Exercises to Accompany Introduction to Functional ... · In your programming language 4.3. Find out...

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ExercisestoAccompanyIntroductiontoFunctionalProgrammingHowtoThinkFunctionallyin(Almost)AnyLanguagewithBarryBurdThislistincludesthreekindsofexercises:

• ExercisesmarkedGeneral:Completetheseexerciseswithoutwritingorreadingcodeofanykind,orexplorefeaturesinaprogramminglanguagethatyoumaynothaveseenbefore.

• ExercisesmarkedUsingpseudocode:Completetheseexercisesbyreadingorwritingsimplifiedsyntaxthatdoesn'tbelongtoanyparticularprogramminglanguage,likethesyntaxthatIuseinthecourse.

• ExercisesmarkedInyourprogramminglanguage:Completetheseexercisesbyreadingorwritingcodeinaprogramminglanguageofyourchoice.(Anddon'tforgettotestyourcode!)

Thereareexercisesforalmosteverysectionofthecourse.Forexample,Exercises1.1,1.2and1.3areforthefirstsection(thesectionentitledAboutThisCourse).Exercises2.1,2.2(andsoon)areforthesecondsection(thesectionentitledSolvingaProblemBothWays).Somesolutionsappearattheendofthelistofexercises.Ifyouhavequestions,sendemailtofunctional@allmycode.com,tweetto@allmycode,orpostonFacebookto/allmycode.Havefunandlearnalot!1. AboutThisCourseGeneral

1.1. ThefollowingprogramiswritteninaveryoldversionoftheBASICprogramminglanguage.Incaseyou'rewondering,thepercentsign(%)isBASIC'smodoperator.So,forexample,6 % 3is0and7 % 3is1.Tracethroughtheexecutionoftheprogramtodeterminetheprogram'soutput.GOTOstatementsarebad,andthisexercisewithGOTOstatementsisintentionallyannoying.Sohavefunwithitbutdon'ttakeittooseriously.

10 LET X = 5 25 PRINT X 20 GOTO 90 30 LET X = X + 8 35 PRINT X 35 IF X % 2 = 0 THEN GOTO 60 40 LET X = X * 5 45 PRINT X 50 IF X < 51 THEN GOTO 30

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60 LET X = X - 1 65 PRINT X 70 IF X % 3 = 0 THEN GOTO 30 85 END 90 LET X = X * 2 95 PRINT X 100 GOTO 40

1.2. TypethecodefromExercise1.1intotheinterpreteratwww.quitebasic.comtofindoutifyourproposedoutputiscorrect.

Inyourprogramminglanguage1.3. Obfuscatedcodeiscodethat'sdifficulttoread.Therearegoodreasonsandbad

reasonsforcreatingobfuscatedcode.Somesoftwaretoolsturnreadablecodeintoobfuscatedcodeinordertokeepthecodefrombeinghacked.Ontheotherhand,someobfuscatedcodeiswrittenforfuntoshowhowstrangethecodecanbe.PeoplepostexamplesofsuchcodeontheInternet.SearchtheInternetforfunexamplesofobfuscatedcode.

2. SolvingaProblemBothWaysPseudocode

2.1. Here'saslightmodificationofthecreditcardcategorizationproblem(calledProblem1inthevideo):Writeimperative-stylepseudocodetodisplaytheitemsintheFoodcategorywhoseamountsare$10ormore.Inyoursolution,don'tusethewordAND.Don'tuseasymbolthatstandsforthewordAND.

2.2. Writefunctional-stylepseudocodetosolvetheprobleminExercise2.1.Inyourprogramminglanguage

2.3. Writeandtestanimperative-styleprogramtosolvethecreditcardcategorizationproblem(calledProblem1inthevideo).

2.4. Writeandtestanimperative-styleprogramtosolvetheprobleminExercise2.1.2.5. Findoutifyourlanguagehasafeaturelikethefilterfunction.Ifitdoes,learnhowto

usethefilterfunctiontosolvethecreditcardcategorizationproblem.3. UsingFilter,MapandFoldPseudocode

3.1. Rewritethefollowingfunctiondefinitionsusinglambdanotation:3.1.1. f(x)=x+13.1.2. f(x)=x

Thisistheidentityfunction–thefunctionthatreturnswhateveryougiveit.3.1.3. nameOf(customer)=customer.name

3.2. Evaluatethefollowinglambdaexpressions:3.2.1. (lx®6*x)(21)3.2.2. (lx®x/2)((lx®x+7)(19))

3.3. Evaluatethefollowingexpressions

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3.3.1. map(timesTwo,[2,4,5])3.3.2. map(timesTwo,[8])3.3.3. map(timesTwo,[])3.3.4. map(addOne,map(timesTwo,[2,2,4,–3]))3.3.5. map(timesTwo,map(addOne,[2,2,4,–3]))3.3.6. foldFromLeft(plus,7,[3,–89])3.3.7. foldFromLeft(minus,7,[3,–8,9])3.3.8. foldFromRight(minus,7,[3,–8,9])3.3.9. foldFromLeft(minus,7,map(timesTwo,[3,0,8]))

3.4. You'regivenalistofcustomers.Eachcustomerhasanameandanamount.Acustomer'samountisthatcustomer'soutstandingbalance.Writeimperative-stylepseudocodetoprintthesmallestnegativebalance.Forexample,ifJoe'sbalanceis–$10,Ann'sbalanceis–$2,andDonna'sbalanceis$5,print–2.(Assumethatthelisthasatleastonecustomerinit,andthatcustomers'balancesrangebetween–$1000and$1000.)

3.5. Youhaveafunction(calledmax)thatfindsthelargeroftwonumbers.Writefunctional-stylepseudocodetosolvetheprobleminExercise3.4.

