A Visual Representation for Functional Programs

Abstract

Functional programming languages have features that make them attractive for software development and for the teaching of programming concepts. Like all conventional programming languages they are tied to an inherently one-dimensional textual programming style. Visual programming seeks to enhance the development and understanding of software, by utilizing the human brain's natural visual/spatial processing abilities. A visual representation for functional program expressions is defined: expressions are represented as rooted, directed acyclic graphs. Various graph nodes represent functions, type information and sub-expressions. Complex expression-graphs are constructed by composing simpler expressions, and functions are defined by collapsing expression-graphs to form new function nodes. This representation is intended to form the basis of a visual programming environment for functional languages. 1 Contents 1 Introduction 4 1.1 Functional expressions as graphs : : : : : :...

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Page 1

AVisualRepresentationforFunctional
Programs
JoelKelso
joel?cs?mur doch?edu?au
December????
Abstract
Functionalprogramminglanguageshavefeaturesthatmakethemattrac?
tivefor softwaredevelopmentandfortheteachingofprogrammingconcepts?
Like allconventionalprogramming languagestheyare tiedtoan inherently
one?dimensional textualprogrammingstyle?Visualprogrammingseekstoen?
hancethedevelopmentandunderstandingof software?byutilizingtheh uman
brain?snaturalvisual?spatialprocessingabilities?
Avisualrepresentationforfunctionalprogramexpressions isde?ned? ex?
pressionsarerepresentedasrooted?directedacyclicgraphs?Variousgraph
nodesrepresentfunctions?typeinformationand sub?expressions?Complex
expression?graphsareconstructedbycomposingsimplerexpressions?andfunc?
tionsarede?nedbycollapsingexpression?graphstoformnewfunctionnodes?
Thisrepresentationisintendedtoformthebasis ofavisualprogramming
environmentforfunctionallanguages?
?

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Contents
?Introduction?
???Functionalexpressionsasgraphs??????????????????????
???Type?directedediting????????????????????????????
?Anatomy of anexpression?graph?
???F unctions??????????????????????????????????
???Types????????????????????????????????????
?????T ypecontexts????????????????????????????
???Pattern?matching???????????????????????????????
?????Patternframes????????????????????????????
???Guardedexpressions?????????????????????????????
???Sub expressionsharing????????????????????????????
???Applyinghigher?order functions???????????????????????
?Expression?graph construction??
???Expressionattachment????????????????????????????
???Functionde?nition??????????????????????????????
???Functiondeclaration?????????????????????????????
???Codehiding??????????????????????????????????
?Summary??
?

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List ofFigures
?Asimple expression?graph?????????????????????????
?Functionnodes???????????????????????????????
?Asignatureexpression?graph????????????????????????
?Sampletype?nodes?????????????????????????????
? Samplepatternno des?????????????????????????????
?Asamplepattern?graph???????????????????????????
?Asamplepatternframe????????????????????????? ??
?Asampleguardedexpression?????????????????????????
?Exampleofashared sub expression?????????????????????
??The?apply?primitive????????????????????????????
??Asimple expression?graphattachment???????????????????
??Exampleofexpression?collapsede?nition??????????????????
??Sub?expressionhidingexample?????????????????????? ??
??A complexexpression?graph?????????????????????????
?

