Advanced Functional Programming

Inthisvolumeyouwill?ndthelecturenotescorrespondingtothepres- rd tationsgivenatthe3 summerschoolonAdvancedFunctionalProgramming, heldinBraga,PortugalfromSeptember12–19,1998. ThisschoolwasprecededbyearlieronesinB?astad(1995,Sweden,LNCS925) andOlympia,WA(1996,USA,LNCS1129). Thegoalofthisseriesofschoolsis tobringrecentdevelopmentsintheareaoffunctionalprogrammingtoalarge groupofstudents. Thenotesarepublishedinordertoenableindividuals,small studygroups,andlecturerstobecomeacquaintedwithrecentworkinthefast developingareaoffunctionalprogramming. Whatmadethisschoolparticularlyinterestingwasthefactthatalllectures introducedusefulsoftware,thatwasusedbythestudentsintheclassestoget hands-onexperiencewiththesubjectstaught. Weurgereadersofthisvolumeto downloadthelatestversionofthissoftwarefromtheInternetandtrytodothe exercisesfromthetextthemselves;theproofoftheprogramisinthetyping. The?rstlecture,onSortingMorphisms,servesasagentleintroductiontothe thingstocome. Ifyouhavealwaysbeenafraidoftheword“morphism”,andyou havebeenwonderingwhatcatamorphisms,anamorphisms,hylomorphims,and paramorphimswereabout,thisisthepapertoread?rst;youwilldiscoverthat theyaremerelynamesforrecursionpatternsthatoccuroverandoveragainwhen writingfunctionalprograms. Thealgorithmsinthepaperareallaboutsorting, andsinceyouarelikelytoknowthosealgorithmsbyheartalready,seeingthem structuredandanalyzedinanovelwayshouldserveasamotivationtoreadon tothesecondlecture. Thesecondlecture,onGenericProgramming,isalmostabookinabook. ThenotescanbeseenastheculminatingpointoftheSTOP-project,sponsored bytheDutchgovernmentattheendofthe80’sandthebeginningofthe90’s. Its overallgoalwasthedevelopmentofacalculationalwayofderivingprograms. The projecthasprovideddeeperinsightintorealfunctionalprogrammingandinto thetheorybehindmanythingscommonlywrittenbyfunctionalprogrammers. Oneofthemainachievementsoftheprojecthasbeentomakepeopleaware ofthefactthatmanyalgorithmscanbedescribedinadata-independentway. ThePolyPsystemintroducedinthesenotesisoneofthetranslationstothe Haskell-worldofthistheoreticalunderpinning. Thethirdlecture,onGenericProgramTransformation,canalsobeseenas anapplicationofthetheoryintroducedinlecturetwo. Manye?ciency-improving programtransformationscanbeperformedinamechanicalway,andthesewould nothavebeenpossiblewithoutinsightintothecorrectnessofsuchtransfor- tionsgainedinthelectureonGenericProgramming. Thefourthlecture,onDesigningandImplementingCombinatorLanguages, introducesaneasytowriteformalismforwritingdownthecatamorphismsint- ducedinearlierchapters. Itisshownhowquitecomplicatedcatamorphisms,that at?rstsightseemratherforbiddingbymakingextensiveuseofhigher-orderdo- VI Preface mains,canactuallybedevelopedinastep-wisefashion,usinganattributegr- marview;itisfurthermoreshownhowtorelatethiswayofprogrammingwith conceptsfromtheobject-orientedworldthusmakingclearwhatthestrengths andweaknessesofeachworldare. The?fthlecture,titledUsingMetaML:AStagedProgrammingLanguage, introducestheconceptofpartialevaluation. Itservesasanotherinstanceof thequestfor“themostgenericofwritingprogramsatthelowestcost”. The stagingtechniquesshowhowcoststhatwereintroducedbyaddingextralevels ofabstraction,maybemovedfromrun-timetocompile-time. Ithasbeencommonknowledgetousersofmodernfunctionallanguagesthat thetypesystemcanbeagreathelpinshorteningprogramsandreducingerrors. Intheextremeonemightseeatypeasapredicatecapturingtheproperties ofanyexpressionwiththattype. InthesixthlectureonCayenne–Spiceup yourProgrammingwithDependentTypesitisshowninwhatdirectionfunctional languagesaremostlikelytodevelop,andwhatmaybeexpectedofthenewtype systemstobeintroduced. Thelastlecture,titledHaskellasanAutomationController,showsthat writingfunctionalprogramsdoesnothavetoimplythatoneisboundtoremain isolatedfromtherestoftheworld. Beingabletocommunicatewithsoftware