Probabilistic Algorithms from a Thermodynamic Perspective

24 Pages·2004·0.33 MB·English

by  Krishna V. Palem

Computational Proof as Experiment:
Probabilistic Algorithms from a
(cid:1)
Thermodynamic Perspective
KrishnaV.Palem
CenterforResearchonEmbeddedSystemsandTechnology,
SchoolofElectricalandComputerEngineering,
GeorgiaInstituteofTechnology,AtlantaGA30332,USA.
[e-mail]
Abstract. Anovelframeworkforthedesignandanalysisofenergy-awarealgo-
rithmsispresented, centeredaroundadeterministicBit-level(Boltzmann)Ran-
dom Access Machine or BRAM model of computing, as well its probabilistic
counterpart, the RABRAM.Usingthisframework, itisshown for thefirsttime
thatprobabilisticalgorithmscanasymptoticallyyieldsavingsintheenergycon-
sumed, over theirdeterministic counterparts. Concretely, we show that the ex-
pectedenergysavingsderivedfromaprobabilisticRABRAMalgorithmforsolv-
ingthedistinctvecto(cid:1)rpro(cid:1)blemintro(cid:2)du(cid:2)cedhere,overanydeterministicBRAMal-
gorithmgrowsasΘ nln n ,eventhoughthedeterministicandproba-
n−εlog(n)
bilisticalgorithmshavethesame(asymptotic)time-complexity.Theprobabilistic
algorithmisguaranteedtobecorrectwithaprobabilityp≥(1− 1)(foracon-
nc
stantcchosenasadesignparameter).Asusualndenotesthelengthoftheinput
instance of the DVP measured in the number of bits. These results are derived
inthecontextofatechnology-independent complexitymeasureforenergycon-
sumptionintroducedhere,referredtoaslogicalwork.Inkeepingwiththetheme
of the symposium, the introduction to this work is presented in the context of
“computationalproof”(algorithm)andthe“workdone”toachieveit(itsenergy
consumption).
1 Introduction
Theword“fact”conjuresupimagesofasenseofdefinitivenessinthatthereisabelief
initsabsolutetruth.Thisnotionistheveryessenceofmodernmathematicaltheories,
with their foundationalframeworkbased on (formal)languagessuch as the predicate
calculus.Thus,followingRussellandWhitehead’sseminalformalizationofmathemat-
icalreasoningembodiedintheirPrincipia[31],theverynotionoftheconsistencyofan
axiomatictheorydisallowsevenahintofadoubtaboutafact,oftenreferredtoasathe-
orem(oritssubsidiarylemma)inmodernaswellasancientmathematicalthought.The
modernfoundationsofverificationasproof,withemphasisonitsautomaticormecha-
nizedform,appliedtoproblemsmotivatedinlargepartfromwithinthedisciplinesof
(cid:1)
ThisworkissupportedinpartbyDARPAunderseedlingcontract#F30602-02-2-0124.
N.Dershowitz(Ed.):Verification(MannaFestschrift),LNCS2772,pp.524–547,2003.
(cid:2)c Springer-VerlagBerlinHeidelberg2003
ComputationalProofasExperiment 525
computerscienceandelectricalengineering(seeMannaforexample[13,14])arealso
boundinessentialwaystothisnotionofanabsoluteordeterministictruth.
Concomitantto this absolute notion of truth, and a significant contribution of the
mathematicaltheoryofcomputing(referredtoinpopulartermsastheoreticalcomputer
science) is the notion of the complexity or equivalently, the “degree of difficulty” of
such a proof. Thus, starting with Rabin’s [23] work as a harbinger with further con-
tributions by Blum [1], the notion of a machine independent measure of complexity
led to the widely used formulationsof Hartmanisand Stearns [7]—essentially within
the context of a deterministic mechanistic approach to proof. Here, a deterministic
algorithm—equivalently, any execution of a Turing machine’s program [28]—upon
halting, is viewed as provinga theoremor fact, stated as a decision problem.For ex-
ample,determiningtheoutcomeofthecelebratedhaltingproblem[14,28]wouldcon-
stitute provingsucha theoremin the contextof a giveninstance,where an answer of
ayeswouldimplythattheTuringmachineprogramgivenastheinputwouldhaltwith
certainty.
Both this notion of absolute truth as well as the deterministic (Turing machine
based) approach to arriving at it mechanically are subject to philosophically signifi-
cantrevisionifoneconsidersalternateapproachesthatarenotdeterministic.Acritical
firststepinvolvesnon-deterministicapproacheswiththefoundationslaidbyRabinand
Scott[25].Basedonthesefoundations,Cook’s[4](andLevin’s[12])characterizations
of NPas a resourceboundedclass ofproofs,whoseremarkablerichnesswasdemon-
strated by Karp [9], elevated NP to a complexity class of great importance, and the
accompanyingP=?NPquestiontoitsexaltedstatus.Here,whiletheapproachtoprov-
ing is not based on the traditional deterministic transition of a Turing machine, the
meaningoftruthoneassociateswiththefinaloutcome—acceptorreject—continuesto
bedefiniteordeterministic.
Movingbeyondnondeterminism,the earlyuse ofstatistical methodswith empha-
sis on probability can be found in Karp’s [10] introductionof average case analysis.
Compelled by the need to better understand the gap between the empirical behavior
andtheresultsofpessimal(mathematical)analysisofalgorithms(oradeterminationof
lengthsofproofsinoursense),inKarp’sapproach,theinputisassociatedwithaprob-
abilitydistribution.Thus,whiletheproofitselfisdeterministic,itsdifficulty,length,or
morepreciselyitsexpectedtimecomplexityisdeterminedbyaveragingoverallpossible
inputs.
A striking shift in the notion of proof as well as the truth associated with it em-
anated from the innovation of probabilistic methods and algorithms. In this context,
boththemethodor“primitive”proof-step(oftheunderlyingprogram)aswellasthecer-
taintyassociated withthe proofundergoprofoundrevision.Schwartz[26] anticipated
the eventualimpactof therole ofprobabilityinthe contextofthese influentialdevel-
opments best: “The startling success of the Rabin-Strassen-Solovay (see Rabin [24])
algorithm,togetherwiththeintriguingfoundationalpossibilitythataxiomsofrandom-
nessmayconstituteausefulfundamentalsourceofmathematicaltruthindependentof,
butsupplementaryto,thestandardaxiomaticstructureofmathematics(seeChaitinand
Schwartz [3]), suggests that probabilistic algorithms ought to be sought vigorously.”
Thus, in this probabilistic context, both the deduction step as well as the meaning of
526 KrishnaV.Palem
trutharebothassociatedwithprobabilitiesasopposedtocertainties.Forconvenience,
letusrefertotheseasprobabilisticproofs(oralgorithmswhenconvenient).
With this as background, we now consider the long and fruitful relationship be-
tweenthenotionsofproofinthedomainofmathematicsanditsremarkableuseinthe
physical sciences over the past several centuries. Historically, mathematical theories
haveservedremarkablywellin characterizinganddeducingtruthsaboutthe universe
inavarietyofdomains,withnotablesuccessesinmechanics(classicalandquantum),
relativityand cosmology,andphysicalchemistryto namea few areas—see von Neu-
mann’s [30] development of quantum mechanics as a notable example. In this role,
knowledgeaboutthephysicalworldisderivedfrommathematicalframeworks,meth-
ods, and proofs, which could include the abovementioned algorithmicform of proof
as well. Thus, in all of the above endeavors, the direction is from (applying) mathe-
maticsto(creatingknowledgeabout)physicalreality.Bycontrast,inthiswork,weare
concernedwiththeoppositedirection—fromusingcomputationaldevicesrootedinthe
realityofthephysicaluniversesuchastransistors,toestablishing(computationallyde-
rived) mathematical facts or theories. Let us, for convenience (and without a careful
andscholarlystudyofthepossibleuseofthisconceptbyphilosophersearlieron),refer
to this opposing perspective as a reversal of ontological direction, wherein the phys-
ical universe and its empirical laws form the basis for all deductionof mathematical
facts through computational proof. To clarify, the reversal in “ontological direction”
whichthiswork(andearlierpublicationsofthisauthoronwhichit isbased[19, 20])
explore,referstothefactthatthephysicaluniverseanditslawsasembodiedincomput-
ing devices, form the basis for (algorithmically)generatingmathematicalknowledge,
bycontrastwiththe traditionalandoppositedirectionwhereinmathematicalmethods
produceknowledgeaboutthephysicalworld.