Inyourprogramminglanguage3.6. Findoutifyourlanguagehasfeatureslikethemap,foldFromLeftand

foldFromRightfunctions.4. ImperativeandFunctionalProgrammingLanguagesGeneral

4.1. MicrosoftExcelhasaFilterfeature.Youcanfinddocumentationaboutthisfeaturebyvisitinghttps://support.office.com/en-us/article/Filter-data-in-a-range-or-table-01832226-31b5-4568-8806-38c37dcc180e.Readaboutthisfeature,andtryusingitinMicrosoftExcel.

4.2. MicrosoftExcelhasanAggregatefunction,whichbehavesabitlikeourfoldFromLeftandfoldFromRightfunctions.Youcanfinddocumentationaboutthisfunctionbyvisitinghttps://support.office.com/en-us/article/AGGREGATE-function-43b9278e-6aa7-4f17-92b6-e19993fa26df.Readaboutthisfunction,andtryusingitinMicrosoftExcel.

Inyourprogramminglanguage4.3. Findouthowtheexpertsclassifyyourprogramminglanguage.Isitimperative,purely

functional,hybrid,orsomeotherkindoflanguage.4.4. Ifyourlanguagesupportssomefunctionalfeatures,readuponthosefeatures.Finda

fewsimplesampleprogramsonthewebandrunthemtofindouthowtheybehave.4.5. Manylanguagescanbeextendedtoincludefunctionalfeaturesthataren'tofficially

partofthelanguage.Groupsofdeveloperscreatetoolsenablingyoutousetheseadditionalfunctionalfeatures.Findoutifyourprogramminglanguagehassuchextensions.

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5. PureFunctionsPseudocode

5.1. Whichofthesepseudocodefunctionsarepure?Whicharen'tpure?Why?5.1.1. f(x)=x+x+x5.1.2.

f(x) { x = x + 7 return x }

5.1.3. f(x)=x+current_day_of_the_monthwherecurrent_day_of_the_monthisanumberfrom1to31

5.1.4. f(x) { integer y = 3 return x + y }

5.1.5. f(x) { integer y = random() return x + y - y }

5.1.6. length(the_string_s)=numberofcharactersinthe_string_s5.1.7.

post(message, URL) { add the message to the message board at the URL }

5.2. Whichoftheseexpressionsarereferentiallytransparent?Whicharen't?Why?5.2.1. inputFromKeyboard(x)

Ifyouexecutey = inputFromKeyboard(x),andtheusertypes7,thenthevalueofybecomes7.

5.2.2. 7+65.2.3. f(x)=x+current_day_of_the_month

wherecurrent_day_of_the_monthisanumberfrom1to315.3. RevisityoursolutiontoProblem3.2tomakesurethatnothinginyoursolutionis

mutable.6. SomeBenefitsofPureFunctionsPseudocode

6.1. Usepseudocodetowritetestsforthefollowingtwofunctions.Identifythesetupstep(s)requiredtotesteachfunction.

6.1.1.

mortgage(principle, ratePercent, numYears) {

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rate = ratePercent / 100 numPayments = numYears * 12 effectiveAnnualRate = rate / 12 payment = principal * (effectiveAnnualRate / (1 - (1 + effectiveAnnualRate)^(-numPayments)))) } Incaseyoucare,my^symbolmeans"tothepowerof."Also,themonthlypaymentona30year,$100000.00mortgagewith5.25%interestis$552.20.

6.1.2. mortgage(principle, numYears) { rate = currentRatePercent / 100 numPayments = numYears * 12 effectiveAnnualRate = rate / 12 payment = principal * (effectiveAnnualRate / (1 - (1 + effectiveAnnualRate)^(-numPayments)))) } ThefunctioninthisexerciseisalmostthesameasthefunctioninExercise6.1.1.Theonlydifferenceisthatthisexercise'sfunctionusesavariable(currentRatePercent)whosevalueissetoutsideofthemortgagefunctionandcanbemodifiedoutsideofthemortgagefunction.

6.2. ThemortgagefunctioninExercise6.1.1mightruncorrectlywhenyoutestitwithparameters100000.00,5.25,30.Doesthismeanthatthefunctionwillruncorrectlywheneveranyonecallsthefunctionwithparameters100000.00,5.25,30?

6.3. ThemortgagefunctioninExercise6.1.2mightruncorrectlywhenyoutestitwithparameters100000.00,30.Doesthismeanthatthefunctionwillruncorrectlywheneveranyonecallsthefunctionwithparameters100000.00,30?

6.4. Definefactorial(n) = 1*2*3*...*n.Here'sapseudocodeprogramtorepeatedlycalculatefactorial(n)andcountthenumberofmultiplicationoperationsdoneduringthecalculation:

loop input n result = 1 count = 0 for i from 2 to n do result = result * i count = count + 1 print result print count

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Thefactorialfunctionispuresoitcanbememoized.Usememoizationtomakevaluesofcountsmallerforvaluesofnlessthanorequalto100.

Inyourprogramminglanguage

6.5. Writeaprogramthatinputsanintegerthat'slessthanorequalto1000(callitn)andthenusesalooptoadduptheintegers1ton.Theprogramdisplaystheresultingsum.

6.6. ModifytheprogramofExercise6.4sothattheprogramrepeatedlyinputsanewvaluefornandthendisplaysthesumoftheintegers1ton.Theprogramstopsrepeatingwhentheuserenters0forn.Tosolvethisproblem,createanarrayofsize1000(callitthetotalsarray).Put1intototals[1].Thenputthesumof1and2intototals[2].Thenputthesumof1,2and3intototals[3].Andsoon.Whatiftheuserinputs950fornandthen765forn?Don'trecalculatethesumofthenumbers1to765.Inonestep,getthatsumfromthetotalsarray.

7. AvoidingRaceConditionsGeneral

7.1. Ann'sparents,BobandCarol,haveajointbankaccount.BobvisitsanATMmachine.Hecheckshisbalance,whichis$300.SoBobrequestsawithdrawalof$100.Thenthemachine'sscreendisplaysamessagesayingthatBobcan'twithdrawthemoney.Whatwentwrong?

7.2. Describeascenarioinwhichthecreditcardtotalproblem(calledProblem2inthevideo)cansufferfromraceconditions.

Pseudocode7.3. Whatoutputsmayresultfromrunningthefollowingcode?

x = 0 three times do { simultaneously do { x = x + 1 } and { x = x + 1 } } print x

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8. EfficientParameterPassingGeneral

8.1. InanoldversionoftheFORTRANprogramminglanguage,anynumericliteralsthatwerepassedasparameterswerestoredasvariableswithvaluesbeforetheywerepassed.Whatunwantedconsequencecanthishaveforparameterpassing?

Inyourprogramminglanguage8.2. Therearemanywaysforlanguagestoimplementparameterpassing.You'veprobably

usedparameterpassinginanimperativelanguage,andyouprobablyknowtherulesthatgovernparameterpassinginyourlanguage.Butdoyouknowhowparameterpassingworksunderthehood?Researchthisquestionforthelanguageofyourchoice.

8.3. Aretheredifferentoptionsforpassingparametersinyourprogramminglanguage?Ifso,whichofthemallowyoutomodifythevaluesoftheparametersinthefunctioncall?Whichdon't?

9. LazyEvaluationPseudocode

9.1. Ineachpartofthisproblem,evaluatetheexpressionslazily,andtheneagerly.Ineachcase,aretheresultsdifferent?9.1.1. Inthisproblem,acalltotheprintfunctionreturnsthenumberofvaluesthat

weresuccessfullyprinted.Forexample,print(x, y)mightreturnthenumber2.

x = 7 if x < 5 & (print(x) = 1) print("x is", x)

9.1.2.

if theArray has an element with index 10 & theArray[10] = 0 print("OK")

9.1.3. Inthisproblem,++xbehavesasbothaninstructionandanexpression.Asaninstruction,++xadds1tothevalueofx.Asanexpression,thevalueof++xisthenewlyobtainedvalueofx.Forexample,thecode

x = 7 print(++x) print(x)

displaysthenumbers8 8.Withthatinmind,evaluatethefollowingcodebothlazilyandeagerly.x = 18 if ++x > 19 & print(++x) print("x is", x)

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print(x)

9.1.4. Inthisproblem,x++isthesameasinProblem9.1.3.

x = 18 if ++x > 18 or print(++x) print("x is",x) print(x)

9.1.5. x=(ifynotEqualTo0then(x/y)elsex+1)9.1.6. firstElementOf([0,3.6.9,12])9.1.7. firstElementOf([0,3,6,9,12,...])9.1.8. sumOfTheNumbersIn([0,3,6,9,12,...])

10. IntroductiontoHigher-OrderFunctionsGeneral

10.1. Letf(x)=x+7,letg(x)=x2andleth(x)=1/x.10.1.1. Findthevalueoff∘g(5).10.1.2. Writeanarithmeticexpressionforthefunctionf∘g.10.1.3. Findthevalueofg∘f(5).10.1.4. Writeanarithmeticexpressionforthefunctiong∘f.10.1.5. Findthevalueofh∘h(5).10.1.6. Findthevalueofg∘f∘h(5).10.1.7. Writeanarithmeticexpressionforthefunctiong∘f∘h.

10.2. Inthevideo,Idescribefunctioncomposition,denotedbythe∘symbol.Iscompositionahigher-orderfunction?Why,orwhynot?

10.3. Ifyou'vetakencalculus,you'veseenthederivative,denotedbyd/dx.Explainwhythederivativeisahigher-orderfunction.

10.4. SymboliclogichastwofunctionsnamedthereExistsandforAll.Hereareexamplesoftheuseofthesetwofunctions:

thereExists(even,[1,3,5])isfalsebecausetherearenoevennumbersinthelist[1,3,5]thereExists(even,[1,2,5])istruebecausethere'sanevennumberinthelist[1,2,5]forAll(even,[1,2,5])isfalsebecausenotallnumbersinthelist[1,2,5]areevenforAll(even,[2,6,10])istruebecauseallnumbersinthelist[1,2,5]areevenArethereExistsandforAllhigher-orderfunctions?Whyorwhynot?

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11. CurryingGeneral

11.1. Inthevideo,Iusethenotation

filter : function, list → list

todescribetheparametersandresulttimeofthefilterfunction.Usesimilarnotationtodescribeeachofthefollowingfunctions:

11.1.1. DescribethefunctionfinExercise10.1.11.1.2. Describethefunctionf∘ginExercise10.1.111.1.3. Describetheresultofpartiallyapplying2tothefirstargumentinthefunction

add(x, y) = x + y.11.1.4. Describecomposition(denotedbythe∘symbol).11.1.5. DescribetheapplyNTimesfunctionwhichappliesafunctionntimes.For

example,

applyNTimes(x + 1, 3) = ((x + 1) + 1) + 1 applyNTimes(x * 2, 4) = (((x * 2) * 2) * 2) * 2

11.1.6. DescribetheaddFfunction,whichtakestwofunctions,f andg,andreturnsanotherfunctionthatsumsupthereturnvaluesfromfandg.Forexample,

addF(x2, 2*x) = x2 + 2*x

11.1.7. Inthevideo,youstartwithaddand,fromit,youcreatecurryAdd.WhatareyoudoingwhenyougofromaddtocurryAdd?

12. ClosuresPseudocode

12.1. What'stheoutputofthefollowingcode?makeNewFunction(factor) { size = 1 f() = multiplysize by factorandreturnthenewsizevalue return f } increaseA = makeNewFunction(5) print( increaseA() ) print( increaseA() ) increaseB = makeNewFunction(10) print( increaseB() ) print( increaseB() )

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print( increaseA() )

12.2. What'stheoutputofthefollowingcode?

createGreeting(interjection) { f(name) = interjection"," name"!" return f } formalGreeting = createGreeting("Hello") casualGreeting = createGreeting("Hi") print ( formalGreeting("Mr. Williams") ) print ( formalGreeting("Ms. Polywether") ) print ( casualGreeting("Joe") )

12.3. What'stheoutputofthefollowingcode?delayDisplay(n) { f(message) = waitnsecondsandthendisplaymessage } g = delayDisplay(1) h = delayDisplay(5) h("Goodbye") g("Hello")

13. IntroductiontoListsPseudocode

13.1. Find head(tail([6,9,12,8]))13.2. Find tail(tail(tail([6,9,12,8])))13.3. Find construct(10,tail([21,15]))13.4. Trueorfalse? tail([18,8]) isequalto 813.5. Find concatenate(concatenate([1,3],[1,8]),[0,0])13.6. With x = [3,2,19],find construct(head(x),tail(x))

14. Recursionand15. MoreRecursionExamplesPseudocode

15.1. Showhowtousethereversefunctionfromthevideotofindoutifalistofcharactersisapalindrome.

15.2. Usethereversefunctionfromthevideotodefinealastfunction.Whenappliedtoalist,thelastfunctionreturnsthelastvalueinthelist.

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last([1,7,5]) = 5 last([2]) = 2

15.3. UsethelastfunctioninExercise15.2towriterecursivecodeforafunctionthatcreatesalistoffactorialsuptoandincludingthenumbergiventoit.Forexample,

factorials[1] = 1 factorials[2] = [1,2] factorials[3] = [1,2,6] factorials[4] = [1,2,6,24]

15.4. Writerecursivecodeforafunctionthatsumsthenumbersinalist.Forexample,

sum([1,3,5])=9 sum([]) = 0

15.5. AfunctionnamedfirstNFromtakesalistandanintegern,andreturnsthefirst

nvaluesfromthelist.Forexample,

firstNFrom([9, 6, 3, 4], 2) = [9, 6] firstNFrom([9, 6, 3, 4], 1) = [9]

WriterecursivecodeforthefirstNFromfunction.(Assumethatnisalwaysgreaterthan0andlessthanorequaltothelengthofthelist.)

15.6. Afunctionnamedalternatestakesalistandreturnsalistcontainingthealternatevaluesfromtheoriginallist.Forexample,

alternates([2,19,81,4]) = [2,81] alternates([1,2,7,5,9]) = [1,7,9] alternates([8]) = [8] alternates([]) = []

Writerecursivecodeforthealternatesfunction.16. ComputationsthatMightFailPseudocode

16.1. FindthevalueofsqrtMaybe(x-10) >>= minus4Maybe >>= reciprocalMaybe >>= plus13Maybewhenx=10.

16.2. FindthevalueofsqrtMaybe(x-1) >>= minus4Maybe >>= reciprocalMaybe >>= plus13Maybewhenx=17.

16.3. FindthevalueofsqrtMaybe(x-10) >>= minus4Maybe >>= reciprocalMaybe >>= plus13Maybewhenx=9.

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16.4. FindthevalueofreciprocalMaybe(x) >>= sqrtMaybewhenx=1.16.5. FindthevalueofreciprocalMaybe(x) >>= sqrtMaybewhenx=0.16.6. ThedefintionMaybefunctiontakesastringofcharacters(suchasaword)and

returnsthedictionarydefinitionofthatstringofcharacters,ifthereisone.ThelengthMaybefunctiontakesastringofcharacters(suchasawordorgroupofwords)andreturnsthenumberofcharactersinthestring,ifthereisone.(Counttheblankspacesandpunctuationinthestring.)16.6.1. ThedefinitionofhouseisAbuildingwhosepurposeistoregularlyshelterthe

samepersonorpeople.FindthevalueofdefinitionMaybe(x) >>= lengthMaybewhenx="house".

16.6.2. Thenon-wordbxbutwisn'tarealwordsoitdoesn'thaveadefinition.FindthevalueofdefinitionMaybe(x) >>= lengthMaybewhenx="bxbutw".

16.7. Describetheparametertypes,returntype,andruleforapplyingthebindfunctioninExercise16.6.

17. MoreMonadsPseudocode

17.1. Usingthefunctionsdefinedinthevideo,defineafunctionwhoseparameterisapersonandwhoseresultisalistoftheperson'sgreatauntsanduncles.

17.2. Ann'ssmallbusinesshas3employees.Aparticularemployeemayormaynothaveclients.DescribethefunctionsforfindingalltheclientsinAnn'sbusiness.

17.3. It'sParents'AppreciationDayfortheemployeesofAnn'ssmallbusiness.(SeeExercise17.2.)DefineafunctiontocreatealistofparentstoinvitetotheAppreciationDayparty.(Don'tbotherinvitingAnn'sparents.They'renofun!)

17.4. Ifyoutrytoperformacomputation,andthecomputationfails,thewordNothingintheoutputwithnootherexplanationmightbeabitfrustrating.Tofixthis,imagineanewkindofmonadthatIcalltheMaybeAFloatWithMessagemonad.LiketheMaybemonadfromthevideo,theMaybeAFloatWithMessagemonadhasoneoftwothingsinit:

• Ifthecomputationsucceeds,themonadcontainsJustavalue.• Ifthecomputationfails,themonadcontainsErrorBecauseOfvalue.