writtenbyothersinauniformway,isprobablyoneofthemostinteresting newdevelopmentsincurrentcomputerscience. Itappearsthattheconceptofa monadtogetherwiththeHaskelltypingrules,isquiteadequatetodescribethe interfacebetweenHaskellprogramsandtheouterworld. Finallywewanttothankeveryonewhocontributedtothisschoolandmade itsuchasuccessfulevent:sponsors,localsystemmanagers,localorganizers, students,andlastbutnotleastthelecturers. Weareconvincedthateveryone presentattheschoolenjoyedthiseventasmuchaswedid,andweallhopethat youwillfeelsomeofthespiritofthiseventwhenstudyingtheselecturenotes. March1999 DoaitseSwierstra PedroHenriques Jos´eOliveira VII Sponsorship Theschoolhasreceivedgeneroussponsorshipfrom: FCT-Fundac˜¸aoparaaCiˆenciaeTecnologia,Minist´eriodaCiˆenciae Tecnologia AdegaCooperativadePontedeLima AgˆenciaAbreu CGD-CaixaGeraldeDep´ositos CIUM-CentrodeInform´aticadaUniversidadedoMinho DI-DepartamentodeInform´aticadaUniversidadedoMinho GEPL-GrupodeEspeci?cac˜¸aoeProcessamentodeLinguagens LESI-Direc¸c˜aodeCursodeEngenhariadeSistemaseInform´atica Enabler Lactolima Latic´?niosdasMarinhas,Lda NovabasePorto-SistemasdeInforma¸c˜aoSA PrimaveraSoftware ProjectoCamila-GrupodeM´etodosFormais Sidereus-SistemasdeInforma¸c˜aoeConsultoriaInformat´icaLda SIBS-SociedadeInterbanc´ariadeServico¸s VieiradeCastro LocalCommittee: Jos´eAlmeida,Minho Lu´?sBarbosa,Minho Jos´eBarros,Minho M. Joao ˜ Frade,Minho PedroHenriques,Minho F. M´arioMartins,Minho F. LuisNeves,Minho CarlaOliveira,Minho JorgePinto,Lix JorgeRocha,Minho CesarRodrigues,Minho Joa˜oSaraiva,Minho M. Joa˜oVaranda,Minho IX TableofContents SortingMorphisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 LexAugusteijn 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 MorphismsonLists. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. 1 TheListCatamorphism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. 2 TheListAnamorphism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2. 3 TheListHylomorphism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2. 4 InsertionSort. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2. 5 SelectionSorts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 LeafTrees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3. 1 TheLeaf-TreeCatamorphism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3. 2 TheLeaf-TreeAnamorphism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3. 3 TheLeaf-TreeHylomorphism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3. 4 MergeSort. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4 BinaryTrees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4. 1 TheTreeCatamorphism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4. 2 TheTreeAnamorphism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4. 3 TheTreeHylomorphism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4. 4 Quicksort. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4. 5 HeapSort. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 5 Paramorphisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 5. 1 TheListParamorphism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 5. 2 InsertAsParamorphism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 5. 3 RemoveAsParamorphism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 6 GeneralizingDataStructures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 6. 1 GeneralizingQuicksort. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 6. 2 GeneralizingHeapSort. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 7 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 GenericProgramming–AnIntroduction–. . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 RolandBackhouse,PatrikJansson,JohanJeuring,LambertMeertens 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .