Table Of ContentDataflow Graphs as Matrices
and Programming with Higher-order Matrix Elements
MichaelBukatin1 and SteveMatthews2
1 NokiaCorporation
Burlington,Massachusetts,USA
6
[email protected]
1
0 2 DepartmentofComputerScience
2 UniversityofWarwick
Coventry,UK
n
[email protected]
a
J
Abstract
6
Weconsiderdataflowarchitecturefortwoclassesofcomputationswhichadmittakinglinearcom-
]
binationsofexecutionruns:probabilisticsamplingandgeneralizedanimation.Weimprovetheearlier
L
techniqueofalmostcontinuousprogramtransformationsbyadoptingadisciplineofbipartitegraphs
P
linking nodes obtained via general transformations and nodes obtained via linear transformations
.
s which makes it possible to develop and evolve dataflow programs over these classes of computa-
c
tionsbycontinuousprogramtransformations. Theuseofbipartitegraphsallowsustorepresentthe
[
dataflowprogramsfromthisclassasmatricesofrealnumbersandevolveandmodifyprogramsby
1 continuouschangeofthesenumbers.
v Wedevelopaformalismforhigher-orderdataflowprogrammingforthisclassofdataflowgraphs
0
basedonthehigher-ordermatrixelements.Someofoursoftwareexperimentsarebrieflydiscussed.
5
0
1
1 Introduction
0
.
1
Because probabilistic sampling and generalized animation are both stream-based, dataflow program-
0
6 mingisanaturalframeworkforthissituation.Earlierwewereabletoleveragetheabilitytotakelinear
1 combinationsof executionruns to obtain the notion of almost continuoustransformation of dataflow
: programs[1].
v
i There were three main sources of benign discontinuitiesin [1]: additionof new subgraphsvia an
X
operationoflimiteddeepcopy,insertionofanewvertexandanedgeduringthefirststageofS-insert,
r andinsertionofanedgeduringthesecondstageofS-insert.
a
Wenowallowtodecoratesubgraphswithweightsandrequirethatnewsubgraphscreatedbylimited
deepcopyappearinitiallywithzeroweightsubjecttosubsequentcontinuousevolution.
Wealsointroducethedisciplineofarrangingandlinkingverticesasabipartitegraph,namelythat
generaltransformsoffixedaritymustpointonlyatlineartransformsofunlimitedarity,andviceversa.
Thisallowstoeliminatethebenigndiscontinuitiesmentionedaboveandobtaincontinuousprogram
transformations.
1.1 DataflowGraphs as Matrices
It is convenientto have a situation where for any trajectory of program developmentor evolution all
partsoftheprogramwhichmightemergepreexistinasilentway.
The discipline of bipartite graphs actually makes this possible. We fix a particular signature by
takingafinitenumberofoperations,eachwithitsownfixedfinitenon-negativearity.
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DataflowGraphsasMatrices BukatinandMatthews
Wetakeacountablenumberofcopiesofeachtemplateoperationfromthesignature.Thenwehave
acountablesetofinputsofthoseoperations,Y ,andacountablesetoftheiroutputs,X.
j i
AssociatewitheachY a linearcombinationofallX withrealcoefficientsa . We requirethatno
j i ij
morethanfinitenumberofelementsofthematrix(a )arenonzero.
ij
We oftenimposeadditionalconditions,e.g. we oftenrequirethatall a are nonnegative,andthat
ij
thesumofelementsa associatedwithaparticularmatrixcolumnY doesnotexceed1.
ij j
Thuswehaveacountable-sizedprogram,namelyacountabledataflowgraph,allbutafinitepartof
which issuppressedbyzerocoefficients. Anyfinite dataflowgraphovera particularsignaturecanbe
embeddedintoauniversalcountabledataflowgraphoverthissignatureinthisfashion.
Hencewerepresentprogramsoverafixedsignatureascountable-sizedreal-valuedmatriceswithno
morethanfinitenumberofnonzeroelements,andanyprogramevolutionwouldbeatrajectoryinthis
spaceofmatrices.
1.2 MultipleTypes ofData Streams
OriginallythisconstructionwasenvisionedforthesituationwheneachnodeX andY hasadatastream
i j
ofthesametype,andallthesestreamsareequippedwiththesamemeaningfuladditionoperation[1].
Themostimportantcasehereiswhenthesearestreamsofrealnumberswhichcanbeconsideredas
one-pointgeneralizedanimations.
