2KazutakaMatsudaandMengWangprograminversiontechniques,obtainaconsisten - PDF document

2KazutakaMatsudaandMengWangprograminversiontechniques,obtainaconsistentparser.We,asusual,manu-allyimplementedaparserandapretty-printerforthelanguage,butwiththehopethatwe,andmanyotherswhoreadthispaper,willnotneedtodoitagainfortheirownlanguageimplementations.Priortothiswork,therehasbeenarichbodyofliteratureonexploringcorrectness-by-constructiontechniquestoautomaticallygenerateoneorbothprogramsoftheprinter/parserpair,notably[2,4,17].Wehaveintentionallyomittedtheprex\pretty-"fromthementioningofprintersherebecausefewoftheexistingworkisactuallyproducingpretty-printersinthesenseofHughes[10]andWadler[22].1Tobemorepreciseaboutwhatwemeanby\prettiness",letusconsiderasubtractionlanguagee::=1je1�e2thathasaconstant(1)andaleft-associativebinaryoperator(�).WerepresentthesyntaxwiththefollowingASTdatatype.dataE=OnejSubEEUsingthelanguageweproposeinthispaper,whichisbasedonWadler'sli-brary[10],onecandeneapretty-printerasbelow.pprOne=text"1"ppr(Sube1e2)=group(ppre1 ㌀ nest2(line ㌀ text"-" ㌀ text"" ㌀ pprPe2))--ThesuxPinpprPstandsforparentheses.pprPOne=text"1"pprP(Sube1e2)=text"(" ㌀ group(ppre1 ㌀ nest2(line ㌀ text"-" ㌀ text"" ㌀ pprPe2)) ㌀ text")"Thepretty-printinglibraryfunctionsareshowninslantsansserif.Roughlyspeak-ing,textsconvertsastringstoalayout,d1 　 d2isaninxbinaryoperatorthatconcatenatestwolayoutsd1andd2,whichbindslooserthanprexapplications,andlinestartsanewline,butitsbehaviorcanbeaectedbysurroundingnestandgroupapplications:nestndinsertsn-spacesaftereachlinesind,andgroupdsmartlychoosesbetweenthelayoutdandotherlayoutsderivablefromdbyse-lectivelyinterpretinglinesassinglespaces.(Inthispaper,wewrite\space"forthespacecharacterandwrite\whitespace"forthespacecharacterandthenew-linecharacter.Otherkindsofspacessuchashorizontaltabsarenotdiscussedastheydonotyieldnewinsight.)Thefunctionpprpretty-printsSub(SubOneOne)(SubOneOne)as1-1-(1-1)or1-1-(1-1)or1-1-(1-1)dependingonthescreenwidththatisusedtorendertheresult.Thisne-grainedcontrolfromusersoverbracketing,spacingandindentationisclearlybeyondanytechniquebasedonmechanicaltraversalsofASTs,whichislikelytorigidly 1TheSynsystem[2]iscapableofhandlingnon-contextuallayouts,whichcanbeseenasalimitedformofprettiness. 4KazutakaMatsudaandMengWanglanguageprovides,andthetranslationfromthesurfacelanguagetothecorelanguageistheidentityoperationinthiscase.Therefore,wefocusonthecoresysteminthissectionandpostponethediscussionofthesurfacelanguagetoSect.4.Asastart,let'srevisitthepretty-printerpprdenedintheprevioussec-tion.Ifthefunctionisinvertedasitis,wecanhopefornomorethanaparserthatonlyrecognizesprettystrings.Thisisneitherthefaultoffunctionpprnoroftheinverter:apretty-printerppr(correctly)producesonlyprettylayouts,andaninvertercannotinventinformationthatisnotalreadycarriedbythefunctiontobeinverted.Toremedythisinformationmismatch,weinstrumentthepretty-printerwithadditionalinformationaboutnon-prettybutneverthelessvalidlayouts.2.1IntroducingUglinessReinterpretationofline.Acommonsourceofprettinessisthecleverinterpre-tationoflineseitherasasinglespaceoranicelyindentednewlinedependingontheenvironment.Thiseectcanbesimplyeliminatedbyreinterpretinglineasoneormorewhitespaces.Usingthisnewinterpretationinthederivationofaparserenablesustoparsecertainnon-prettylayouts.Forexample,nowtheinverseofthepretty-printercanparsethefollowingstrings.1-1or1-1Thesestringsdonotsatisfyournotionofprettinessdenedbyppr,andwillnotbeproducedbythepretty-printer,butwillbeacceptedbythegeneratedparserthroughthereinterpretationoflines.Alsonotethatthisreinterpretationalsomeansthatwecansafelyignoregroupandnestduringinversion,becausetheirsolepurposeistoaectthebehavioroflines.Still,thissolutionaloneisnotenough.Stringslike1-1and(1)-((1))remainunparsable:thepretty-printerhasdictatedthatthereisonlyasinglespacebetweentheoperatorandthesecondoperandbyusingtext""insteadofline,andthatthereshouldn'tberedundantparentheses.Weneedtondawaytoalterthesebehaviorsinparsingwithoutlosingpretty-printing.BiasedChoice.Toannotatepretty-printerswithinformationaboutnon-prettylayouts,weintroducethechoiceoperator + .Inpretty-printingtheoperatorbehavesase1 + e2=e1,ignoringthenon-prettyalternativee2;inparserderiva-tiontheoperatorisinterpretedasanondeterministicchoice,whichacceptsbothbranches.Theoperator + bindslooserthan 　 andhasthefollowingalgebraicproperties.Associativitye1 + (e2 + e3)=(e1 + e2) + e3Distributivity-L(e1 + e2) ㌀ e3=e1 ㌀ e3 + e2 ㌀ e3Distributivity-Re1 ㌀ (e2 + e2)=e1 ㌀ e2 + e1 ㌀ e3Forexample,onecandenevariantsof(white)spaceswiththechoiceoperatorasfollows.nil=text"" + space--(zero-or-morewhitespacesinparsing)space=(text"" + text"\n") ㌀ nil--(one-or-morewhitespacesinparsing) 6KazutakaMatsudaandMengWang prog::=rule1;:::;rulenrule::=fp1:::pn=ep::=xjCp1:::pne::=text"string"je1 ㌀ e2jlinejnestnejgroupe(Wadler'sCombinators)je1 + e2(BiasedChoice)jfx1:::xn(TreelessCall) Fig.2.Syntaxofthecorelanguage:frangesoverfunction,Crangesoverconstructors,xandxisrangeovervariablesandnrangeovernaturalnumbers.presentation).Ppr!Ppr f$1gj"("NilPprNil")"f$3gPpr !1fOnegjPprLine0"-"Space0PprPfSub$1$5g:::Thecorrectnessoftheparserconstructioncomesfromourpreviouswork[15].SinceFliPprproducesaCFGwithactions,usershavethechoiceofusinganyparsergeneratorthatsupportsfullCFG.Inourimplementation,weuseFrostetal.[8]'stop-downparser.3CoreLanguageandParserConstructionInthissection,wegivetheformaldenitionofthecorelanguageofFliPpr,anddiscussparserconstructionbyprograminversion.3.1SyntaxandSemanticsFigure2showsthesyntaxofourcorelanguage,arst-orderfunctionallanguagesimilartoonefoundintheintroduction.WeincludeWadler'spretty-printingcombinators[22]andthebiasedchoiceasprimitiveoperators,andplacetworestrictionsforlaterinversion:{Functioncallsmustbetreeless[21]:theytakeonlyvariablesasarguments.{Variableusemustbelinear:everyboundvariableinaruleisusedexactlyonceontheright-handside.Anotableexceptioniswith + .Fore1 + e2,thetwobranchesaresupposedtobebothlinear.Thus,theycontainthesamesetoffreevariables.Forexample,assumingfislinear,thengx=fx + fxislinear,buthx=line + fxandkx=line + text"s"arenot.Forsimplicity,weoftenomittheruleseparator\;"ifnoconfusionwouldarise.Weusevectornotationexforasequencex1;:::;xn.Weabusethenotationtowritefexforfx1:::xn.Theformalpretty-printingsemanticsofthelanguageisshowninFig.3.Wewrite�`e+vifunderenvironment�,expressioneevaluatestovaluev.Values 8KazutakaMatsudaandMengWang3.2ParserConstructionbyInversionToinvertprogramswritteninthecorelanguage,werstlyperformasemantic-preservingtransformationtoremovethepretty-printingcombinators,andobtainasyntaxthatisrecognizablebyourgrammar-basedinversionsystem[15].ConvertingtoNondeterministicProgramsThisstepisdoneby\forgettingsmartlayoutingmechanism",throughthefollowingrewritingrules.text"s"�!"s"nestne�!egroupe�!eline�!spacee1 　 e2�!e1++e2e1 + e2�!e1?e2Here,spaceisarewrittenversion(accordingtotherulesabove)ofitsdenitioninSect.2,i.e.thefunctiondenedbyspace=(""?"\n")++nilnil=""?spaceandtheoperator?isanondeterministicchoice.Theformalsemanticsoftheobtainednondeterministicprogramsisdenedstraightforwardlybyaddingthefollowingrules. �`"s"+"s"�`ei+v �`e1?e2+vi=1;2f�`ei+vigi=1;2 �`e1++e2+v1++v2Theirbehaviorsof"s",?and++arethesameasthereinterpretationsoftext"s", + and 　 ,respectively;weusedierentsymbolstoclarifythattheconversiondiscardsthepretty-printingsemantics.Notethat,sincethelanguageislinearandtreeless,thecall-timechoiceandtherun-timechoice[19]donotdier.Wewritef ande astherewrittenversionoffande.Thefollowinglemmastatesthattherewritingissemanticpreserving.Lemma1(SemanticPreservation).�`e+NDsi�`e +s.utGrammar-BasedInversionTherewrittenprogramscanbeprocessedtoob-tainagrammarwithactions4thatcomputestheinverseoftherewrittenprogrambyusinggrammar-basedinversion[15].Thebasicideaoftheinversionistoreadaruleofaprogramasaproductionruleofagrammar,andtousesemanticactionstotrackhowvariables(i.e.,inputs)arepassed.Intheinversion,weconstructtwosortsofnon-terminals:Ffforfunctionsf andEeforexpressionse .Forafunctionf thattakest1;:::;tnandreturnss,Ffisusedtoparsestrings,andthesemanticactionreturnsoriginalinputs(t1;:::;tn).Foranexpressione suchthat�`e +s,Eeisusedtoparsestrings,andthesemanticactionreturnstheoriginalenvironment�.ThegenerationoftheproductionrulesandsemanticsactionsarepresentedinFig.5.ThegrammarinSect.2isasimpliedversionofthegrammarobtainedbythisgeneration. 4Intheoriginalpaper[15],transformationsonparsetrees(ormoreprecisely,deriva-tiontreesofproductions)areused,insteadofsemanticactions. 10KazutakaMatsudaandMengWang4.1ProblemswithProgrammingintheCoreLanguageLetusconsiderextendingthesubtractionlanguagewithdivisionandvariables.dataE=


jDivEEjVarStringRecallthatweusedtwomutuallyrecursivefunctionspprandpprPtocontrolbracketingissuesaround\-".Ingeneral,whentherearemanyoperatorswithdierentprecedencelevels,itsucestouseafunctionforeachprecedencelevel.Forexample,assuming\-"hasprecedence-level6and\/"hasprecedence-level7astheydoinHaskell,apretty-printercanbewrittenasfollows.pprx=ppr5x--5isthelowestprecedencelevel:::ppr 5(Subxy)=:::ppr6x:::text"-":::ppr7y:::--(1)ppr 5(Divxy)=:::ppr5x:::text"/":::ppr6y:::--(2):::ppr 6(Subxy)=text"(" ㌀ nil ㌀ :::{-theRHSof(1)-}::: ㌀ nil ㌀ text")"ppr 6(Divxy)=:::{-theRHSof(2)-}::::::ppr 7(Subxy)=text"(" ㌀ nil ㌀ :::{-theRHSof(1)-}::: ㌀ nil ㌀ text")"ppr 7(Divxy)=text"(" ㌀ nil ㌀ :::{-theRHSof(2)-}::: ㌀ nil ㌀ text")"Therearealotofundesirablerepetitionsintheabovedenitionlargelyduetothetreelessrestriction.Anotherproblemthatitisnon-trivialtoseparatevariablenameswithpre-denednames.Forexample,letusconsiderpretty-printingforVarx.Onemaybetemptedtowriteppr (Varx)=textxbutaparserderivedfromtheabovewillparse\-"asVar"-",becausethereisnoinformationintheabovedenitionthatspeciesvalidvariablenames.Wecanimprovethepretty-printerasfollows.ppr (Varx)=fxf('a':x)=text"a" ㌀ gx:::f('z':x)=text"z" ㌀ gxg[]=text""g('a':x)=text"a" ㌀ gx:::g('z':x)=text"z" ㌀ gxNotethatstringsarerepresentedaslistsofcharactersasinHaskell.Thisfunctionppr ispartialandintentionallyundenedforVar"-".Inthisdenition,wehavesuccessfullyrestrictedvariablenamestorangeoverlower-caseEnglishalphabets,butinaverycumbersomeway.4.2AnOverviewToreducetheprogrammingeort,weproposeasurfacelanguage,whichhasrelaxedlinearityandtreelessnessrestrictions,andisequippedwithashorthandnotationforexpressingnameranges.Inthislanguage,apretty-printerforthe 12KazutakaMatsudaandMengWang ;�;
`e: ;�;fx:g`x: ;�;;`x:�(x) ;�;;`c:St;�;
`e:Doc ;�;
`nestne:Docf;�;
i`ei:Docg1inop=text"s";group;( ㌀ );line ;�;U1in
i`ope1:::en:Docf;�;
`ei:Docgi=1;2 ;�;
`e1 + e2:Doc ;�;fx:ASTg`text(xasr):Docf;�;;`ei:Stg1inf;�;
`eb:gb=t;f ;�;
`ifprede1:::enthenetelseef:f;�;
i`ei:ig1in(f)=1!


!n!Doc ;�;U1in
i`fe1:::en:Doc `fp1:::pn=e (f)=1!


!n!Doc9�;
1;:::;
nf�;
i`pi:ig1indom(�)U1infv(pi);�;U1in
i`e:Doc `fp1:::pn=e �;
`p: �(x)=St �;;`x:St2fAST;Docg �;fx:g`x:f�;
i`pi:g1in2fAST;Stg �;U1in
i`Cp1:::pn: Fig.7.Typingrules:here]representsdisjointunion.operationsfoundintheprevioussubsection.Usedasarguments,suchconstantscanbeeliminatedatcompilationtimethroughpartialevaluation;wecallsuchconstantsstaticinformation.Theifbranchingsinspectstaticinformation,andareeliminablestaticallyaswell.Weuseatypesystemtodistinguishstaticinformation(oftypeSt)fromotherkindsofvaluessuchastheinputASTs(oftypeAST)andthepretty-printingresults(oftypeDoc).Thetypesystemensuresthatstaticinformationareeliminablethroughpartial-evaluation,andvariableusesarelinear.Formally,primitivetypesandfunctiontypesaredenedby:::=ASTjStjDoc::=1!