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J?K?KelsoAVisualRepresentationforFunctionalLanguages?
?Intro duction
Thisreportformspartofaprojecttheaimofwhichistoproduceavisual program?
mingenvironmentforfunctional programminglanguages? In viewof thisend?the
reportintro ducesavisual representationforfunctionalprogramexpressions?and
outlinesavisualeditingsystemforthemanipulationoftheseexpressions?
Aoverviewofthe representation andediting systemisgivenbelow?alongwith
itsrelationtotheexistingworkonvisual functionallanguages?
Throughout thereportweuseHaskell??? asanarchetypalfunctionallanguage?
althoughtherepresen tationcouldbeusedforsimilarlanguagessuchMiranda
?
???
orHope????
???Functional expressionsasgraphs
Therationalebehindusinggraphsfortherepresen tationoffunctionalprograms
restsontwo observations?Firstly? programming language expressions?because their
syn taxisde?nedbyagrammar?areessentiallytree?like?and secondly?thatthe
commonmodelof functional programexecution isgraphre duction? Representing
functional programsasgraphsallowstheprogrammertoperceiveandeditthepro?
gram?ssyn tacticalstructurespatially?ando?ersinsightin totheprogram?sunder?
lyingsemantics?
Werepresent individualfunctionsasgraphnodes?theapplicationofa function
to an argument sub?expression isrepresentedbyanarcbetweennodes?This isthe
mostcommonrepresentationforfunctionallanguagesintheVisualProgramming
literature?andisused invariousformsin???????and????
Thebasicunitofprogram constructioninourrepresentationistheexpression?
graph?afunctionthat hasbeenappliedtozeroor morearguments? Anexpression?
graphhasanimplicitvalue?whichmaybeagrounddatavalueormaybefunction
valued?andanexplicittype? Twoadv an tagesofthis represen tationare?
??Since mostfunctionalprogram objects? includingconstan ts?structured data
andspecial forms?eg conditionalsand caseexpressions?canberepresentedas
functions?the syntaxof the visualexpressionsisvery simple?
??Function de?nition ?whichis the basicprogram constructionoperationoffunc?
tionallanguages?canbeaccomplished simplybyassociatinganamewithan
expression?graph?whichthenbecomesthe de?ningexpressionofa newfunc?
tion?
?
Miranda
TM
isatrademarkofResearch Softw areLtd?

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J?K?KelsoAVisualRepresentationforFunctionalLanguages?
???Type?directedediting
Weareinterestedinrepresentingstronglytypedfunctionallanguages?Thet ype?
checkingmec hanismof stronglyt yped languagesassignsat ype toeach ob jectat
compile time?sothatnotypecheckingneedbedone atexecutiontime?
Conventionalimplementationsoftheselanguagesareusuallyimplicitlytyped?
theyuseatypeinferencesystemto generatetypesforeverysubexpressionina
program?Expressionsneednotbegiv enexplicittyp esby theprogrammerexcept
in casesofam biguity? Anytypeannotationsthatare madebytheprogrammerare
checkedagainsttheseinferredtypes?discrepanciesindicatethatanexpressionis
notbeinginterpretedinthewaytheprogrammerintended?Strongtypingenables
sucherrors tobelocatedatcompiletime?
Therepresentationdescribedhereissuitableforastronglytypedlanguage?but
isnotimplicitlytypedinthesensedescribedabove?Stronglyt yp ed languagestend
tobeimplicitlyt yp ed?relieving the programmerfrom ha vingto pro videat ype
declarationfor ev ery subexpressioninthe program?T ype informationis? how ever?
anintegralpartoftheproposedrepresen tation?and isusedtodirecttheediting
process?Thetedioustaskofk eeping track oft ypes foreverysubexpressionishandled
automatically?theonlytimeat ype needbeexplicitlydesignatedis whena function
isusedb efore itis de?ned ?eg? whende?ninga recursivefunction?? Thisisnot seen
asaseriouslimitation?sincemakingtype annotationsforsuchfunctionsisregarded
asgood programmingpractice in any case?
VisaVis?intro ducedin???? hasbeenextendedwithanimplicitincrementalt ype
systemthatallowsaformofparametricpolymorphism andfunctionoverloading
????Show?and?Tell?a visualteachinglanguage? hasalsob eenextended toinclude
anexplicitpolymorphict ype system????
Theprop osed representationis based on Haskell?st ypesystem?with itspara?
metricpolymorphism andt ypeclassoverloadingsc heme?
? Anatomy of anexpression?graph
Inthis sectionwe describe thevisualsyn taxofour representation?starting with the
basicarc?node structure?Figure? showsa simpleexpression?graph?
Theno des of an expression?graph canbeclassi?edinto threegroups?
??Ev ery expression?graphhasaro ottype?node? which showsthet ype ofthe
entireexpression and represen tsthev alueofexpression?
?? Ev eryexpression?graphhasoneormorefunction?nodes? whichformthe bo dy
oftheexpression? Expression?graphscan also con tainpatterngr aphswhich