However,wemightwanttoconsidersituationswhentherearemultiplestreamtypesassociatedwith
nodes. ConsiderasituationwhenY andX areofdifferenttypes. Ifa =0,thisisfine,becausethen
j i ij
a ·X doesnotaffectY . IfthereisadefaultadapterT betweenthesetypes,thenothervaluesofa are
ij i j ij ij
allowed,anda ·T (X)iscontributedtothesum. Otherwisetheconditiona =0hastobeenforced
ij ij i ij
(a areclampedatzero).
ij
1.3 MultipleTypes ofAddition
One might want to also consider a situation where more than one meaningful addition operation is
possible within the same program (e.g. point-wise addition of generalized animations and stochastic
sumofthestreamsofprobabilisticsamples).
(Notethattheseadditionsaredifferentfromtemplateoperations:additionshaveunlimitedarityand
map infinite tuples of X’stoY’s, and templateoperationshave finite fixed nonnegativearity and map
finitetuplesofY’stoX’s.)
In the case ofmultiple typesof addition, each inputY ofa template operationfromthe signature
j
needstobemarkedwiththetypeoftheadditionoperationituses. Ifnecessary,similartemplateopera-
tionswithdifferentadditiontypesshouldbeincludedinthesignatureseparately.
Onewouldtypicallyincludeanidentitytransformforeverytypeofdatastreamandforeverytype
ofadditionapplicabletothisparticulartypeofdatastreaminordertoprovideacapabilitytogroupthe
sumshierarchically.
1.4 String-based Indices
Rather than indexingour countablesets {X} and {Y } with numberswe are going to fix an alphabet
i j
andtoindexthemwithstringsfromthatalphabet. Ofcourse, letterscanbeconsideredasdigitsinan
appropriate base, and strings as the corresponding natural numbers, but by indexing with strings we
are trying to de-emphasize the order associated with this natural-numbers-basedinterpretation and to
encourageanimplementationofsparsearraysbasedondictionaries(hashtables),withtheexpectation
thatzeroelementsofarraysandmatriceswouldtypicallybeomittedfromtheirrespectivedictionaries.
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DataflowGraphsasMatrices BukatinandMatthews
InordertomaintainourconventionthattheassociationbetweenaparticularX anditstemplateoper-
j
ationisfixed,andthattheassociationbetweenX anditscorrespondingargumentsY ,...,Y (n >=0)
j k1 knj j
is fixed and that all flexibility is concentratedsolely in the values a we adoptthe following naming
ij
conventions.
We call a multiset (bag) of strings prefix-free if none of the strings in that multiset is a prefix of
anotherstringinthatmultiset. (Notethataprefix-freemultisetcannothavemultipleoccurrencesofthe
samestring.) Alltemplateoperationsaregivenstringnamesinsuchawaythatthemultisetconsisting
ofallthesenamesandthestring′arg′ isprefix-free. AnodeY isindexedwithastring j whichstarts
j
withthenameofthetemplateoperationassociatedwiththisnode. Wesaythatthestring jisthename
ofY . AnodeX whichisoneoftheargumentsforY isindexedwiththestringiwhichisobtainedby
j i j
concatenatingtheprefix′arg1′(′arg2′etc.) withthenameofY . Wesaythatthestringiisthenameof
j
X.
i
1.5 Higher-order Programming
Because matrices, their columns, and their elements are vectors themselves, this is a platform with a
varietyofopportinutiestocreatetechniquesforhigher-orderprogramming.
In this text we focus on one particular avenue for doing so, and this approach is centered around
makingelementsa higher-order(seeSection2).
ij
To quote from Section 1.1 of [2] “A lot of expressive power of this architecture comes from the
ability to have non-standard secondary structures on the set of points. Points can be associated with
vertices or edges of a graph, grammar rules, etc. One should be able to formulate mechanisms of
higher-orderanimationprogrammingviavariableilluminationofelementsofsuchstructures.”
Whatisbeingdoneinthepresentpaperisaninstanceofthisapproach.
Amoreprecisemathematicaldescriptionofthewaythiskindofcomputationalenginefunctionsis
inSection3.
Ourfirstseriesofexercisesinmatrix-baseddataflowprogrammingwhichhavebeenopen-sourced
simultaneouslywiththereleaseofthispreprintisdescribedinSection4.
2 Higher-order Matrix Elements
2.1 Degrees ofOrder
Hereweclassifyvarietiesofa .
ij
2.1.1 NotProperlyHigher-orderElements
Anotproperlyhigher-orderelementa isnotassociatedwithastreamofanyparticularnodeX .
ij k
Zeroelements(zero-order). Thesecoefficientsarezeroandarenottypicallyincludedinthedictio-
nariesimplementingthematrixoritscolumns.
First-orderelements. Constantelementsofthematrix(a )arecalledfirst-orderelements.
ij
Variable elements (first-and-a-half order, sesquialteral-order elements). Elements of the matrix
(a )whichvarywithtimebelongtothisclass. Ourexercisesinhigher-orderdataflowhavebelongedto
ij
thisclasssofar.
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DataflowGraphsasMatrices BukatinandMatthews
2.1.2 ProperlyHigher-orderElements
Aproperlyhigher-orderelementisassociatedwithadatastreamofaparticularnodeX . Thethreetypes
k
abovearestillrepresentablewithhigher-orderelements. Whenanodeisnotturnedon(andtypicallyis
notintheappropriatedictionary),thisisthecaseofzero-orderelement. Whenanodeisconstantvalue
withnoarguments,thisisthecaseoffirst-ordermatrixelement. Whenanodeisavariablestream,but
doesnothaveargumentnodes(sowearetalkingaboutapredefined/externalvariablestream),thisisthe
caseofsesquialteral-orderelement.
Specializedhigher-orderelements. WhenanodeX iscomputedbyatemplateoperationwithatleast
k
oneargument,thisisthecaseofahigher-orderelement. We callitspecialized, becausetheoperation
whichcomputesthiselementisfixed,althoughitcan becontrolledviaits argument(s). Evaluationof
suchanoperationisimmediatelyeffective,i.e. thenewlycomputedcoefficientisthenimmediatelyused
tocomputetheappropriatelinearcombination.
Fullyhigher-orderelements. InthiscasethenodeX iscomputedbyatemplateoperationwhichis
k
theidentitytransform(oneoftheidentitytransforms,ifmultipletypesofadditionofstreamsofrealsare
includedinthesignature).Inthiscase,thevalueinquestioncanbecomputedinaveryflexiblemanner
from any linear combination of any transformations. However, the value is essentially computed on
the“downswing”usingthepreviousvaluesof(a ),andbecomestheneweffectivevalueofthematrix
ij
only after the identity transformis applied to it. So if one needs to implementchangingthe a via a
ij
particularspecializedtemplateoperation,aone-cycledelayisinvolvedbeforethenewvalueofa goes
ij
viaalinearcombinationandviaanidentitytransformandbecomeseffective.
2.2 Embedding the Set of Matrix Coefficient Indices into the Set of Column In-
dices
An interesting question is whether one can have all matrix elements to be properly higher-orderele-
ments. Theansweris“yes”,anditisessentiallybasedonthecountabilityoftheunionofcountablesets.
Thiscountabilityallowsustotakethesetofindicesofmatrixcoefficients,(ij), andembeditintothe
setofindicesofmatrixcolumns, j.
Herewearegoingtogiveanexampleofsuchanembeddingwhichissufficientlydetailedtoenable
computerimplementation.
Without properly higher-order matrix elements, it was possible to avoid giving unique names for
matrix elements. For example, one could represent matrix columns as separate dictionaries and then
justusetherownamesformatrixelements,likewedointheexampleofSection4.
Withproperlyhigher-ordermatrixelementsoneneedsuniquenamesformatrixelements. Westart
with the naming scheme for columns and rows given in Section 1.4 and given the name N for col-
j
umn j and the name M for row i, we define the name for the matrix element (ij) as concatenation
i
′(′+N +′)#(′+M+′)′. Nowweshouldjustreserveparenthesesfortheusewithinthenamesofmatrix
j i
elements only, but not within the “originally present names”, and this should be enough to avoid the
nameclashes.
3 A More Precise Description of the Abstract Machine
Weonlydescribethemachineforthesituationwhereallnodesarestreamsofreals,andwhereonekind
ofadditionoperationexistswiththesumbeingcomponent-wiseaddition.Itisnotdifficulttogeneralize
appropriately.
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DataflowGraphsasMatrices BukatinandMatthews
DefineanalphabetS whichdoesnotcontainparenthesesandthenumbersign,′()#′.
DenotethesetofallfinitestringsoverS asS ∗.
DefineanoperationasatupleF containingname(astringN ∈S ∗), arity(anon-negativeinteger
F
AF),andamathematicalfunction fF :RAF →R(ifAF iszero, fF issimplyarealnumber).
Defineasignatureasafinitesetofoperations,S={F ,...,F }.