Forexample,ifyoutrytodivideby2,yougetJust 1/2.Butifyoutrytodivideby0,yougetErrorBecauseOf 0.The0intheresultmightnotbeveryinformative,butit'sprobablybetterthannothing.DescribethedetailsoftheMaybeAFloatWithMessagemonad.

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SomeSolutions1. AboutThisCourse

1.1. Theoutputis510505857653253243323312. SolvingaProblemBothWays

2.1. for each purchase in purchasesList if purchase.category == Food if purchase.amount >= 10 print purchase

2.2. hasCategoryFood(purchase) = purchase.category == Food tenOrMore(purchase) = purchase.amount >= 10 print(filter(tenOrMore,filter(hasCategoryFood,purchasesList)))

3. UsingFilter,Map,andFold

3.1. Rewritethefollowingfunctiondefinitionsusinglambdanotation:3.1.1. lx®x+13.1.2. lx®x3.1.3. lcustomer®customer.name

3.2. Evaluatethefollowinglambdaexpressions:3.2.1. (lx®6*x)(21)=6*21=1263.2.2. (lx®x/2)((lx®x+7)(19))=(lx®x/2)(19+7)=(lx®x/2)(26)=13

3.3. Evaluatethefollowingexpressions3.3.1. map(timesTwo,[2,4,5])=[4,8,10]3.3.2. map(timesTwo,[8])=[16]3.3.3. map(timesTwo,[])=[]3.3.4. map(addOne,map(timesTwo,[2,2,4,–3]))=

map(addOne,[4,4,8,–6])=[5,5,9,–5]

3.3.5. map(timesTwo,map(addOne,[2,2,4,–3]))map(timesTwo,[3,3,5,–2])=[6,6,10,–4]

3.3.6. foldFromLeft(plus,7,[3,–89])=7+3+(–8)+9=113.3.7. foldFromLeft(minus,7,[3,–8,9])=((7–3)–(–8))–9=33.3.8. foldFromRight(minus,7,[3,–8,9])=3–((–8)–(9–7))=133.3.9. foldFromLeft(minus,7,map(timesTwo,[3,0,8]))=

foldFromLeft(minus,7,[6,0,16])=((7–6)–0)–16=–153.4.

smallestNegativeBalance = -1000 for each customer in customersList if customer.balance < 0 if customer.balance > smallestNegativeBalance

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smallestNegativeBalance = customer.balance print smallestNegativeBalance

3.5. getBalance(customer) = customer.balance isNegative(number) = number < 0 balancesList = map(getBalance, customersList) negBalancesList = filter(isNegative, balancesList) smallestNegBal = foldFromLeft(max, -1000, negBalancesList) print(smallestNegBal) Withoutnamingsomanyintermediatefunctions:getBalance(customer) = customer.balance isNegative(number) = number < 0 print(foldFromLeft(max, -1000, filter(isNegative, map(getBalance, customersList))))

Withoutnaminganyintermediatefunctions:print(foldFromLeft(max, -1000, filter(lnumber -> number < 0, map(lcustomer -> customer.balance , customersList))))

1. invisibleline2. invisibleline3. invisibleline4. invisibleline

5. PureFunctions5.1.

5.1.1. Thisfunctionispurebecauseitdoesn'tuseanyvalueotherthanitsparameterx,anditdoesn'tmodifyanyvalue(s)declaredoutsideofitself.

5.1.2. Thepurityorimpurityofthisfunctiondependsonthewaytheprogramminglanguagehandlesparameters.Insomelanguages,modifyingaparameter'svaluehasnoeffectontheparameterinthecallingcode.Insuchalanguage,thefollowingcodewoulddisplaythevalue10andthefunctionwouldbepure.

x = 10 y = f(x) print x

Inotherlanguages,modifyingaparameter'svaluechangestheparameterinthecallingcode.Insuchalanguage,thesamecodewoulddisplaythevalue17andthefunctionwouldbeimpure.

5.1.3. Thisfunctionisimpurebecauseitusesanumberthatitobtainsfromthesystemclock,andthesystemclockisn'tinternaltothefunction.

5.1.4. Thisfunctionispure.Itdoesn'tuseanyinformationthatcomesfromoutsideofthefunction.Itdefinesanadditionalvariableybutthatvariableisfullycontainedinsidethefunction.Thisfunctionisequivalenttothex + 3function.

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5.1.5. Thisfunctionisimpure.Thefunction'sreturnresultdoesn'tdependonthevalueobtainedbycallingrandom()sointhatsense,thefunctiondoesn'treallyuseanoutsidevalue.Butasubsequentcalltorandom()(afterexitingacalltof(x))willbedifferentbecauseonecalltorandom()hasbeen"usedup"bythecalltof(x).So,inasubtlesense,thisfunctionchangessomethingexternaltoit.Sothisfunctionisimpure.

5.1.6. Thisfunctionispure.Itdoesn'tchangeanythingoutsideofitself,andit'sreturnresultdependsonlyontheinputparameter(the_string_s).

5.1.7. Thisfunctionisimpure.ItsexecutionchangeswhateverwebsitetheURLpointsto.

5.2. 5.2.1. Thisexpressionisreferentiallyopaque(theoppositeofreferentially

transparent).IfyoucallinputFromKeyboard(x)twice,thevalueofinputFromKeyboard(x)mightbe7thefirsttimeand123897thesecondtime.

5.2.2. Thisexpressionisreferentiallytransparent.Theexpression7 + 6means13,nomatterwhereitappearsinaprogram.

5.2.3. Thisexpressionisreferentiallyopaque(theoppositeofreferentiallytransparent).Ifyoucallf(x)twice,thevalueoff(x)mightbe7thefirsttimeand31thesecondtime.