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J?K?KelsoAVisualRepresen tation forFunctionalLanguages?
allowpattern?matc hing ofstructureddata?
??Expression?graphsmay havezeroormorep arametertyp e?no des?which repre?
sentun?lledparameters?theyareplace?holders whereotherexpressions may
beattac hed?
???F unctions
F unctions are representedbyfunction?nodes?boxeslab eled withthefunction?sname
andpossibly containing otherpertinentinformation? Thenameis assumedtobean
ASCIIstring?
?
A singleconnector leadsfromtheb ottom ofthefunction? Thisconnector repre?
sen tsthevalueofthe expressionro otedin thefunction?An umb erofconnectorslead
upwardsfromthetopedgeofthefunction?oneforeachfunctionparameter?Fig?
ure? showssomeexamplefunctionnodes?Constructorfunctions?andliteraldata
valuessuch asnumbersorcharacters?are distinguishedby ha vingdiagonal?sided
boxes?and specialformsbyusing adi?erentfont fortheir names?
Notethat function?nodes? like quarksandU?S? presidents? never actuallyappear
by themselves?The smallestusefulexpression?graphisa singlefunctionnodewithits
attendanttypenodes?Anexpression graphof thist ype isreferred toasa signature?
anditshowstheintrinsictypeofits function?Figure?showthesignature ofthe
additionfunction?
???T ypes
Eachoftheconnectorsmentionedabove?ifnotconnectedtoanotherfunction?node?
willbeterminatedbyatype?node?Inallcases?atype?nodedenotesthetypeofthe
expressionthatapp ears?or couldapp ear? above it?
Ro ott yp e?nodes?terminatingthelowerconnectorofafunction?no de?showthe
t ypeoftheentire expression?graph?Thesenodesareused asahandle for the entire
expression?graphduringan attac hment op eration?seelater??Parametert yp e?nodes?
terminatingtheupp erconnectors? showthetype?and?optionally?thename? ofa
function?s argumen ts?These nodes are targetsforother expression?graphs during
attachmen t?
?
Itw ouldbepossibletouseagraphic to identifythe functionratherthanatextual name?but
thisraises thequestionof wherethis graphicwouldcome from?Either the programmerwould
haveto produce oneforeach function?abadidea sinceit isrecognizedthat icon?design isbest
lefttographic artists?orautomaticallygenerated? In the lattercase the graphicwould have to
be at leastacomplexasthe functionitself? whichratherdefeatsthe purposeofusing namesfor
abstraction?

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J?K?KelsoAVisualRepresentationforF unctionalLanguages?
2
4
a
aa
Num a => a->a->a->a
parameter
nodes
function
nodes
root node
-
^*
*
Figure ??Asimpleexpression?graph

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J?K?KelsoA VisualRepresentationfor FunctionalLanguages ?
+
if-then-else
42
ordinary
function
special formnumeric
literal
constructor functions
for the list data type
[]:
Figure??Functionnodes
Num a => a->a->a
aa
+
The minimal expression-
graph for the addition
function
Figure??A signature expression?graph

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J?K?KelsoA VisualRepresen tationforFunctionalLanguages?
TheactualtypeisrepresentedsimplybytheASCIIstring ofaHaskelltype
declaration?thet ypenotation used? includingt ypev ariablesandthecurriedrepre?
sentationoffunctionswithm ultiplearguments? isdescribed intheHaskellreport
?????Itispossibletorepresenttypespictorially?but thissu?ersfromthesameprob?
lemasforfunctionnames?asmentionedinthefootnotetotheprevioussection?
Figure?sho wssomesampletypenodes?
af::a->b
Num a=>a->a->a
Int
parameter of
integer type
parameter of
variable type
parameter of
function type,
with name ‘f’
expression-graph root node,
with type class context
Figure??Sampletype?nodes
?????Type contexts
P arametricpolymorphism allo ws functions witht yp esthatare parameterizedby
typev ariables?The scope oft ypevariables inapolymorphicfunction isa single
expression graph?
F orexamplethetypevariableaapp earing intwodi?erenttypenodesinan
expressiondenotesthesamet ype?Thesametypevariable app earing inadi?erent
expression is notrelatedtothe?rst?onlytot ypev ariablesinitso wn expression?
Whenoneexpressionis attached toanother?thisnamingconvention is taken into
account andv ariablesinthecompositeexpressionareconsistentlyrenamed?
Typeclasspolymorphism?asfeaturedinHaskell?allo wstypevariablestobe
quali?edbyaclasscontext?Weallowthescopeofatypeclasscontexttobea
singleexpression?andthe contextbindingsaresho wnintheroott ype?no defor the
expression?Figure?sho wsanexample ofaroot node withatypeclasscon text?it
bindsthevariable atothe Numnumericclass?