To reiterate, in all of this work, the meaning we associate with proof will be that
associatedwiththeexecutionofaTuringmachineprogram,andwewillbeinterestedin
the“complexity”ofrealizingsucha(mechanizedproof)inthephysicaluniverse.Thus,
to reiterate, we will consider a concrete and physically realizable form of a proof—
suchasthatgeneratedbyatheorem-proverexecutingonaconventionalmicroprocessor,
or perhaps its Archimedian predecessor—as a physical counterpart of Putnam’s [22]
“verificationist” approach by contrast with (as observed by him [22]) the “Platonic”
approachwith“evidencethatthemindhasmysteriousfacultiesofgraspingconcepts”
(or“perceivingmathematicalobjects...”).
Continuing,afirstandimportantobservationabouttheuniverseofphysicalobjects
suchasmodernmicroprocessorsisthattheirinherentbehaviorisbestdescribedstatis-
tically.Thus,allnotionsofdeterminismare“approximations”inthattheyareonlytrue
with sufficiently high probability. (See Meindl [15] and Stein [27] for a deterministic
interpretation of the values 0 and 1 within the context of switching based computing
through electrical devices, to better understand this point.) Building on this observa-
tion, the work described in this paper characterizes the (somewhat oversimplified in
thisintroduction)factthattheprocessofcomputationalproofentailsphysical“work”,
which in turn consumes energy described in its most elegant form through statistical
thermodynamics. The crux of our thesis is that since nature at its very heart, or our
perceptionofit,asweunderstandittoday,isstatisticalata(sufficiently)small,albeit
ComputationalProofasExperiment 527
classical scale—side-stepping the debate whether “God does or does not play dice”
(attributedtoEinsteintowhomastatisticalfoundationtophysicalrealitywasasource
of considerable concern)—themost naturalphysical models for algorithmic proof or
verificationusingfine-grainedphysicaldevicessuchasincreasinglysmalltransistors,
areessentiallyprobabilistic,andtheirenergyconsumptionisacrucialfigureofmerit!
Thus,anydeterministicformofcomputationalproofbasedonusingmoderncomputing
devicesareessentiallyapproximationsderivedbyinvesting(sufficiently)largeamount
of energy to make the probability of error small [15]. For completeness, we reiterate
herethatfollowingtheprincipleofreversalofontologicaldirection,weareonlycon-
cernedwiththediscoveryofmathematicalknowledgeviacomputationalproofsrealized
throughthedynamicsof a physicalcomputingdevice,suchas the repeatedswitching
ofsemiconductordevicesinamicroprocessor.
Now,consideringthespecifictechnicalcontributionsofthiswork,first,inorderto
describeandanalyzethesephysicallyrealizedproofsoralgorithms,weintroduce(Sec-
tion 2) a simple energy-aware model for computing: the Bit-level (Boltzmann) Ran-
dom Access Machine or BRAM , as well as its probabilistic variant, the RABRAM (in
Section 2.4). Specifically, each primitivestep or transitionof these modelsinvolvesa
change of state—realized in a canonicalway througha transition function associated
withafinitestatecontrolasinTuringmachines[28]—thatmirrorsacorrespondingand
explicitchangeinsomephysicallyrealizabledevice.Onevariantofsucharealization
isthroughthenotionofaswitchingstep[15,20]whereasanearliermoreabstractvaria-
tionisthroughthenotionofanemulation[19]ofthetransitioninthephysicaluniverse.