1 n
Werequirethatthemultiset{N ,...,N ,′arg′}isprefix-freeperSection1.4.
F1 Fn
WetakeF tobetheidentityoperation,h′id′,1,x7→xi.
1
3.1 AbstractMachine WithoutProperly Higher-order Matrix Elements
The first case we consider is the case of sesquialteral-order matrix elements, which will be further
illustratedbytheexampleinSection4.
WeintroducethepartiallydefinedmapOpfromS ∗toS(operations)andtwopartiallydefinedmaps
X andY fromS ∗ toN→R(datastreams).
Wedenotestringconcatenationwith+andstringrepresentationofanon-negativeintegerkasstr(k).
Forw∈S ∗andforeveryF ∈Sintroducei=N +′ ′+w. Forallsuchi,Op(i)andX(i)(denoted
m Fm
asX)aredefinedandOp(i)=F .
i m
For all such i, for all k∈{1,...,A }, introduce j= ′arg′+str(k)+′ ′+i. For all such j, Y(j)
Fm
(denotedasY )isdefined.
j
Nowwehaveadataflowgraphrepresentingthemachine,andweneedtodefinethevaluesofdata
streamsN→RassociatedwithgraphnodesX andY ,whichwilldefinehowthemachineworks.
i j
Timestartswithvalue0andincreasesby1ateachmachinestep. Foreveryiand j definedabove
there is a stream of values N →R associated with a . Given that we are considering the case of
ij
sesquialteral-ordermatrixelements,weassumethatstreamsassociatedwitha areexternaltothepro-
ij
gramandarejustgeneratedbytheirexternallyprogrammedgenerators.
At time 0 we start with all X andY being the streams of one element, which equals to 0 for all
i j
those streams. The valuesof a at the moment0 are not importantat this levelof consideration, be-
ij
causeweonlygiveamathematicaldefinitioninthissectionandignorethequestionofsparsenessasan
implementationdetail(ofcourse,nothingwouldevenfitafinitemachinewithoutthisimplementation
detail).
Then given streams X,Y , and a of lengtht, here is how the componentst+1 are defined(that
i j ij
is,computedbytheabstractmachine;sothisisadefinitionbyinductionwithrespecttotime,andthis
definitiondescribeshowthemachineworks).
Step1. First,foralli,thevalueofX attimet+1iscomputedasfollows. OneconsidersOp(i),and
i
forallk∈{1,...,A }and j = ′arg′+str(k)+′ ′+ionetakesthevaluey asbeingthevalueofthe
Op(i) k jk
streamY atmomentt.ThevalueofX attimet+1theniscomputed(defined)as f (y ,...,y ).
jk i Op(i) j1 jAOp(i)
Step2. Thenthevaluesofstreamscorrespondingtoa atthemomentt+1areexternallygenerated.
ij
Theconditionthatonlyafinitenumberofthosearedifferentfrom0atthemomentt+1isobserved.
Step3. Finally,forall j,thevaluesofstreamsassociatedwithY atthemomentt+1arecomputedas
j
thelinearcombinationsS a ·X, wherea andX arethevaluesofthecorrespondingdatastreamsat
i ij i ij i
themomentt+1.
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DataflowGraphsasMatrices BukatinandMatthews
3.2 AMore AbstractView
Hereisamoreabstractwaytodescribewhatisgoingonintheprevioussubsection.
Therearetwo(ingeneral,countablydimensional)vectorspacesX andY.Thereisafixednon-linear
function F :Y →X induced by the signature. There is a family of linear transformationsL :X →Y
t
parametrizedwithtime(thiscorrespondstosesquialteral-orderofL).
We define streamsof vectorsx, y fromX,Y parametrizedwith time by inductionovertime. We
t t
definex andy tobezerovectorsandL isirrelevant.
0 0 0
Attimet+1,firstwetakex =F(y(t)),thenwenotethecurrentL andtakey =L (x ).
t+1 t+1 t+1 t+1 t+1
InthesubsectionwhichfollowsweusetheembeddingfromSection2.2toincorporateLintoX.
Thenattimet+1 we take x =F(y(t))and amongotherthingsthisyieldsL . Thenwe take
t+1 t+1
y =L (x ).
t+1 t+1 t+1
We occasionally parametrize F as F as well, to allow the dynamic tuning of the system and to
t
accountforthestochasticfactorsinrandomizedtemplateoperations.