5.3. Here'sacopyofoneofmysolutionstoProblem3.2:

getBalance(customer) = customer.balance isNegative(number) = number < 0 balancesList = map(getBalance, customersList) negBalancesList = filter(isNegative, balancesList) smallestNegBal = foldFromLeft(max, -1000, negBalancesList) print(smallestNegBal) Inthissolution,noticethatnoneoftheprogram'svariables(balancesList,negBalancesList,smallestNegBal)havevaluesthatvary.Youdon'thavetothinkoftheexpressionbalancesList = map(getBalance, customersList) asassigningavaluetobalancesList.Instead,youcanthinkofitasthedefinitionofbalancesList.

6. SomeBenefitsofPureFunctions6.1. Testingfunctions:

6.1.1. Thisfunctionispuresothetestrequiresnosetup.

if mortgage(100000.00, 5.25, 30) = 552.20 return "Passed"

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else return "Failed"

6.1.2. Thisfunctionisn'tpureso,beforethetest,youhavetosetupthevalueofcurrentRatePercent.

currentRatePercent = 5.25 if mortgage(100000.00, 30) = 552.20 return "Passed" else return "Failed" Inthisexample,thesetupinvolvesonlyonestatement.Thesetupforafunctioninareal-lifeapplicationtypicallyinvolvesmanymorestatements.

6.2. ThefunctioninExercise6.1.1ispure,soitalwaysyieldsthesameresultwhenitrunswithparameters100000.00,5.25,30.(I'mignoringthingslikedifferencesinthewaycomputersperformarithmetic,orpeopletrippingoverpowercordswhilethefunctionexecutes.)Soifthefunctionpassesyourtestwithparameters100000.00,5.25,30,theprogramisguaranteedtoruncorrectlywiththoseparameters.

6.3. ThefunctioninExercise6.1.2isn'tpuresoitdoesn'talwaysyeildthesameresultwhenitrunswithparameters100000.00,30.ThefunctionmightbecorrectwhenthecurrentRatePercentis5.25,butnotwhenthecurrentRatePercentis6.00.

6.4. largestKnownFactorial = 1 for n from 1 to 100 do knownFactorials[n] = 1 loop input n count = 0 if n > largestKnownFactorial for i from largestKnownFactorial + 1 to n knownFactorials[i] = knownFactorials[i - 1] * i count = count + 1 largestKnownFactorial = n print knownFactorials[n] print count

Here'saJavaprogramtoimplementthepseudocode:

import java.math.BigInteger; import java.util.Scanner; public class Main { Scanner keyboard = new Scanner(System.in);

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public static void main(String[] args) { new Main(); } Main() { int n; BigInteger[] knownFactorials = new BigInteger[101]; int largestKnownFactorial = 1; for (n = 1; n <= 100; n++) { knownFactorials[n] = new BigInteger("1"); } for (;;) { System.out.print("n: "); n = keyboard.nextInt(); BigInteger result = new BigInteger("1"); int count = 0; for (int i = 2; i <= n; i++) { result = result.multiply( new BigInteger(Integer.toString(i))); count++; } System.out.println("Without memoization:::Result: " + result + " Count: " + count); count = 0; if (n > largestKnownFactorial) { for (int i = largestKnownFactorial + 1; i <= n; i++) { knownFactorials[i] = knownFactorials[i - 1] .multiply( new BigInteger(Integer.toString(i))); count++; } largestKnownFactorial = n; } System.out.println("With memoization:::Result: " + knownFactorials[n] + " Count: " + count); } } }

7. AvoidingRaceConditions

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7.1. Bobisavictimofaracecondition.WhileBobwaspreparingtorequest$100,CarolwaswithdrawingmoneyfromthesameaccountatadifferentATMmachine.BythetimeBobcompletedhisrequest,therewasn'tenoughmoneyintheaccounttocoverBob's$100withdrawal.

7.2. DividetheFoodpurchasesintotwothreads(perhapstwocoresonamulti-coreprocessor).ThreadAtotalsuphalfoftheFoodpurchaseswhileThreadBtotalsuptheotherhalf.Thegrandtotalismaintainedinacentrallocationthat'supdatedbybothThreadAandThreadB.Intheend,thegrandtotalisincorrect.Inthefunctionalversionoftheproblem,there'snototalvariable,onlyasumorfoldexpression.Sothecodedoesn'tlenditselftotheupdatingofmutablevariables.

7.3. Thevariablexismutableandtherearetwosimultaneousthreads(callthemThreadAandThreadB).Inascenariothatsuffersfromnoraceconditions,ThreadAexecutesallofitsstatementsandthenThreadBexecutesallofitsstatements.Inthisscenario,thefinalvalueofxis6.Inascenariothatsuffersthemostfromraceconditions,thefollowinghappens: ThreadAgetsthevalueofx,whichis0. ThreadBgetsthevalueofx,whichisstill0. ThreadAadds1to0andassigns1tox. ThreadBadds1to0andassigns1tox. ThreadAgetsthevalueofx,whichis1. ThreadBgetsthevalueofx,whichisstill1. ThreadAadds1to1andassigns2tox. ThreadBadds1to1andassigns2tox. ThreadAgetsthevalueofx,whichis2. ThreadBgetsthevalueofx,whichisstill2. ThreadAadds1to2andassigns3tox. ThreadBadds1to2andassigns3tox.

Theprogramprints3.

8. EfficientParameterPassing8.1. Considerthefollowingpseudocode:

f(2) print 2 print 2 + 2 f(x) { x = x + 1 } InanoldversionofFORTRAN,theoutputofthiscode(writteninFORTRANsyntax)wouldbe3 6,not2 4.