Anycomputationalproof(orequivalentlyalgorithm)describedinsuchamodelhas
an associated technology-independentenergy complexity, introduced as logical work
in Section 3 for the deterministic as well as the probabilistic cases. Historically, the
interestandsubsequentlythesuccessofprobabilisticalgorithmswithinthecontextof
algorithm design, was to derive (asymptotically) faster algorithms. Assuming that all
steps take (about)the same amountof energy,traditionalanalysisof time-complexity
will triviallyimply thata probabilisticalgorithmmightconsumeless energy,because
itcomputesandsolvesproblemsfaster—shorterrunningtimeimplieslesserswitching
energy.Incontrasttothese obviousadvantages,we showin Section4 thatthe energy
advantages offered by probabilistic algorithms can be more subtle and varied. Con-
cretely,weprovethatforthedistinctvectorproblemorDVP ,aprobabilisticalgorithm
anditsdeterministiccounterparttakethesamenumberof(time)stepsasymptotically,
whereastheprobabilisticapproachyieldsenergysavingsthatgrowasn→∞.
SolvingtheDVPinvolvescomputationally(intheBRAMorRABRAMmodel)prov-
ingthatagivenn−tupledefinedonthesetofsymbols{0,1}hasthesymbol1inallof
itsnpositions;theanswertothisdecisionquestion(ortheorem)isYESifindeedallpo-
sitionsoftheinputn−tuplehavethesymbol1andtheanswerisNOotherwise.Inthis
paper,weareinterestedinthefollowingdensevariantofthe DVP :theinputn−tuple
eitherhasno0symbolinit,orifitdoeshavea0symbol,ithaslog(n)suchsymbols.
Forthis(dense)versionoftheDVPproblem,whichforconveniencewillbereferredto
asthe DVP problemthroughout(definedinSection4.1),weprovethatanovelproba-
bilisticvalueamplificationalgorithm,provesthe(algorithmic)theorem,orresolvesthe
associateddecisionquestionwithanerrorprobabilityboundaboveby 1 (foraconstant
nc
528 KrishnaV.Palem
(cid:1) (cid:3) (cid:4)(cid:2)
cchosenasadesignparameter)usinganexpected(2n+logk(n))κTln 2 1−εlogn
n
Joules,where0<ε<1andk>2areconstants.Thealgorithmanditsassociatedanal-
ysisareoutlinedinSection4.
Inanearlierpublication,thisauthorproved[19]thatanydeterministicBRAMalgo-
rithm for solving the DVP consumes at least (2n−log(n)+1)κTln2 Joules; this is a
lowerbound.Bycombiningthesetwofacts,weshowthatthroughtheuseoftheprob-
abilisticalgo(cid:1)rithm(cid:1)introduce(cid:2)d(cid:2)here,theexpectedsavingsinenergymeasuredinJoules
growsasΘ nlog n , foraconstant0<ε<1,andforannbitinputtothe
n−εlog(n)
DVP .Thusboththesavingsaswellastheerrorprobabilityarerespectivelymonotone
increasinganddecreasingfunctionsofn.Tothebestofourknowledge,thisresultisthe
firstofitskindthatestablishesanasymptoticimprovementintheenergyconsumed.
Thesemodelsand analysismethodologybuild on the followingresults (from[18,
20])thatbridgecomputationalcomplexityandstatistical thermodynamicsforthefirst
time: a single deterministic computationstep, which correspondsto a switching step,
consumesatleastκTln(2)Joules,andthisisalowerbound.Furthermore,usingprob-
abilisticcomputationalsteps(orswitching),theenergyconsumedbyeachstepcanbe
shownto be aslow asκTln(2p)Joules,where p≥ 1 isthe probabilitythatthetran-
2
sition is correct; (1−p) is the per-step error probability. Also, κ is the well-known
Boltzmann’sconstant,T isthetemperatureofthethermodynamicsystem,andlnisthe
naturallogarithm.Inallofthiswork,thephysicalmodelsarebasedonthestatisticaland
hence probabilistic generalizations of switches formulated originally by Szilard [11]
within the contextof clarifying the celebrated Maxwell’s demon paradox[11, 29]. A
detailedcomparisonandbibliographyofrelevantworkfromtherelatedfieldreferredto
astheThermodynamicsofComputingcanbefoundin[20].Additionally,Feynman[5]
providesa simple and lucid introductionto the interplaybetween thermodynamically
based physical models of computing, mathematical models, and abstractions such as
Turingmachines.
2 The Bit-Level (Boltzmann) Random Access Machine - BRAM
Inthissection,wewillintroduceourmachinemodelforcomputing,exclusivelyoper-
atinginthelogicaldomain.However,toreiterate,afundamentaltheoremofthiswork
isthateachofitsstatetransitions—explainedbelow—willbeassociatedwithdefinite
amountsofenergyexpenditure.Furthermore,thisenergyconsumptionwillbeprecisely
relatedtotheinherentamountofenergyneededtocompute,usingthismodel.Signif-
icantly, a BRAM model will allow us to abstract away all aspects of the underlying
physics and characterize energypurely in the world in which models of computation
suchasTuringmachinesarerealized.Weanticipatethisasbeingveryhelpfulfromthe
perspectiveof algorithmanalysisand design—anexercisewhich,in a BRAM , can be
decoupledfromthespecificitiesofphysicalimplementations.
The BRAM however does provide a bridge to the physical world through the en-
ergycostsassociatedwiththetransitionsofitsfinitestatecontrol(definedbelow).This
bridgetotheworldofimplementationandenergyallowsustodefinethenovelcomplex-
ity measure of logical work as detailed in Section 3, which characterizesthe “energy
complexity”ofthealgorithmbeingdesigned.
ComputationalProofasExperiment 529
2.1 InformalIntroductiontoaBRAM
Informally,a BRAM (abit-levelrandomaccessmachine1)hasaprogramwitha finite
numberofstates.Thetransitionfromacurrentstatetothenextinvolvesevaluatingthe
associatedtransitionfunctionleadingtothe“reading”ofoneormorebitsofaninput
fromaspecificmemorylocation,transitioningtoanewstateandwritinganewvaluein
adesignatedmemorylocation.Thenumberofbitsreadisdependentofthesizeofthe
alphabet,tobedefinedbelow.EveryexecutionstartsinauniqueSTARTstate,andhalts
uponreachingauniqueSTOPstate.
Toextendsuchmodelstobeabletoaccountfortheenergyconsumed,wedefinea
BRAM(somewhat)formally.Foracomputerscientist,definingaBRAMbasedonwell-
understoodelementsofarandomaccessmachine(orRAM)iselementary;however,we
defineithereforcompleteness.ThetextbookbyPapadimitriou[21]providesarigorous
andcompleteintroductiontomodelssuchasTuringmachinesandrandomaccessma-
chines including definitions of conventionalmeasures of complexity for representing
time and space. This book also providesa comprehensiveintroduction to the numer-
ouswell-understoodinterrelationshipsbetweenclassesof(timeandspace)complexity,
andcan serveas anexcellentguideto the topicof definingmodelsof computationin
classicalcontexts,notconcernedwithenergy.
2.2 DefiningaBRAM
A BRAM consists of several components,which will be introducedin the rest of this
section.
The BRAM Program Following convention,the program P is representedas a five-
tuple{PC,Σ,R,δ,Q}.Notethatconventionally,variantsoftheprogramarereferredto
asthefinitestatecontrol.
