Alligator Eggs!
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Transcript of Alligator Eggs!
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7/19/2015 Alligator Eggs!
http://worrydream.com/AlligatorEggs/ 1/12
Materials
Step1:PrintoutthisPDFontosixorsodifferentcoloredsheetsofpaper.(Evenbetter,photocopyitontocardstock.)
Step2:PrintoutthisPDFontoacouplewhitesheetsofpaper.
Step3:Cutoutthepieces!
Pieces
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7/19/2015 Alligator Eggs!
http://worrydream.com/AlligatorEggs/ 2/12
Thesearehungryalligators:
Hungryalligatorsarehungry.They'lleatanythingthat'sinfrontofthem!Buttheyarealsoresponsiblealligators,sotheyguardtheirfamilies.
Theseareoldalligators:
Oldalligatorsarenothungry.They'veeatenenough.Alltheydoisguardtheirfamilies.
Theseareeggs:
Eggshatchintonewfamiliesofalligators!
Families
Here'sasmallfamily:
Agreenalligatorisguardinghergreenegg.
Here'saslightlybiggerfamily:
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7/19/2015 Alligator Eggs!
http://worrydream.com/AlligatorEggs/ 3/12
Agreenalligatorandredalligatorareguardingagreeneggandaredegg.Or,youcouldsaythatthegreenalligatorisguardingtheredalligator,andtheredalligatorisguardingtheeggs.
Here'sahugefamily!
We'vegotyellow,green,redalligatorsguardingthisfamily.Theyareguardingthreethings:agreenegg,anoldalligator,andaredegg.Theoldalligatorisguardingayelloweggandagreenegg.
Noticethateggsonlyusethecolorsofthealligatorsguardingthem.Youcan'thaveablueeggwithouttherebeingabluealligatoraroundtoguardit.
Eating
Now,thingsaregoingtogetmessy.Herearetwofamilies,sidebyside:
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7/19/2015 Alligator Eggs!
http://worrydream.com/AlligatorEggs/ 4/12
Thatgreenalligatorsureishungry.Andthere'sayellowfamily,rightinfrontofhermouth,anditlookstasty!
Ithinkyouknowwhathappensnext.
Unfortunately,thegreenalligator'seyeswerebiggerthanherstomach.Sheatetoomuch!
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7/19/2015 Alligator Eggs!
http://worrydream.com/AlligatorEggs/ 5/12
Andsoshedies,andgoesofftoalligatorheaven.But,thestorydoesn'tendthere.Becausethegreenalligatordied,thegreeneggstartstohatch...
Amazingly,ithatchesintoexactlywhatthegreenalligatorate.It'sthemiracleoflife!
Now,we'redowntoonefamily.Wehavearedalligatorguardingayellowalligatorandaredegg,andtheyellowalligatorisguardingheryellowegg.
Butthatyellowalligatorsureishungry,andthere'satastyredegginfrontofher.Herewegoagain...
Pooralligator.Evenasingleeggistoomuchforherstomach!
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7/19/2015 Alligator Eggs!
http://worrydream.com/AlligatorEggs/ 6/12
Theyellowalligatordies...butonceagain,theyelloweggstartstohatch...
Andithatchesintoexactlywhattheyellowalligatorate!
Now,there'snothingforanyonetoeat,sowestop.
TheEatingRule
Thatwasanexampleofthefirstruleofthegame:theeatingrule.
Theeatingrulesaysthatiftherearesomefamiliessidebyside...
thetopleftalligatoreatsthefamilytoherright.
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7/19/2015 Alligator Eggs!
http://worrydream.com/AlligatorEggs/ 7/12
Then,thatalligatordies.Butifshewasguardinganyeggsofthesamecolor,eachofthoseeggshatchesintowhatsheate.
TheColorRule
Continuingwiththeexampleabove,theorangealligatoreatstheyellowfamily,leavingthis:
Now,thegreenalligatoratthetopleftwantstoeatthefamilytoherright.Butbeforeshecandoso,weneedtolookatthecolorrule.
Thecolorrulesaysthatifanalligatorisabouttoeatafamily,andthere'sacolorthatappearsinbothfamilies,weneedtochangethatcolorinonefamilytosomethingelse.
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7/19/2015 Alligator Eggs!
http://worrydream.com/AlligatorEggs/ 8/12
Inthepictureabove,greenandredappearinboththefirstandsecondfamilies.So,inthesecondfamily,weswitchallofthegreenstocyan,andalloftheredstoblue.