9. LazyEvaluation

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9.1.1. Withlazyevaluation,thecodedoesn'texecute(print(x) = 1)because,withxnotlessthan5,theifconditioncannotpossiblybetrue.Becausetheentireifconditionisfalse,sothecodedoesn'tprintanything.Witheagerevaluation,thecodeevaluatesbothx < 5and(print(x) = 1).Sothecodeprintsthevalueofx(whichis7).Becausetheteststillmakestheifstatement'sconditionfalse,thecodedoesn'tdisplayx is 7.Note:Inanextreme,counterproductiveversionofeagerevaluation,thecodewouldevaluatetheprint("x is", x)insidethebodyoftheifstatementeventhoughtheifstatement'sconditionisfalse,andthusdisplayx is 7.

9.1.2. AssumethattheArrayhasonly4elements,theArray[0],theArray[1],theArray[2],andtheArray[3].Withlazyevaluation,thecodeneverperformsthetesttheArray[10] = 0andthecalltoprintisn'texecuted.Witheagerevaluation,thecodeperformsthetesttheArray[10] = 0eventhoughtheoutcomeofthattesthasnoeffectonthevalueoftheifstatement'scondition.Insomelanguages,therequestforthevaluetheArray[10]overrunsthespaceallocatedtothearray(whichisn'tgood).Inotherlanguages,therequestfortheArray[10]generatesanerror.

9.1.3. Theexpression++x > 19adds1tox,makingthevalueofxbe19.Sothat++x > 19expressionisfalse.Withlazyevaluation,thecodedoesn'tbothertoexecuteprint(++x).Andbecauseit'sinsidethebodyoftheifstatement,thecodedoesn'texecuteprint("x is", x).Sotheonlyprintingthecodedoesisthefinalprint(x),andtheentireoutputisthenumber19.Witheagerevaluation,thecodeexecutesprint(++x).Executionofthatfunctioncalloutputsthevalue20.Thentheexecutionofthefinalprint(x)outputs20asecondtime.

9.1.4. Theevaluationoftheexpression++x > 18setsthevalueofxto19.So++x > 18istrue.That'senoughtomaketheentireifconditionbetruenomatterwhatcomesafterthewordor.Sothere'snoneedtoevaluateprint(++x).Withlazyevaluation,thecodedoesn'tevaluateprint(++x),sothecallprint("x is",x)displaysx is 19,andthefinalprint(x)calldisplays19again.So,overall,theoutputofthecodeisx is 19 19.Witheagerevaluation,thecodeevaluatesprint(++x)eventhoughthatevaluationdoesn'tchangethevalueoftheentireifstatementcondition.Evaluationofprint(++x)changesthevalueofxto20.andprints20.Thenthecodeexecutestheothertwoprintstatements.So,overalltheoutputofthecodeis20 x is 20 20.

9.1.5. Assumethatyisequalto0.Withlazyevaluation,thecodeignoresthethen (x /y)partandgoesstraighttotheelse xpart.Sotheoveralleffectislikeexecutingx = x + 1.Witheagerevaluation,thecodedoesn'tignorethethen (x /y)partanddividesxby0,whichisn'tagoodthingtodo.Insomelanguages,thisgeneratesanarithmeticerrorandtheprogramcrashes.

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9.1.6. Withbothlazyandeagerevaluation,thevalueofthisexpressionis0.9.1.7. Withlazyevaluation,thecodefiguresoutwhatfirstElementOfmeansand

looksforonlythefirstelementof[0, 3, 6, 9, 12, ... ]whichis0.Witheagerevaluation,thecodetriestofindallelementsof[0, 3, 6, 9, 12, ... ]andthattakesforever.Sothecodenevergetstolookforthefirstelementofthelist.

9.1.8. Withbothlazyandeagerevaluation,thecodehastofindallnumbersinthelist[0, 3, 6, 9, 12, ... ].Thattakesforever,soinbothcases,thecodenevercomesupwithananswer.

10. IntroductiontoHigher-OrderFunctions

10.1. 10.1.1. f∘g(5)=f(g(5))=f(25)=3210.1.2. f∘g(x)=f(x2)=x2+710.1.3. g∘f(5)=g(12)=14410.1.4. g∘f(x)=g(x+7)=(x+7)210.1.5. h∘h(5)=h(1/5)=1/(1/5)=510.1.6. g∘f∘h(5)=g(f(1/5))=g(1/5+7)=(1/5+7)210.1.7. g∘f∘h(x)=g(f(h(x)))=g(f(1/x))=g(1/x+7)=(1/x+7)2

10.2. Compositionisahigher-orderfunctionbecausecompositiontaketwofunctions(asitsparameters)andreturnsathirdfunctionasitsresult.Forexample,inExercise10.1.2,compositiontakesthefunctionsx2andx+7andcreatesthefunctionx2+7.

10.3. Thederivativetakesafunctionasitsparameterandreturnsanotherfunctionasitsresult.Forexample,d/dxtakesthefunctionx2andreturnsthefunction2xasitsresult.

10.4. ThefunctionthereExiststakes,asitsparametersafunction(suchaseven)andalist(suchas[1,3,5]).Thefirstparameter,even,isafunctionbecauseeventakeanumber(suchas1)andreturnstrueorfalse,dependingonwhetherthenumberisevenornot.Therefore,thereExistsisahigher-orderfunction.Similarly,forAllisahigher-orderfunction.