TheSetofStates-PCisthesetofstates.Eachstate pc ∈PChasdesignatedloca-
i
tionsinmemory,definedbelow,thatserverespectivelyasitsinputandoutput.Without
lossofgenerality,letthestatesbe labeled1,2,3,...,|PC|. ThesetQ consistsofthree
specialstates,START,STOPandUNDEFINED-STATEnotinPC.
TheAlphabetofthe BRAM -Σisafinite alphabet,andwithoutlossofgenerality,
we will use the set {1,2,...|Σ|}, which includes the empty symbol φ to denote this
alphabet. From the standpoint of algorithm design, in most cases, it suffices to work
with an alphabet drawn from the set Σ={0,1}. However, as we will discuss in this
paper, the size of the alphabet |Σ| has important consequences to the precise energy
behavior of the associated state transitions. Therefore,the contextswherein the more
restricted alphabet is used need to be distinguished from those contexts in which the
moregeneralalphabetofsize|Σ|>2isused.
The Address Registers of the States in PC - These registers are places where the
inputandoutputaddressesofastatearestored.Inconventionalcomputerscienceand
engineering parlance, a BRAM uses a form of accessing memory that is referred to
1GivenaBRAM’seventualconnectionwithenergyanditsstatisticalinterpretation,onecanalso
interprettheacronymtomeanaBoltzmannrandomaccessmachine.
530 KrishnaV.Palem
as indirect addressing. We shall return to a discussion of the role of these registers
in Section 2.3. The address registers, represented by the set R is partitioned into two
classes Rin and Rout; these are both sets (of registers) where each register ρin ∈Rin
j
(ρout∈Rout)isa(potentiallyunbounded)linearlyorderedsetofelementsreferredtoas
j
cells<sj,1,sj,2,...,sj,k >(<tj,1,tj,2,...,tj,k>).Eachofthecellssl (tl)isassociated
withavaluefromtheset{0,1,φ}.Wenotethateventhoughtheoverallalphabetmay
beofsize|Σ|>2,eachcellintheregisterseitherstoresasinglebit,orisempty.Fur-
thermore,ifthevalueassociatedwithsuchanelementisφ(notdefined)forsomevalue
ofk(cid:5)≤k,thenthevalueassociatedwithallsj,k(cid:5)(cid:5) (tj,k(cid:5)(cid:5))isφforallk(cid:5)≤k(cid:5)(cid:5)≤k;thus,in
thegeneralcase,thevaluesstoredinanyoftheaddressregistersareacontinuous“run”
ofvaluesfromtheset{0,1}followedbyarun,possiblyoflengthzero,ofthesymbolφ.
We associate the pair ρin ∈Rin and ρout ∈Rout uniquely with the state pc . For a
j j j
givenstate,intuitively,thesepairofregistersyieldtheaddressesfromwheretheinput
σ is to be read, and to where the output σ(cid:5) (if any) is to be “written” respectively.
Itisimportanttonotethattheseaddressescaninfactbetheregistersthemselves.The
potentiallyunboundedlengthsoftheregistersdenotethefactthattherangeofaddresses
being accessed (correspondingto the length of a Turing machine’stape for example)
couldbeunbounded2.
The Transition Function - We are now ready to define the transition function δ,
whichwillplayacentralroleincharacterizingtheenergybehaviorofcomputations.In
itsmostgeneralform,atransitionfunctionisbasedonanalphabetofsize|Σ|≥2.
Syntactically,δ:(PC∪{START})×Σ→(PC∪Q−{START})×Σisthetransition
function.Wheneverδ(pc,σ∈Σ)=(pc ,σ(cid:5)∈Σ),wesaythatδtransitionsfrom pc to
i j i
thenext-state pc withσasinputandσ(cid:5) astheoutput.
j
Some useful remarks about the transition function follow. First, we note that the
stateUNDEFINED-STATEisintherangeofδ.Givenastate pci,letνidenotethenumber
of symbols from Σ for which δ transitions into a state in PC∪{ STOP }, as opposed
into the UNDEFINED-STATE . For the remaining(|Σ|−νi) symbols, δ transitions into
UNDEFINED-STATE .(Thisisonewayofdefiningtransitionsofvarying“arity”νi as-
sociatedwithstate pc,thusallowingstateswithvaryingnumberofsuccessorswithan
i
alphabetoffixedsize).Inthissetting,itistrivialtoverifythatthereisnolossofgener-
alityindefiningδsuchthatthefirstν symbolsfromthelinearlyorderedsetΣrepresent
i
defined transitions whereas symbols νi+1,νi+2,...|Σ| represent undefined transitions.