!n!Typingjudgment;�;
`e:readsthatunderfunction-typeenvironment,non-lineartypeenvironment�andlineartypeenvironment
,ehastype.Similarly,wedene�;
`p:and`fp1:::pn=eforpatternsanddeclarations.Figure7showsthetypingrules,whicharemostlyself-explanatory.Notably,theusesofvariablesoftypeASTandDochavetobelinear,asdictatedbytherules.ThelinearityrestrictionofASTvariablesisinheritedfromthecorelanguage,whilethatofDocvariablesisrequiredforthecorrectnessoffusion;itisknownthatthedeforestationisnotcorrectfornon-linearandnon-deterministicprograms[1].Aprogramisassumedtohaveadistinguishedentrypointfunction 14KazutakaMatsudaandMengWangasbelow.pprx=go5x:::go 5(Subxy)=:::go5x:::go6y:::go 5(Divxy)=:::go6x:::go7y::::::go 6(Subxy)=:::go5x:::go6y:::go 6(Divxy)=par(:::go6x:::go7y:::):::go 7(Subxy)=par(:::go5x:::go6y:::)go 7(Divxy)=par(:::go6x:::go7y:::)Roughlyspeaking,thankstothetypeAST!Docoftheentrypointfunction,thetypesystemguaranteesthateverySt-typeexpressionmustbeaconstantitselforapartofsomeconstantobtainedbypattern-matching,andthuscaneliminatedbypartial-evaluation.FusingFunctionstoObtain1-tierProgramsWeshowthetransforma-tionof2-tieredprogramsto1-tieredprograms,withtheunderstandingthattheprocedurecanbeappliediterativelytotransformm-tieredprogramsto1-tieredprograms.Thetransformationisdonebydeforestation[21].Roughlyspeaking,defor-estation(or,supercompilation[20]5)performscall-by-nameevaluationofexpres-sions;butinsteadofcomputingavalue,itproducesanewexpressionthathasthesamebehaviorastheoriginalonebutwithintermediatedatastructureselim-inated.Withoutlossofgenerality,weassumethatASTargumentsappearbeforeDocargumentsinfunctioncalls.ThedeforestationprocedureDJeKisdenedasfollows.{DJope1:::enK=opDJe1K:::DJenK,whereoprangesovertext"s",( 　 ),line,nesti,groupand( + ).{DJfexeeK=feeexez.AssumingexhavetypeAST(recallthatonlyvariableshavetypeAST),eehavetypeDoc,andfezgarethefreevariablesinee,thenewlygeneratedfunctionfeeisdenedasfeeepez=DJe[ey7!ee]Kforeachcorrespondingrulefepey=einthedenitionoff(withproper-renaming).Here,wedonotrepeatedlygeneraterulesoffeeiftheyarealreadygenerated(uptorenamingofthefreevariablesinee).Theaboveprocedurefollowsfromtheoriginalone[21],andissimpliedtosuittherestrictedsurfacelanguage.Theprocedureterminatesifthenumberoffunctionsfeegeneratedinthelattercaseisnite.ByusingDJeK,wereplaceeverytier-2rulefepey=ewithfepey=DJeK.Example1.Wedeforestthepretty-printerdenedinSect.4.2. 5Becauseofthelinearity,Wadler'sdeforestation[21]and(positive)supercompila-tion[20]coincideforthesurfacelanguage. 16KazutakaMatsudaandMengWangWeleaveoutnestandtext"s"forsimplicity.Inthedatatype,weuseEVarbothforvariablesandfunctioncallstoavoidambiguityingrammars.Theoverallprincipleofourpretty-printingistoinsertbreaksafter=,andbefore�and,with2-spaceindentation.Westartwithlistsofrules,andinsertseparatorswithoptionalwhitespacesnil 　 text";" 　 line0betweenindividualrules.pprx=pprRulesxpprRules[]=nilpprRules(r:rs)=nil ㌀ pRulesrrs ㌀ nilpRulesr0[]=pprRuler0pRulesr0(r:rs)=pprRuler0 ㌀ nil ㌀ text";" ㌀ line0 ㌀ pRulesrrsForeachrule,itsright-handsidemaystartanewline.pprRule(Rulefpse)=group(varf ㌀ space ㌀ pprPatsps ㌀ space0 ㌀ text"=" ㌀ nest4(line0 ㌀ pprExpe))varx=text(xas[a-z][ a-zA-Z0-9]*'*)Alistofpatternsistreatedinasimilarwaytoalistofrules.pprPats[]=text""pprPats(p:ps)=pPatsppspPatsp0[]=pprPatp0pPatsp0(p:ps)=pprPatp0 ㌀ space ㌀ pPatsppsRedundantparenthesesinpatternsareadmissibletothegeneratedparser,butwillnotbeproducedbythepretty-printer.pprPatp=manyPars(pprPat p)pprPat (PVarx)=varxpprPat (PConc[])=concpprPat (PConc(p:ps))=par(conc ㌀ space ㌀ pPatspps)conf=text(xas[A-Z][ a-zA-Z0-9]*'*)Expressionsareprintedaccordingtotheprecedence-levelsandassociativitiesoftheoperators.pprExpe=go4egoie=manyPars(go ie)go i(EConc[])=concgo i(EConc(e:es))=parIf(i9)(conc ㌀ space ㌀ pExpsees)go i(EOpOAlte1e2)=parIf(i5)(group(go5e1 ㌀ nest2(line0 ㌀ text" ㌀ space0 ㌀ go4e2)))go i(EOpOCate1e2)=parIf(i6)(group(go6e1 ㌀ nest2(line0 ㌀ text�" ㌀ space0 ㌀ go5e2)))go i(EVarf[])=varfgo i(EVarf(e:es))=parIf(i9)(varf ㌀ space ㌀ pExpsees)Finally,alistofexpressionsprintedinasimilarwaytoalistofpatterns.pExpse0[]=go9e0pExpse0(e:es)=go9e0 ㌀ space ㌀ pExpsees 18KazutakaMatsudaandMengWangintheoperands,anddenefx7!vg]fx7!vg=fx7!vg.ThisnaivesolutionworkseectivelyforXML,becausethenumberofpossibleASTsisusuallynite.However,ingeneralparsingbecomesundecidablewithnon-linearuseofASTvariables,asshownin[13](Theorem4.4).Thus,forthiskindofnon-linearuses,amethodthatchecksthenitenessofparsetreesisrequired.Thenon-linearityofDocvalueshasnon-trivialinteractionwithnondetermin-ism.Intheabsenceoflinearity,thecall-by-valueandthecall-by-nameseman-ticsmayceasetocoincide.Thisisaproblembecausecall-by-valueissuitableforgrammar-basedinversion[15],butcall-by-nameissuitablefordeforestation[18].WealsoneedtoresorttogrammarsbeyondCFGs,whichmayposedicultiesininversion.Itisachallengingproblemtondasweetspotbetweenobtainingecientinversesandsupportingfusioninthesurfacelanguage.7RelatedWorkDierentapproacheshavebeenproposedtosimultaneouslyderiveaparserandaprinterfromsomeintermediatedescriptions.Inparticularly,onecouldstartfromanannotatedCFGspecicationtoderivebothaparserandapretty-printer[2].Comparedtothesesystems,FliPproersnercontroloverpretty-printing.Inparticular,weareabletodealwithcontextualinformationandtodeneauxiliaryfunctionslikeparinprinting,whichismadeconvenientlyavailablebythesurfacelanguage.Otherapproachesincludeinvertiblesyntaxdescriptions[17]basedoninvertibleprogramming,andBNFC-meta[4]basedonmetaprogramming.Bothworkrecognizestheimportanceofgoodprinting,butisnotabletosupportpretty-printing.Therearealsogeneral-purposebidirectionallanguages[3,6,11]thatinthe-orycanbeusedtobuildtheprinter/parserpairfromthedenitionofoneofthem.Notablyquotientlenses[7]aredesignedtoincludearepresentativeofaquotientbeforeperformingbidirectionalconversions;inourcase,roughlyspeak-ingthisquotientoperationistheerasureofredundantwhitespacesandparen-theses.However,thereisagapbetweenthetheoreticalpossibilityandpracti-callyexecution.Inparticular,thepretty-printinglibrariesofWadler[22]andHughes[10]arenotonlyuser-friendlybutalsohighlyoptimized.Moreover,forecientparsingwehavetoperformwhole-programanalysis(asinconventionalparsingalgorithmslikeLR-k)orusesophisticateddatastructuresandmemo-ization[8,16].Itisnotobvioushowthesesophisticatedimplementationscanbepackedintoabidirectionalprogram.Inourapproach,weavoidthisproblembyusinggrammar-basedinversion[15],whichgeneratesgrammarsandoutsourcestheparsingalgorithmstoselectedparsergenerators.Therearealotofdiscussionsonhowtomakedeforestation(supercompila-tion)terminate(e.g.,[12])forTuring-completelanguages.Theseapproachesuseconditionstogiveupfusion,andreusethealready-generateddeforestedfunc-tions.Asaresult,theseapproachesmayfailtofusesomefunctions,andthusarenotsuitableforourpurpose.Thecompletenessofdeforestation,inthesensewhetherallthenestedcallsarefusedaway,hasnotbeenthefocusofstudyin 