3.3 AbstractMachine withallFullyHigher-order MatrixElements
HerewemodifytheconstructionofSection3.1inordertodeliverwhatwaspromisedattheendofthe
previoussubsection.WestartwithsetsofnamesI={i}andJ={j}definedinSection3.1.
ThenwedefineI =I andJ =J,andorganizetheinductionasfollows.
0 0
GivenI andJ wefirstusetheschemefromSection2.2todefinesetofnamesofmatrixelements:
n n
A ={′(′+j+′)#(′+i+′)′|i∈I ,j∈J }.
n+1 n n
ThenweusethefactthatoursignaturehastheidentityoperationanddefineIn+1=IS{′id′+a|a∈
An+1}andJn+1=JS{′arg1id ′+a|a∈An+1}.
Nowconsiderthelimitofthisprocess,I¥ =SIn andJ¥ =SJn.
Then this limits is the fixed point of this process, namely that if we define
A¥ ={′(′+j+′)#(′+i+′)′|i∈I¥ ,j∈J¥ },thenI¥ =IS{′id′+a|a∈A¥ }andJ¥ =JS{′arg1id′+a|a∈
A¥ }.
Now the steps at the end of Section 3.1 are modified as follows. PerformingStep 1 yields values
ofstreamscorrespondingtoa atthemomentt+1. Ifonlyfinitenumberofthosevaluesaredifferent
ij
from0atthemomentt,thenonlyafinitenumberofthemwouldbedifferentfrom0atthemomentt+1
(forthedetailedexplanationofthereasonsforthatseethelastparagraphofthenextsubsection).
ThenwecangoaheadandperformStep3.
3.4 Sparseness Considerations
The discussion of the abstract machines would not be complete without discussing how to keep the
amountofcomputationsateachstepfinite(as,ifpossible,minimal).
ForthesetupofSection3.1,therequirementisthatifamatrixelementa isnonzero,thenitslinear
ij
combinationneedstobeevaluatedandthetemplateoperationcomputingX andthetemplateoperation
i
usingY asanargumentbothneedtobeevaluated,sotheappropriatenodesshouldbeincludedintothe
j
dictionariesin question. Whenforatemplateoperationwith inputsY ,...,Y andoutputX allmatrix
1 k i
elements related to nodesY ,...,Y and output X are zero, it is not recommendedto evaluate such a
1 k i
templateoperationortoincludethenodesinquestionintodictionaries(althoughdoingsoislegal).
Thismeansthatwhena becomesnonzeroforthefirsttime,theY nodeandthetemplateoperation
ij j
usingthisnodeasaninputtogetherwiththeotherverticesofthistemplateoperationmustbeaddedto
theappropriatedictionaries,ifitisnotalreadythere. Similarly,theX nodeandthetemplateoperation
i
computingittogetherwiththeotherverticesofthattemplateoperationmustbeaddedtotheappropriate
dictionaries,ifitisnotalreadythere,andmoreover,itneedstoberetroactivelycomputed(byevaluating
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DataflowGraphsasMatrices BukatinandMatthews
thecorrespondingtemplateoperationwithzerovaluesofitsarguments)beforethelinearcombination
usinga canbeevaluated.
ij
The same considerationsare valid for the setup of Section 3.3. At any given momentof time we
only have a finite number of operations computing a being in the dictionaries and subject to being
ij
evaluated. So only finite number of them might become nonzero for the first time at any given time
moment.Hencenomorethanafinitenumberofnewnodesneedtobeaddedtothedictionariesoneach
timestep.
4 A Simple Example
This architecture is applicable to probabilisticsampling and to generalizedanimation. We created an
opensourcesoftwareprototypedemonstratingtheuseofthosetechniquesforstreamsofrealnumbers
implementingafamilyofcontinuouscellularautomata.
The software and associated videos have been released simultaneously with this preprint as
aug 20 15 experimentintheProjectFluidrepository[3].
Note that we cannot make these video previews quite faithful; video compression software intro-
ducessomedistortionattheinitialstageofuploadingavideo,thenitnoticestheproblemswithquality
andofferstoautomaticallyfixlighting,theoverallresultmightbeworseinsomeaspectsandbetterin
someotheraspectsthanthevideonativelyproducedbythesoftware.
This is an example of a situation when a matrix-based dataflow program is used in combination
withothersoftware. Inthisparticularcase,thevideooutputfromthedataflowprogramisbeingpassed
throughanexternallighting-enhancingsoftware.