ThesenotionsareillustratedinFigure1.Inthesequel,wewill(mostly)beconcerned
withBRAMprogramswhosetransitionfunctionhasamaximumarityoftwo.(Itistriv-
ial to verify that any BRAM program with transition function of arity more than two
canbereplacedwitha BRAM programwithtransitionfunctionwhosemaximumarity
istwoalthoughitsenergybehaviorneednotbepreserved).Furthermore,anytransition
withanarityoftwowillhenceforthbereferredtoastheBRANCHinstruction.
For convenience, drawing upon graph theoretic terminology, let us refer to ν as
i
thefanoutof pc andfurthermore,refertostate pc asbeingasuccessorof pc ifand
i j i
onlyifthereexistsasymbolσ∈Σsuchthatδ(pc,σ)yields pc asthenextstate.Let
i j
successors denotethesetofallsuccessorsofstate pc fromPC.
i i
2Inany terminatingcomputation, therewillbe alimiton thisbound, typicallyspecified asa
functionofthelengthoftheinput[21].
ComputationalProofasExperiment 531
1
Input symbols 1,2,(cid:133)ν are legal
i
pc 2 νisuccessors
Current state
ν
i
Fig.1.Illustratingthelegalandillegalcasesofatransitionfunctionwithanalphabetof
size|Σ|≥ν
i
Successors of PC
pc'
1 Input Transition to
1
Input
2 pc‘
2 1 pc'
pc 3 1
pc‘
4 3 2 pc'
Current state 2
pc‘
4 3 pc'
3
4 pc'
Alphabet∑= {1,2,3,4} U φ 4
Fig.2.Astate,itssuccessorsandrelatedtransitions
The Memory Each BRAM has a (potentially unbounded) MEMORY denoted as the
setof L=(2|PC|+1)linearlyorderedsets orbanks,eachpotentiallyunbounded.As
showninFigure3,elementsIand(I+1)inMEMORYaredenotedMI andMI+1where
1≤I≤2|PC|arerespectivelyusedasregistersρin∈Rinandρout∈Rout,wherei=(cid:8)I(cid:9).
i i 2
Additionally, the last set ML of MEMORY, denoted M is a potentially unboundedset
M<m ,m ,...,m >.Eachcellm ofmemoryisassociatedwithanelementfromthe
1 2 k j
set{0,1,φ}.Informally,M isthesetoflocationswheretheinputsandoutputsvalues
beingcomputedbytheBRAM“program”arestored—itistheworkspace.
Recallthattheinputargumentstothetransitionfunctionδarethecurrentstate pc
andtheinputvaluefromthealphabetΣ.SincetheinputcanonlybeasymbolfromΣ,a
maximumoflog(|Σ|)bitsareneededtostorethisvalue3.Therefore,forconvenience,
3Unlessspecifiedotherwise,alllogarithmswrittenaslogaretothebasetwo
532 KrishnaV.Palem
J → 1 2 3 4 (cid:133) 2|PC|-1 2|PC| L=2|PC| + 1
ρ1in ρ1out ρ2in ρ2out ρpcin ρpcout log∑bits
location1 Scratch/
Working
location2
Memory
Addresses for PC Addresses for PC Addresses for PC
1 2 pc
Fig.3.TheMemoryStructureoftheBRAM
eachM willbepartitionedinto“locations”wherelocationL forJ≥1ismadeupof
I J
log(|Σ|)constituentcells;lets=(log|Σ|(J−1)).ThenLJ =<ms+1...m(s+log|Σ|)>
The Memory Access Unit The value at a location L is the concatenation of the
J
valuesinitsconstituentcells.Sincethevalueofalocation,whendefined,isanatural
numberfromtherange{1,2,...|Σ|},itisdeterminedbyabinaryinterpretation,ofthe
concatenationofsymbolsfromtheset{0,1}.Ifoneofthevaluesassociatedwithany
ofthecellsinL isφ,thenthevalueofthislocationisundefined.