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J?K?KelsoAVisualRepresen tationforFunctionalLanguages ??
???Pattern?matching
Thevisual pattern?matchingrepresentationdescribedhereisageneralizationofthe
pattern?matchingsc hemesin textualfunctionallanguages?Itp erformsthefunctions
of testingand decomposing structureddata?
P atternmatching isrepresentedbya visualpattern?graph? Whereanormal
expression?graphcomposes incomingdatausingfunctions?apattern?graphdecom?
posesincomingdata?Theconstituentsofpattern?graphsarepattern?nodes??upside?
down?functionnodesthathaveasingleupp erconnectorandmultiplelow erconnec?
tors?Figure?sho wssome samplepattern?no des?includinganumeric literal pattern?
a ?wildcard?pattern andthepatterns forthelistdatat ype?
[]
’a’
:
42
patten-nodes for
the list data type
wildcard
Pattern-nodes for an
numeric literal, a
character literal, and
a wildcard pattern
Figure??Sample pattern nodes
At thetop ofapattern?graph is thepattern?sroot?no de?whichshowsthet ype
ofthe incoming data? Thebody ofapattern isbuiltfrompattern?nodesinm uch
thesame way as anexpression?graph is buildfrom functionnodes? Theleav es ofa
pattern?graph?app earingattheb ottom? are eitherconstant datav aluesorpattern
t ype?nodes?whichshowthet ype of the decomposeddata? Thesecorresp ondto the
variables app earinginatextual patternexpression?andcanbeoptionallynamed?
Figure?showsaexamplepattern?graph?
Theexactsemanticsof theevaluationofanexpression containingapattern?
graphisdep endentontheunderlying language?butgenerallywhena pattern is
encountered?the incoming expressionis evaluated?rst?and then itsstructure is
matc hed against that of thepatternexpression?

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J?K?KelsoAVisualRepresentationforFunctionalLanguages??
:
1
[Int]
:
2
[Int]
that will match any
list beginning [ 1, 2, ... ]
An pattern-graph
Figure??A samplepattern?graph
?????P atternframes
Patternmatchingisalsousedinfunctionallanguages to specify conditionaldef?
initions?Toallowthis isour visualrepresentationwein troduceaspecial form
of expression?graph?thepatternframe?whichappears asanoutlinearoundan
expression?graphthatcontainsa pattern?graph?Thepatternframeisageneral?
izationofthecaseexpressioninHask ell?
Apattern frameis similar toan ordinary expression?graph?inthatithasa root
no de andpossibly sev eralparametert yp e?no des?Theexpression?graph that forms
thebo dyofthepattern framemustcontainatleastonepattern?graph? Apattern
frame de?nesthevalueofthe expressioninthecasethat the incoming datamatches
the pattern?Onceapattern frame hasb eenintro duced?additionalpattern frames
canbe addedtodeal withalternative patterns of incomingdata? Thesepattern
framesapp ear stack ed ina pile ?withone showingatatime?? andmust havethe
sametype as the?rst?
Themechanismbywhichapattern frame isc hosen from thealternatives tobe
ev aluateddep endsonthe patternmatc hing seman ticsof theunderlyingfunctional
language?Dep endingontherequiremen tsofthe mechanism?someadditional syntax
mayberequired?F orinstance? in ordertocorresp ond toHaskell? thealternativ es

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J?K?KelsoAVisualRepresen tation forFunctionalLanguages??
couldbenumberedby theprogrammer?andtestedinorderuntilamatchis found?
Alternative patternmatching schemes arep ossible?for instance?acheckforoverlap?
pingpatternsor pattern completenesscouldbein troduced? or someformofbest??t
patternmatchingcouldbeemployed?Figure?shows twoviewsofapileofpattern
frame alternatives?each witha di?erent pattern upp ermost?Thebo dy androot of
the pattern frame isnotsho wn?
[a]
:
[a]
[]
pattern 1 pattern 2
A pattern frame with two alternative patterns
Figure??A samplepattern frame
??? Guardedexpressions
Analternative methodof sp ecifyingaconditional expression? featuredinHask ell
andMiranda?isa setof guarded expressions?
Guardedexpressions are intwo parts? theguard?which isitselfabooleanex?
pression? andavalue expression?The usualsemantics inv olve ev aluatingthe guard
expressions inorder un til one isfoundtobe true? whereuponthe corresp onding
v alueexpressionb ecomesthevalue oftheexpression?
Thevisual represen tation ofaguarded expression isthe guard frame?Aguard
frameissimilar toapattern frame?withthero otnodeprotruding fromtheb ottom
andparameterno desfrom thetop?Inside the framearetwoexpression?graphs? on
theleftisthe guardexpression?which hasthe sameparameters as theguard frame

