimensional becausetheentire system

canbereadilydisplayedatdifferent time increments in itsentirety.

Two-cell system

To beginthedevelopment ofthe_CNN, firstconsider_{the very simply}case ofa two-cellnetwork. Let eachcellberepresented_bythe_followingstateequations:

a =

~xa+(1+H)ya-syb+ za (3.12)

xb=

-xb+ sya +(l+ n)Vb +*b (3-13)

where s andpare_arbitrary(for_now)constants. Thesestate equations represent

only the reaction part ofthe RD equation (3.10). It has been proven _[2] that thefirststeptowardpatternformation isthegeneration of a stable limitcycle.

A limitcycleissimplyperfectoscillatorybehaviorofthe stateovertime.

The two-cell system above will reach _{this oscillatory} behavior for certain

values ofs andjjl. _{Specifically,} it has beenshown thata stablelimit cycle will be achieved about an equilibrium point if 0 <p< s. _Furthermore, theradius

or _{amplitude, R,} ofthelimitcycle will beapproximatelyequalto _{1 + n +}s. To visualize all of _this, the two-cell system is simulated for p. = _0.7 and

s = ₁ (as in _[2]) over 30 seconds with the initial conditions that xa = _{1 and}

3-i.Cellularnon-linear networks 24

Time(s

Figure3.2. Stateand graphs of cell a _(dashed) and cellb _(solid) fromthe two-cell network _implementing thereaction state equations. Calculated for s = 1 and_p= 0.7.

stablelimitcycle.

Extension to CNNs

Each ofthe cells ofthetwo-cellsystem isnowextendedtobe a full 2D CNN. Thestate equations ofthe cellsin these twonetworks are givenas,

Xa;i,j Xa-ij

-\-\1~t~

H)ya\i,j SVb;i,j <

Za-\-Da{Va;i-l,j+_Va.i+l.j + Va.i.j-l +Va-.i.j+l ~

^Va-.i.j),

xb;i,j=

-xb-ij+(1+_n)yb;i,j+_sya]ij+zb+

Db(yb;i-l,j +_Vb-.i+l.j + yb;i,j-l +Vb-ij+l

-tyb;i,j),

(3-14)

(3-15)

whereeach networkismadeupofNxMcells and 1 <i<N and 1<_j <M. The neighborhoods aredefined_by,

N _(i,j) =

{c(r,s)

{s-j)2<\}

That_is, onlyadjacent non-diagonal cells on a rectangular grid.

Each cell in the networks represents a geometric _location, and its state

representstheconcentrationoftheelementalatthatlocation. _Again,thereare twoelementals atwork, a andb,and_theyoppose cachother.

Thequantities_{Vi-ij+yi+ij +yl,j-i +Vi,j+i-4y;j} representdiscretespace

versions ofthe Laplacianoperator. Ifone considers theaxis_containingNcells

tobethen-axis (analogoustothex-axisin thebrief Laplaciandiscussion),and

3-i. Cellularnon-_{linearnetworks 25}

Figure3.3. Stategraphforthetwocells ofthereaction stateequations_showing a stable limitcycle. The initial conditions_{that xa} = 1 and_xb=0 are shown.

Calculated fors = _{1 and}

p= 0.7.

at geometric position _(i,j) has beenapproximated as:

d2y

gn2

- (Vi-hj ~

Vij) -(ViJ~

Vi+i,j)

d2y

dm2 (Vij-i

~ Vi,j)

-{Vij -Vi,j+i)

(3-17)

(3-i8)

Itisimportanttonotethatthe twoCNNscan not_{trulydirectly}observeeach other. That _is, the neighborhood of a cell on thea network does not include

anycells onthe6network. _Onlyin thereactive part ofthestate equations can a cellfromone network observe another cell from theother network. _Further, this "other"

cellislimitedtoitsdual

-thecell ontheother network atthesame geometric position asthefirst. Later (in Chapter3), theuse of dualcells will

beencapsulated intothenotion of cells with multiple states. Thisabstraction will, however, havetowait.