The set of template operations for this series of experimentsis very simple: zero-arity constants,
blackandwhite(-1and1),andaunaryrandomizedpropagator,copyingthevalueofitsinputY toits
j
outputX withprobability p(atypicalvalueofwhichinthisseriesofexperimentsis0.995)andsetting
i
theoutputtozerootherwise. ThesetupisessentiallythatofSection3.1, exceptthatwe allowseveral
copiesofatemplateoperationtoshareaninputwhenconvenient(mathematicallythisisequivalentto
constrainingseveralcolumnsofamatrixtobethesame).
In this series of experiments we use the dynamic nature of matrix elements very sparingly, first
havingtheweightsconcentratedontheoutputoftheconstants(initializationphase)andthenswitchingit
(continuously,or,asinthisseriesofexperiments,abruptly)totheoutputoftherandomizedpropagators
formingastructuredeterminingtheconnectivityofthecontinuouscellularautomatoninquestion.This
architecturealsoallowstocontinuouslymorphbetweenavarietyofconnectivitystructures,butwehave
leftthisasasimpleexercisetothereader.
WeareobservingavarietyofemergingTuringstructureswithinterestingdynamicsinthisseriesof
experiments,dependingmostlyontheconnectivitypattern,andalsoontheinitializationconfiguration.
In the spirit of non-equilibriumthermodynamics, even when one starts with a uniform initial pattern
(all black or all white), the structures first emerge driven by the randomized propagatorsand by the
connectivity pattern, and then fade out to gray (zero value, the thermodynamic equilibrium). If one
includes the amplification during the visualization stage (scaling the brightness of the image so that
the maximalabsolute value of a renderedpoint reaches 1), then one can observe that the meaningful
fine structures of the dynamic patterns tend to persist indefinitely (in some cases the main motive is
preserved,inothercasesthepatternisshiftingandevolvingunpredictably).
SeetheFluidrepositoryforfurtherdetails.
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DataflowGraphsasMatrices BukatinandMatthews
4.1 AnExamplewithProperly Higher-order MatrixElements
Theaug 24 15 experimentimplementsthesameexampleusingfullyhigher-orderelements(allbut
twomatrixelementsinthatexperimentarefullyhigher-order,whiletwoofitselementsarenotproperly
higher-order,andtheirchangeiscontrolledfromtheoutside).
Thisexperimentalsofeaturespause/resumefacilitycontrolledbythemousekeyandthespacebar,
improved brightness amplification, and an adjustment mechanism to stabilize the values and prevent
theirrelaxationtozero.
Werecommendthatthereaderusesthisexperimenttofurtherexplorethesystem.
5 Conclusion
This architecture allows us to evolve dataflow programs in continuous fashion while those evolving
programsarerunning.Thismakesitpossibletosamplecontinuoustrajectoriesinthespaceofdataflow
programs,inadditiontotheusualpracticesofsamplingthesyntaxtreesofprograms.
The representation of programs as matrices of real numbers makes the task of program learning
moresimilartothetaskofmachinelearningformorenarrowandconventionalclassesofmodels.
Onewaytointroducesomehigher-ordermechanismsindataflowprogramming(andinneuralnet-
works)isvialettinganyparticulardatastreamtopropagatealongmultipledirectionsinthegraph,andto
dynamicallycontroltheactualdirectionsofpropagationbysettingthemultipliersalongsomeofthose
pathstozero.
Asomewhatunexpectedresultofthepresentpreprintisthatthisparticularhigher-ordermechanism
seems to be universal: all higher-orderprogrammingfor a given class of dataflow graphsover datas-
treamswithlinearcombinationsofexecutionrunsbuiltontopofafixedsignatureoftemplateoperations
seemstobeexpressibleviasettingmultipliersalongthepathsofdatapropagation(inparticular,setting
allbutthefinitenumberofthemtozero).
Acknowledgments. WewouldliketothankRalphKopperman,LenaNekludova,andJoshTenen-
baumforhelpfuldiscussionsandtoacknowledgecollaborationwith LenaNekludovaonsoftwareex-
periments.
References
[1] M.BukatinandS.Matthews. Almostcontinuoustransformationsofsoftwareandhigher-orderdataflowpro-
gramming,Preprint(2015). http://www.cs.brandeis.edu/~bukatin/HigherOrderDataFlow.pdf.
[2] M. Bukatin and S. Matthews. Linear models of computation and program learning, Preprint (2015).
http://www.cs.brandeis.edu/~bukatin/LinearModelsProgramLearning.pdf.
[3] Fluid. Project“Fluid”githubrepository,2015. https://github.com/anhinga/fluid.
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