11. Currying

11.1.1. f:number->number11.1.2. f∘g:number->number11.1.3. Theresultisanewfunctionadd2,withtheformulaadd2(y)=y+2.

add2:number->number11.1.4. Whenyoucomposeonefunctionwithanotherfunction,yougetyetanother

function.∘:function,function->function

11.1.5. TheparameterlistforapplyNTimesisafunction(suchasx + 1)andanumber(suchas3).Theresultisafunction(representedbyanexpressionsuchas((x + 1) + 1) + 1).applyNTimes : function, number -> function

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11.1.6. TheaddFfunctiontakestwofunctionsasitsparametersandreturnsyetanotherfunctionasitsresult.addF : function, function -> function

11.1.7. Ididn'tfocusonthispointinthevideo,butwhenyougofromaddtocurryAdd,you'reapplyingafunctionthatyoucancallthecurryfunction.Inthisexample,thecurryfunctiontakestheaddfunctionasitsargumentandreturnsthecurryAddfunctionasitsresult.curry : function -> function

12. Closures12.1. Theoutputis

5 25 10 100 125

12.2. TheoutputisHello, Mr. Williams! Hello, Ms. Polywether! Hi, Joe!

12.3. TheoutputisHello(afterwaiting1second)Goodbye(afterwaiting4moreseconds)

13. IntroductiontoListsGeneral

13.1. head(tail([6,9,12,8])) = head([9,12,8]) = 913.2. tail(tail(tail([6,9,12,8]))) = tail(tail([9,12,8]))

= tail([12,8]) = [8] 13.3. construct(10,tail([21,15])) = construct(10,[8]) = [10,8]13.4. falsebecause tail([18,8]) isequalto [8] (thelistwhoseonlyentryisthe

number8)whichisn'tquitethesameasthenumber8.13.5. concatenate(concatenate([1,3],[1,8]),[0,0]) =

concatenate([1,3,1,8],[0,0])=[1,3,1,8,0,0]13.6. With x = [3,2,19],construct(head(x),tail(x)) =

consttruct(3, [2,9]) = [3,2,9] = x

14. Recursionand15. MoreRecursionExamples

15.1. isAPalindrome(aList) = (aList equals reverse(aList))

15.2. last(aList) = head(reverse(aList))

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15.3. factorials(1) = [1] factorials(n) = concatenate(factorials(n-1),[last(factorials(n-1))*n])

15.4. sum([]) = 0 sum(h::t) = h + sum(t)

15.5. firstNFrom(h::t, 1) = [h] firstNFrom(h::t, n) = construct(h, firstNFrom(t, n-1))

15.6. alternates([]) = [] alternates(h::[]) = [h] alternates(h::t) = h :: (alternates (tail t))

16. ComputationsthatMightFail16.1.

sqrtMaybe(10-10) is Just 0 Binding Just 0 with minus4Maybe yields Just -4 Binding Just -4 with reciprocalMaybe yields Just -1/4 Binding Just -1/4 with plus13Maybe yields Just 12.75

16.2. sqrtMaybe(17-1) is Just 4 Binding Just 4 with minus4Maybe yields Just 0 Binding Just 0 with reciprocalMaybe yields Nothing Binding Nothing with plus13Maybe yields Nothing

16.3. sqrtMaybe(9-10) is Nothing Binding Nothing with minus4Maybe yields Nothing Binding Nothing with reciprocalMaybe yields Nothing Binding Nothing with plus13Maybe yields Nothing

16.4. reciprocalMaybe(1) is Just 1/1 whichis Just 1 Binding Just 1 with sqrtMaybe yields Just 1

16.5. reciprocalMaybe(0) is Nothing Binding Nothing with sqrtMaybe yields Nothing

16.6. 16.6.1.

definitionMaybe("house") is Just "A building whose purpose is to regularly shelter the same person or people."

Binding Just "A building whose purpose is to regularly shelter the same person or people."to lengthMaybe yields Just 75.

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16.6.2. definitionMaybe("bxbutw") is Nothing BindingNothingtolengthMaybeyieldsNothing.

16.7. bind : Maybe a string, f -> Maybe an integer withf : string -> Maybe an integer Theruleforapplyingthisexercise'sbindisthesameastheruleforthebindinthevideo:"IfyouhaveJust y,applyftoygettingJust f(y)orNothing.IfyouhaveNothing,getNothing."

17. MoreMonads17.1.

parentsOf >>= parentsOf >>= siblingsOf 17.2.

employeesOf : person -> list of people clientsOf : person -> list of people bind : list of people, f -> list of people Theruleforapplyingthisexercise'sbindisthesameastheruleforthebindinthevideo:"Applyftoeachpersoninthelist,andthenflattentheresultinglist."

17.3. employeesOf >>= parentsOf

17.4. Definetwofunctions,sqrtMaybeMessandreciprocalMaybeMess. sqrtMaybeMess(4) is Just 2 sqrtMaybeMess(-4) is ErrorBecauseOf -4 reciprocalMaybeMess(5) is Just 1/5 reciprocalMaybeMess(0) is ErrorBecauseOf 0 Theruleforapplyingbindisasfollows:"IfyouhaveJust y,applyftoygettingJust f(y)orErrorBecauseOf y.IfyouhaveErrorBecauseOf y,getErrorBecauseOf y."So,forexample,intheexpressionsqrtMaybeMess(x-7) >>= minus4MaybeMess >>= reciprocalMaybeMess >>= plus13MaybeMesswithx=23,sqrtMaybeMess(23-7) is Just 4 Binding Just 4 with minus4MaybeMess yields Just 0 Binding Just 0 with reciprocalMaybeMess yields ErrorBecauseOf 0 BindingErrorBecauseOf0withplus13MaybeMessyieldsErrorBecauseOf 0

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sqrtMaybeMess : float -> MaybeAFloatWithMessage reciprocalMaybeMess : float -> MaybeAFloatWithMessage bind : MaybeAFloatWithMessage, f -> MaybeAFloatWithMessage