Nowthattheydon'tshareanycolors,wecaneat!
Andeat!
Andeat!
Andeat,untilthere'snothingmorethatcanbeeaten.
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7/19/2015 Alligator Eggs!
http://worrydream.com/AlligatorEggs/ 9/12
OldAlligators
There'sonlyonemoreruleinthisgame,andithastodowitholdalligators:
Thetopleftalligatorhereisnothungry.She'snotgoingtoeatanything.Allshecaresaboutisherfamily.So,howdoesshedie?
Shedieswhenshe'sonlyguardingonething.Rightnow,she'sguardingbothagreenfamilyandaredfamily.Theyneedhertotakecareofthem.Butthatgreenalligatorishungry.Thegreenalligatoreatstheredfamily...
Now,theoldalligatorisonlyguardingasinglefamily.Thatfamilycantakecareofitself.She'snolongerneeded.So,shegrowsoldanddies.
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7/19/2015 Alligator Eggs!
http://worrydream.com/AlligatorEggs/ 10/12
Thatistheoldagerule.Whenanoldalligatorisjustguardingasinglething,itdies.
Now,theredalligatoreatstheyellowfamily,andthere'snomorethatcanbeeaten.
Gameplay
Thegamewouldconsistofaseriesofpuzzles,challengingtheplayertodeviseafamilythat,whenfedX,producesY.Forexample:
Herearetwofamilies.Theyarenamed"True"and"False":
Andhereisafamilycalled"Not":
When"Not"eats"True",itproduces"False".Andwhen"Not"eats"False",itproduces"True".Whatcolorsshouldthetwoeggsatthe
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7/19/2015 Alligator Eggs!
http://worrydream.com/AlligatorEggs/ 11/12
bottombe?
(Forthistoworkwell,we'dneedabettercolorrule,toexplainthatfamilieswithdifferentcolorsinthesame"pattern"areequivalent.)
Thesepuzzlescouldbeembeddedintoastory.Theplayerwouldhavetosolveeachpuzzletoseewhathappensnext.Alternatively,thiscouldbemadeintoaboardgame.Eachplayerwouldrolladie,travelforwardanumberofspaces,andsolvethepuzzleatthedestination.
Theory
Thisgamerepresentstheuntypedlambdacalculus.Ahungryalligatorisalambdaabstraction,anoldalligatorisparentheses,andeggsarevariables.Theeatingrulecorrespondstobetareduction.Thecolorrulecorrespondsto(overcautious)alphaconversion.Theoldagerulesaysthatifapairofparenthesescontainsasingleterm,theparenthesescanberemoved.
Variations
OliverSteelepointedoutthatfamilyrolesareveryimportanttochildren,andmorecouldbemadeoftheconceptofgrandparentalligators,parents,babies(eggs),andsoon.Currently,variablenamesarerepresentedwithcolors.Botharearbitrary,andrequirerenaming/recoloringrulestoavoidcapture.AnalternativemaybesomethinglikedeBruijnindices,correspondingtobirthorderandfamilyrelation.
Oliveralsosuggestedthatkidsmightnotliketheideaofparentsdying,andperhapsalligatorremovalcouldbeexplainedasmovingaway.
Bychangingthegamesoalligatorsaregreedy,andeateverythingtotheirright,werepresentarightassociativelambdacalculus.Thisseemstoeliminatealotofoldalligators(parentheses)inChurchnumerals,theYcombinator,etc.Unfortunately,itmeansthatoldalligatorshavetobebornwhenanalligatoreatsmorethanonething,amongotherproblems.I'dstillliketofigureouthowtomakeChurchnumerals(repeatedapplication)looklessugly.
SchematicForm
I'vefoundthataschematicformofalligatorcalculusisactuallyratherhandyforcalculatinglambdatermsbyhand.Wedrawalambdaasalinewithamouth.Parenthesesarealinewithoutamouth.Here'stheidentityfunction:
HerearesomeChurchnumerals:
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7/19/2015 Alligator Eggs!
http://worrydream.com/AlligatorEggs/ 12/12
HerearebooleanANDandOR(assumingthestandarddefinitionsforTRUEandFALSE):
TheYcombinator:
Idon'tknowiftheseareanyeasiertoreadthanthestandardnotation,butI'vefoundthemtobeeasiertoworkwith,usingpencilandpaper.Iimaginethetermseatingoneanotherandhatchingbelow.Idon'tgetlostinalongchainofsymbols,losingtrackofwhat'sapplyingtowhat.