7/19/2015 Alligator Eggs!

http://worrydream.com/AlligatorEggs/ 1/12

Materials

Step1:PrintoutthisPDFontosixorsodifferentcoloredsheetsofpaper.(Evenbetter,photocopyitontocardstock.)

Step2:PrintoutthisPDFontoacouplewhitesheetsofpaper.

Step3:Cutoutthepieces!

Pieces

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:

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:

7/19/2015 Alligator Eggs!

http://worrydream.com/AlligatorEggs/ 4/12

Thatgreenalligatorsureishungry.Andthere'sayellowfamily,rightinfrontofhermouth,anditlookstasty!

Ithinkyouknowwhathappensnext.

Unfortunately,thegreenalligator'seyeswerebiggerthanherstomach.Sheatetoomuch!

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!

7/19/2015 Alligator Eggs!

http://worrydream.com/AlligatorEggs/ 6/12

Theyellowalligatordies...butonceagain,theyelloweggstartstohatch...

Andithatchesintoexactlywhattheyellowalligatorate!

Now,there'snothingforanyonetoeat,sowestop.

TheEatingRule

Thatwasanexampleofthefirstruleofthegame:theeatingrule.

Theeatingrulesaysthatiftherearesomefamiliessidebyside...

thetopleftalligatoreatsthefamilytoherright.

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.

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.

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.

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

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:

7/19/2015 Alligator Eggs!

http://worrydream.com/AlligatorEggs/ 12/12

HerearebooleanANDandOR(assumingthestandarddefinitionsforTRUEandFALSE):

TheYcombinator:

Idon'tknowiftheseareanyeasiertoreadthanthestandardnotation,butI'vefoundthemtobeeasiertoworkwith,usingpencilandpaper.Iimaginethetermseatingoneanotherandhatchingbelow.Idon'tgetlostinalongchainofsymbols,losingtrackofwhat'sapplyingtowhat.