j
AVALUEinMEMORYisafunctionfrom(N+×N+)intothesetΣ∪{φ}definedas
follows:
1. If1≤I≤2|PC|namelyifindexIcorrespondstoaregister,thenVALUE (I,J)=§
where§isthevalueattheJth locationofM .
I
2. IfI=ML,VALUE (ML,J)=§isthevalueattheJth location(LJ)ofM.
ThefunctionVALUEthatisimplementedthroughthememoryaccessunitofaBRAM
yields the value associated with the Jth location in one of the registers in R or at the
locationL fromMdependingonthevalueofI.
J
Theaddressinregisterρin(orρout)istheuniquenon-negativeintegerwhosevalue
i i
isu,wheretheaddressisrepresentedinunary.The MAU isafunctionthatusesthese
(pairof)addressesasanargument.Throughouttherestofthispaperwewillconsider
analphabetwhere|Σ|=2,andthisunaryrepresentationwillacrosslocationsLbeused
to analyzethe energyadvantagesofprobabilisticcomputing.Alternatealphabetsizes
aswellasbinaryrepresentationswillbethetopicoffuturestudyasdiscussedbrieflyin
Section6.
Wedefinefunctionsreadandwritewithaddressesastheirdomain.Thus,usingcon-
ventionsinspiredbyTuringmachinesasoriginallydefined[28],read (I,LOCATION)
ComputationalProofasExperiment 533
and write (Σ,I,LOCATION) are respectivelyused to read the value or (over)writethe
valuesassociatedwiththeconstituentcellsoflocationLinMI.The MAU istheunion
of the read and write functions. It will be used to evaluate the transition function as
explainedinSection2.3below.
2.3 TheComputationofaBRAM
Buildingontheelementsintroducedabove,wewillnowintroducetheoperationalbe-
haviorofaBRAM.GivenanarbitraryBRAMprogramP,initially,allcomputationsstart
in the START state. All the registersand the memorycells are initialized from the set
{0,1,φ}. It is convenientto define the operation of the BRAM inductivelyas follows.
The START statetransitionsto,withoutlossofgenerality,state pc1 atwhichpointthe
computationstartswheretheconcatenationofthecellsinM isinterpretedasanumber
L
inunaryrepresentationandisreferredtoastheinputI toP.Now,state pc issaidto
1
bethecurrentstate.Moregenerally,let pc bethecurrentstate.Instate pc ∈PC,the
l l
transitionfunctionisevaluated.
The input to the transition function is a symbol from Σ, which is accessed using
σ=read(M,LOCATION),whereLOCATIONistheaddressstoredinunaryinρin.These
l
notionsareillustratedinFigure4.
Continuingwith the evaluation of the transition function δ(pc,σ) yields the next
l
state pcl(cid:5) whichthen becomesthe currentstate. Furthermore,the outputsymbolσ(cid:5) is
written(usingwrite )intothe LOCATION whoseaddressisstoredinregisterρout.The
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computationhaltswheneverpcl(cid:5) ≡STOP.
Next state pc Output value σ(cid:146)
l'
Inputσ
pc
l
δ(pc ,σ)
l
Evaluate
READ NEXT STATE? CURRENT STATE = pc
l'
OUTPUT VALUE WRITE σ(cid:146)
Location
Location
from ρin from ρIout
I
Fig.4.IllustratingtheEvaluationoftheTransitionFunction