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J?K?KelsoA VisualRepresentationforFunctionalLanguages ??
butevaluatestoabo oleanv alue?andontherightisthevalue expression? which
shares thesame parameters buthasthetypeofthe guardframe? Aswithpattern
frames?alternativeguard?value pairsarestackedinapile?withonevisible atatime?
Figure?showsaguardframeexpression?
Int->Int
The top frame of a guarded expression.
Int
*
-1
guard 1
<
0
Int->Bool
Figure??Asampleguardedexpression
???Sub expressionsharing
T extuallanguagesgenerally havea method ofin troducing lo calde?nitionsfor nam?
ing subexpressions that app earmore than onceinanexpression? Apartfrom their

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J?K?KelsoA VisualRepresentation forF unctionalLanguages??
valueinabbreviatingandclarifyingexpressions? this allo wslanguagesthatemploy
lazy evaluation to ensure thatthecomputation of sharedsub expressionoccurs only
once? In somecasesitisv aluabletobeablegivethesesub expressionsmeaningful
names? inothercasesprogrammersuse ?place?holder?names?
Sincein ourrepresen tationa sub expressionisalsoasub?graph? thesharing ofa
subexpressioncanbe representedbythem ultiple useofa graphconnector? Thisis
shownbythe useofa sharing no de?whichhasasingleupp erconnector?which is
attached to thesub?graphb eingshared?andmultiple low er connectors?allofwhich
sharethevalueof the sub expression?
Figure?sho wsanexample of anexpression?graph withasharedsubexpression?
Thismec hanism issimilar to the whereandletclausesintextuallanguages? but
do es not allow recursive lo calde?nitions?Anextensiontothevisualsyn tax allowing
arbitaryrecursive local de?nitions isb eingconsidered?
Float->Float
+
sqrt
Float
sharing node
Expression-graph
equivalent to:
\x -> let y = sqrt x in y + y
Figure ?? Exampleofa sharedsubexpression
??? Applyinghigher?orderfunctions
Thesystem describedsofar allowsan expression?graphtorepresentahigher?order
function? andallo wshigher?orderfunction tobepassed asparameters tofunctions?

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J?K?KelsoAVisualRepresentationforFunctionalLanguages ??
Ho
as
This
w
the
ever?
compose
?gure
there is
function?
in
noway toactuallyapplyahigher?order function parameter toan
argument? In particular?thereis nowaytospecifythe?apply?function?
applyfx?fx
Ifwesupplyaspecialapplyprimitive?writingsuchfunctionsbecomespossible?
?
The
Figure
apply
??
primitiv
sho
e
construction
makesp ossible
of
the represen
compose
tation
function
higher?order
from
functions
apply
such
compose ???b??c????a??b???a??c
composefgx?f?gx?
w thethetwo primitives?
alsotroducesexpression?graph constructionop erations?which are the
sub jectofthenext section?
? Expression?graphconstruction
Having giventhevisualsyntaxofexpression?graphs? we nowdescribe the operations
bywhichtheyare constructed?andhow they areusedtode?ne newfunctions?We
assumethatindividual no des?functions? patterns? applynodesetc? areav ailablein
a con tainerof somesortfromwhich theycanbe dragged on toaw orksheetarea?
???Expression attac hment
Thebasicexpressionbuilding operationisattachment? wherebytwoexpression?
graphs
that
are
programmer
joined? This
can
tak esplace
ecify
when
the
the
needed
programmer
when
drags
apply
a ro
is
ot no deof
oneexpression?graph on toaparameter nodeof another? Ifthet ypesofthetwo
expressionsarenot compatible?then noactiontak es placeand the programmer is
warnedthatthe operationis illegal?In thisway? onlyexpressiongraphs thatare
syntactically andtypecorrect canbe constructed?
Figure ??shows attachment takingplaceb etween the additionfunctionanda
n umeric literal?Theattac hment operationhasthreemain e?ects?
?
Actually? di?erentapply primitiv esareneeded for functionsof di?erent arities?Weassume
theselector sparitythenodecreated