20KazutakaMatsudaandMengWang4.Duregard,J.,Jansson,P.:EmbeddedParserGenerators.In:Haskell2011:Proceed-ingsofthe2011ACMSIGPLANHaskellSymposium,pp.107{117.ACM(2011)5.Duregard,J.,Jansson,P.,Wang,M.:Feat:FunctionalEnumerationofAlgebraicTypes.In:Haskell2012:Proceedingsofthe2012ACMSIGPLANHaskellSympo-sium,pp61{72.ACM(2012)6.Foster,J.N.,Greenwald,M.B.,Moore,J.T.,Pierce,B.C.,Schmitt,A.:CombinatorsforBidirectionalTreeTransformations:ALinguisticApproachtotheView-UpdateProblem.ACMTrans.Program.Lang.Syst.29(3)(2007)7.Foster,J.N.,Pilkiewicz,A.,Pierce,B.C.:QuotientLenses.In:ICFP2008:Pro-ceedingsofthe13thACMSIGPLANInternationalConferenceonFunctionalPro-gramming,pp.383{396.ACM(2008)8.Frost,R.A.,Haz,R.,Callaghan,P.:ParserCombinatorsforAmbiguousLeft-RecursiveGrammars.In:Hudak,P.,Warren,D.S.(eds.)PADL.LNCS,vol.4902,pp167{181,Springer(2008)9.Fulop,Z.,Vogler,H.:Syntax-DirectedSemantics:FormalModelsBasedonTreeTransducers.1stedn.Springer-VerlagNewYork,Inc.,Secaucus,NJ,USA(1998)10.Hughes,J.:TheDesignofaPretty-PrintingLibrary.In:Jeuring,J.,Meijer,E.(eds.)AdvancedFunctionalProgramming.LNCS,vol.925,pp.53{96,Springer(1995)11.Jansson,P.,Jeuring,J.:PolytypicDataConversionPrograms.Sci.Comput.Program.43(1),35{75(2002)12.Jonsson,P.A.,Nordlander,J.:PositiveSupercompilationforaHigherOrderCall-by-ValueLanguage.In:POPL2009:Proceedingsofthe36thACMSIGPLAN-SIGACTSymposiumonPrinciplesofProgrammingLanguages,pp.277{288,ACM(2009)13.Kobayashi,N.,Tabuchi,N.,Unno,H.:Higher-OrderMulti-ParameterTreeTrans-ducersandRecursionSchemesforProgramVerication.In:POPL2010:Proceed-ingsofthe37thACMSIGPLAN-SIGACTSymposiumonPrinciplesofProgram-mingLanguages,pp.495{508,ACM(2010)14.Kuhnemann,A.,Gluck,R.,Kakehi,K.:RelatingAccumulativeandNon-AccumulativeFunctionalPrograms.In:Middeldorp,A.(ed.)RTA.LNCS,vol.2051,pp.154{168,Springer(2001)15.Matsuda,K.,Mu,S.C.,Hu,Z.,Takeichi,M.:AGrammar-BasedApproachtoInvertiblePrograms.In:Gordon,A.D.(ed.)ESOP.LNCS,vol.6012,pp.448{467,Springer(2010)16.Might,M.,Darais,D.,Spiewak,D.:ParsingwithDerivatives:AFunctionalPearl.In:ICFP2011:Proceedingofthe16thACMSIGPLANInternationalConferenceonFunctionalProgramming,pp.189{195,ACM(2011)17.Rendel,T.,Ostermann,K.:InvertibleSyntaxDescriptions:UnifyingParsingandPrettyPrinting.In:Haskell2010:Proceedingsofthe2010ACMSIGPLANHaskellSymposium,pp.1{12,ACM(2010)18.Sands,D.:ProvingtheCorrectnessofRecursion-BasedAutomaticProgramTrans-formations.Theor.Comput.Sci.167(1&2),193{233(1996)19.Sndergaard,H.,Sestoft,P.:Non-DeterminisminFunctionalLanguages.Comput.J.35(5),514{523(1992)20.Srensen,M.H.,Gluck,R.,Jones,N.D.:APositiveSupercompiler.J.Funct.Program.6(6),811{838(1996)21.Wadler,P.:Deforestation:TransformingProgramstoEliminateTrees.Theor.Comput.Sci.73(2),231{248(1990)22.Wadler,P.:APrettierPrinter.InGibbons,J.,deMoor,O.,eds.:TheFunofProgramming.PalgraveMacmillan(2003)