The so-called zero-flux_boundary conditions are implemented for this net work. This _boundarycondition states that the state of _non-existingcells be yondboundariesare equalto thestatesofthosecells attheboundaries. _Thus,

a_boundaryrepresents abarriertomotion

-zero-flux.

Using all ofthese _ideas, a simulation oftwo 64 x 64 CNNs is performed.

Again valuesof 1 and0.7 are used for s and_{p, respectively,} and the twodif fusion coefficients, Da and _Db areboth set to 0.1. In additionto the diffusion

coefficients, thebiasesare now used: za issetto0.3and_{zb is}setto0.3. This isthesame configuration as usedinthefirstexamplein _[2], withtheexception

thatthe CNNsare 64x 64 insteadof44x 44.

3-i. Cellularnon-linear networks 26

t=0s t=10s t=_20s r.=30s

,

II

i=_{40s t}=_{50s i}=60s t=70s

Figure 3.4. Snapshots of cell outputs of the a layer in the two-layer

two-dimensionalsystem. Blackareas represent cells with outputs equalto 1 while

white areas represent cells with outputs equal to 1. _Gray areas represent outputs inbetweentheseextremes.

small bandsprotruding fromand edge. Oneofthesebands hasthe states all

setto 1 andtheotherhas themall set to0. Alltheinitialstates ofthe blayer

are set to 0. Snapshots ofthe a layer are takenat 10 second intervals over a

period of70seconds. This data is presented asFig. 3.4.

Even though it is proven that patterns can form given the parameters

above _[2], such phenomena as spirals are not _easily created. These complex

patterns requirecertaininitialconditionstoform. For_instance,ifthesystemis

initializedwithrandom state values(intherange_[1,1]), thesystemillustrated

inFig._3.5 could emerge.

BoththesystemsofFig. 3.4 andFig. 3.5continuetoevolve overtime. The

first system spirals foreternity and the second system "bubbles" for eternity.

However, this reactive steady-state canbe neutralized. Ifthere exist avariety

ofhard barriers

-that _is, instantaneous (ingeometry) changes in state -the

system will be dominated _by thediffusepart ofthe equation. _Essentially, the twoelementals mixslowly toforma uniformfinalstate.

3.1.4. Higher _meaning

Some time has just been spent onthe analysis of a particular application of

CNNs. It remainstobeseen whatthat analysis revealsandhow thatanalysis

can bemimickedtoproduce solutionsforseparate problems.

Ifoneispresented with a set of spatial-temporal partialdifferentialequations

and desires to simulate the action of_{those equations,} a few criteria must be

3-i. Cellularnon-linear networks 27

t=_{0s t}=_{10s t}=20s t=30s

cffiift.Q

t=_{50s t}=60s t=_70s

Figure3.5. Snapshotsof celloutputs ofthealayerin the two-layersystem with

random initialstatesforall cells. Utilizesthesamenetworksasin Fig. 3.4but

begins with randominitialstates.

equationscanbemanipulatedintotheform of a cell's state equation (3.4).

1. Thetimederivativesofvariablesmayonlybe first-order. That is, dx/dt

isallowable whilednx/dtn foranyn greaterthan 1isnot.

2. Thetimederivativeof a variable mustbe dependentonlyon_{the geometry}

ofthe system. Thatis, a system such as

Xl =_x2

X2 =

Xl

(319)

(3-20)

is strictlynot allowed while a system such as

1 =_V2x2

x2 =_Xl

(3-2i)

(3-22)

is allowable.

3. The A andB functionsof _(3.4) generalizethestateequationtobe thor

oughly non-linear. Whilethis is perfectly acceptable when _using CNNs in a simulation environment, generalized nonlinear function will signif

icantly increase the complexity of the hardware implementation. The designermust decide whether this increasein _complexity outweighs the

benefitsofCNNs.

