the combinatorics of merge and workspace right-sizing

TheCombinatoricsofMergeandWorkspaceRight-Sizing*

*SupportedbyJapanSocietyforthePromotionofScienceGrant-in-AidforScientificResearch#16K02769

Sandiway Fong RobertBerwick JasonGinsburgUniversityofArizona MassachusettsInstituteof

TechnologyOsakaKyoiku University

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Being human: our brains• Popularconception,e.g.,Gazzaniga(2008),pg.8:wecongratulateourselvesonourbrainpower:• "I want to begin by simply

recognizing the huge differences between the human mind and brain and other minds and brains […]"

• Weposit:• Themarvelwecallthehumanbrainisactuallytheweaklinkinourcognitiveapparatus.

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SensoryApparatus

• Oursensoryapparatusprovideswaytoomuchinformationforthebraintoprocess.Thereisamplebiologicalevidenceforthis:• Oureyes:

• havesensitivitydowntothesinglephotonlevel(Tinsleyetal.2016)• havepeakacuityof77cycles/degree(Curcioetal.1990)

• Ournose:• sensitivitycanbeoftheorderofpartsperbillion(ppb)(Wackermannová etal.,2016).

• Ourears:• eardrumscandetectvibrationssmallerthanthediameterofahydrogenatom(Fletcher&Munson,1933).

• Allcompletelyunnecessaryforsurvival.• [Dynamicrangeishuge:yetneuronscanfire…]

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Whysuchsensitivity?

• Evolutionarypressureunknown:• couldbeasimplematterofchemistry

• Anyway:• It'sclear,incaseaftercase,thebraindoesnotmakeuseofthefullresolutionofoursensoryinputs.

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Throwingawayinformation

• Sincewecannotcopewiththesensorydeluge,thebraintossesoutmostofthesensoryinformation:• (attentionpuzzle)howtosievethesensoryinformation?• (dynamicrangepuzzle)yetwecanreportonsinglephotons(abovechance)

• Ourhypothesis:• Samegoesforlanguage,thebrain(must)economizewhereitcan:• Chomsky(2005)termsthepressureforcomputationalefficiencyaThirdFactorconsideration.• ThisappliestothereductionofMergetosimplest(binary)Merge(cf.Komachietal.,2019).

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ResolutioninLanguage

• SimplestMergemayfollowdirectlyfromWorkspace(WS)sizingconstraints.• AlthoughsimplestMerge(byitself)hasdemonstrablyundesirablecombinatorics(e.g.fromaninitialWSofjusttwolexicalitems,about8milliondistinctsetscanbeformedinjust8Merges),languagedoesnotmakefulluseofthisresolution.

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WorkspaceSize

• Merge(Chomsky1995,2013,2015,etc.)• Mergeisfree:

• In{α,{β,ɣ}},ɣ canundergointernalSetMergeorPairMergetoform:• {ɣ,{α,{β,ɣ}}}• <ɣ,{α,{β,ɣ}}>

• IfMergeisfree,thenhowdoyoublockaninfinitenumberofMerges?• Intheory,anySOcanundergointernalorexternalSetMergeorPairMergeaninfinitenumberoftimes.• Intheory,whengeneratingaphrase,youhaveaninfinitenumberofpossiblederivations.

*FromFongandGinsburg(2018) 7

Simple example: the book

# of possible Syntactic Objects as # Merges grows

Log-scale graph of possible SOS as # Merges grows

Free Merge must be constrained.

#Merges #SOs

1 3

2 7

3 29

4 161

5 1,423

6 18,144

7 318,480

8 7,396,976

*FromFongandGinsburg(2018)8

WorkspaceRight-Sizing

• Accessibletermsinaworkspace(NoamChomsky,p.c.;Chomsky2017Readinglecture)• WS=Workspace• SO=SyntacticObject• DefineWSSizeas#SOs+#Accessibleterms

• #accessibletermsofaSO=numberofpropersubsetsoftheSO+lexicalitems• SubjecttoMinimalSearch(lowercopiesnotconsideredaccessible)

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• {a,{b,c}}isaSyntacticObjectSO• dandeareSOs

Workspace

de

#SOs=3#acc.terms=4WSSize=#SOs+#acc.termsWSSize=3+4WSSize=7

Accessibleterms=4

SOs=3

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• ExternalMerge(EM)ofdand{a,{b,c}}#SOs=2#acc.terms=6WSSize=2+6WSSize=8Accessibleterms=4


#SOs=2

ExternalMerge(EM)decreasesthe#ofSOsby1,butincreasesthenumberofaccessibletermsby2.

EMincreasesWSSizeby1

d

ee

#SOs=3#acc.terms=4WSSize=7

#acc.terms=6

SOs=3

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d

• InternalMergeofband{a,{b,c}}

#SOs=3#acc.terms=6WSSize=3+6WSSize=9

#acc.terms=6

SOs=3

InternalMerge(IM)keepsthe#ofSOsthesame,andincreasesthenumberofaccessibletermsby2

IMincreasesWSSizeby2

e d

e


#SOs=3#acc terms=4WSSize=7

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Countercyclic MergeJohnhaseatendinner.l AssumePerfectiveMergescounter-cyclicallywithT [{a,{b,c}},d]à [{a,c},{a,{b,c}},d](Huijbregts 2019)

{a,{b,c}}{a,c}

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Countercyclic Merge

SOs

d


d

WSSizeincreasesby1WSSizeincreaseisokay

Isthereanywaytoblockthis?

acc.terms

a is 1 acc. Termc is 1 acc term 14

Proposal 1: Merge cannot decrease WS Size.• You canʼt remove anything from the WS.

Proposal 2: Copies are not accessible terms.

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d

• InternalMergeofband{a,{b,c}}


#acc.terms

SOs

IMkeepsthe#ofSOsandaccessibletermsthesame.

IMdoesnotchangeWSSize

e d

e


WorkspaceRight-SizingMinimalSearch:excludelowercopiesfromsetofaccessibleterms• Copyofbisindicatedasb

notaccessible16

Countercyclic MergeJohnhaseatendinner.l AssumePerfectiveMergescounter-cyclicallywithT [{a,{b,c}},d]à [{a,c},{a,{b,c}},d](Huijbregts 2019)

{a,{b,c}}{a,c}

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#SOs=3#acc.terms=2WSSize=#SOs+#Acc.WSSize=5

Countercyclic Merge

#SOs=3

d

acc.terms


d

WSSizedecreasesThisisn’tallowed

notaccessible

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#SOs=3#acc.terms=2WSSize=#SOs+#Acc.WSSize=5

SidewardMovement

#acc.terms=2


WSSizedecreasesThisisn’tallowed

notaccessible

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[{a,b},{c,d}]—>[{a,c},{a,b},{c,d}](Huijbregts 2019)• Notsure if examples likethis exist,butthey stillneed to be blocked.

SidewardMovement• RelativeClauseadjunction(followingNunes 2001)

• WhichclaimthatJohnmadewashewillingtodiscuss?(Chomsky1993:36,Nunes2001:316)

[{a,b},c,d]à [{a,c},{a,b},d](Huijbregts 2019){a,b}

{a,c}

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[CP washewillingtodiscuss[whichclaim]1][CP [whichclaim]1[OP2 thatJohnmadeOp2]]

• SidewardMerge


c d

d


#SOs=3#acc.terms=2WSSize=5 NotaccessiblebecauseitisanSO

notaccessible

Give Example phrase

WSSizeremainsthesameSimilartoIM• Notethatthismaybeaproblem• TheWSSizedoesn’tdecrease.• Thisisaproblemifwewanttoban

sidewardmovement

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Conclusion

• UnconstrainedSimplestMergeleadstoanundesirablecombinatorialexplosioninthenumberofpossibleSyntacticObjects.

• ResearchQuestion:whatarethewell-motivatedconstraintsthatdon'tinvolvenewmachinery(aproblemforevolution)?

• PossibleAnswer:thereisnonewmachinery,onlythesimplestpossibleboundonWSSize,i.e.,WSSizemustnotdecreaseandWSSizemustnotexpandbymorethan1.

Furtherconsequences(notdiscussedhere):

• Canreplaceotherconstraintssuchasonvacuousmovement(e.g.iteratedIM)– nowcanbeblockedbyWSSizenarrowlimits– normallywouldrequireloopdetectionmachinery.

• CanalsoexplainwhylanguagecannotcountdespitethefreeavailabilityofIM,i.e.thesuccessorfunctionissimplyaloop

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ReferencesChomsky,Noam.(1993).AMinimalistprogram for linguistic theory.InTheview from Building20:Essaysinlinguistics inhonor of SylvainBromberger,ed.By KennethHaleand SamuelJayKeyser,1-52.Cambradige:MA:MITPress.

Chomsky,Noam.(2005).Three Factors inLanguageDesign.Linguistic Inquiry,36,1–22

Chomsky,Noam.(2017)Lecture atUniversityof Reading.

Curcio,C.A.,Sloan,K.R.,Kalina,R.E.,&Hendrickson,A.E.(1990).HumanPhotoreceptor Topography.TheJournalof Comparative Neurology,292,497–523.

Fong,Sandiwary,&Ginsburg,Jason(2018).Onconstraining FreeMerge.The43rdMeetingof the Kansai Linguistics Society.KonanUniversity:Kobe,Japan.

Fletcher,H.and Munson,W.A.(1933).Loudness,its definition,measurement and calculation.TheJournalof the Acoustical Societyof America,5,82–108.

Gazzaniga,M.(2008).Human:TheScienceBehindWhat Makes Us Unique.NewYork,HarperCollins.

Huijbregts,Riny.(2019)Specialand general theory of Merge with special and general thanks to Noam.SyntaxInterfaceLectures2019,Utrecht.

Komachi,M.,Kitahara,H.,Uchibori,A.,&Takita,K.(2019).Generativeprocedure revisited.Reportsof the Keio Instituteof Culturaland Linguistic Studies,50,269–283.

Nunes,Jairo.(2001).SidewardMovement.LinguisticInquiry,32,303-344.

Tinsley,J.N.,Molodstov,M.U.,Prevedel,R.,Wartmann,D.,Espigulé-Pons,J.,Lauwers,M.,&Vaziri,A.(2016).Direct detection of asingle photon by humans.NatureCommunications, 7,12172.

Wackermannová,M.,Pinc,L.,&Jebavý,L.(2016).Olfactory sensitivity inmammalian species.PhysiologicalResearch,65,369-390.

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the combinatorics of merge and workspace right-sizing

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