4. High-order geometricderivatives shall require a network whose cells are

3.2. Extensionofthemodelto thehypercube 28

fidelity (highlyprecise,highly accurate)statevalues. _Again,this isnot a greatissue insoftwaresimulations, but does put additional requirements on the hardware implementations. Ifthe third derivativeof states with respect to _{geometry is needed, then the} neighborhood of each cell will have to be at least N2 and concern will begin to plague the minds of the designer of the adders. For _instance, Chua reports _[5] that their implementation of CNNs _using integrated circuits is able to achieve a precision of_{approximately} 7bits. This_{is probably}sufficientfor firstand second order_derivatives,but_anythinggreater_mayrequire more precision.

Itis,then,reasonabletosay thatCNNscan model alarge_varietyof physical phenomena(thosephenomenathatcanbeexpressed as sets of partialdifferen tialequations). Once a mathematical modelhas been _formulated, its applica bilityto_beingsimulated _{(or analyzed) by} CNNs canbereadilyascertained.

That_is,CNNs_maybeused as aformoffinite-elementanalysis where space is discretized. Each_elementary volumeof space _maybe represented_by a cell with multiplestates (as dictated_bythemathematical modelofthephenomena tobeobserved or modeled).

However, the focus of this paper now turns to a different scenario -one

in which each CNN cell _is, in _fact, a direct analogue to a physical device. The neighborhood and observability of each cell is represented physically as

interconnects between cells _{(tunnelingjunctions,} chemically active _{sites, &x.)} In this scenario, the utilityof CNNs as partial-differential equation solvers is abandoned.

3.2. Extension ofthe model to the hypercube

ThediscussionofCNNs has revealedadirectmethodologyfor simulatingsys temswith large amounts ofidentical components. The CNN behavior is very promisingas ananalogsystem abletoproduce emergent computationalbehav

ior. Becausesuch a use ofthesystem (its design and _analysis) requires afirm

understanding ofthe technology used to develop the network while the goal

ofthis chapter is to produce a methodology for simulating logic circuits in a technology-agnostic manner, this field of emergent computation issuperseded witha moredirect approach.

One aspect ofCNNsis focused upon: the use a state-space representation oflarge latticesofidenticalor_{nearly identical devices.} Particularly, we extend theCNNmethodologysuchthateach cell(ornodeas_theywillsoonbe_termed) canbe represented with multiple states. Thesecellswillthenbeusedtodesign digital logic circuitsusing the hypercube configuration.

Within the hypercubeconfigurationofthecircuit,nodes are connected with

whathavebeentermed datapaths

3.2. Extensionofthemodeltothehypercube 29

To simulate circuits constructedin the_hypercube, one must select proper

state equationstomodelthenodes. In additionto_this, thegate control ofthe datapaths must alsobemodeled. GiventheCNNparadigm, thislast modelis

incorporated intothestate equationsofthenode. Toseethis,a simple example

isconsidered.

2

Figure3.6. Example node with multipleinputsanda single output.

Imagineasinglenode x withtwoinputsuxand_u2and oneoutputy. Letthis

nodebe defined_bythe twostates xx andx2 given _bythe_followingequations:

1 =_x2,

(3.23)

x2=

^-x2 -xx+ f(ui,u2), _(3.24)

V=

xu _(3-25)

where _{, r,} and k are arbitrary constants. The function _/ must account for

the two inputs Ui and u2 and _{is completely} dependent on the nature ofthe system. Forour _purposes, let us assumethat the inputsareunderthecontrol of_{complementary}gates. _Therefore,whenoneinput'sgateisopenfor dataflow,

theotherisclosed. Thefunction _/must capturethisidea.

Ifthestate ofthegates are represented_byvaluesin therange0to 1 where1 represents "openfor dataflow''

and0represents "closed", thenonepossibility

for _defining/ is

f{g,ui,u2) -(1

-g)ui +gu2, _(3.26)

where _g represents the state of the gate _controlling u2. Under _{this model,}

the gate canbe variablyopened and closed _allowing fragmentsofinformation to pass through. While we will see later that there can be physical _meaning attachedto thisconceptduetosuch phenomenaasquantum_{tunneling, for}now

we will consider this function as simply a reasonable and convenient method

forchoosingbetweeninputs.

With this definitionof_/, thesecond state ofthenodex canbe