Who's Afraid of George Kingsley Zipf?

16 Pages·2009·0.14 MB·English

Who’s Afraid of George Kingsley Zipf?
CharlesYang∗
DepartmentofLinguistics&ComputerScience
UniversityofPennsylvania
[email protected]
June2009
Abstract
WeexplorethestatisticaldistributionsofnaturallanguageknownasZipf’slawandde-
velopanewapproachtoassessthepropertiesoftheunderlyinggrammargivenasample
oflinguisticproduction. Focusingonlanguageacquisition,weshowthattheitemorusage
basedapproachtolanguageandlanguagelearningfailstoprovideadequatestatisticaltests
oflinguisticproductivity,andthatevenveryyoungchildren’sgrammarisabstract,system-
atic,andfullygenerative.
1 Introduction
Einsteinwasaverylatetalker.Aslegendhasit,thefirstthingtheyoungEinsteineveruttered,at
theageofthree,was“Thesoupistoohot”.Apparentlytheboygeniushadnothinginterestingto
saybeforethat.
The credibility of such tales aside — similar stories with other famous subjects abound —
theydocontainakerneloftruth: achilddoesn’thavetosaysomething,anything,justbecause
he can. And this poses a challenge for the study of child language, when children’s linguistic
productionisoftentheonly,andcertainlythemostaccessible,dataonhand.Languageuseisthe
compositeoflinguistic,cognitiveandperceptualfactorsmanyofwhich,inthechild’scase,are
stillintheprocessofdevelopmentandmaturation;anditisdifficulttodrawinferencesaboutthe
learner’slinguisticknowledgefromhislinguisticbehavior.Thismuchhasbeenwellappreciated
eversinceChomsky[1]drewthecompetence/performancedistinction.Thepioneeringworkon
childlanguagethatsoonfollowed,mostnotablyRogerBrown’slandmarkstudy[2],recognized
such potential gaps between what the child knows and what the child says and proposed to
interpretchildlanguageintermsofadult-likegrammaticaldevices. (See,inparticular,Brown’s
∗Forhelpfulcomments,IwouldliketothankVirginiaValian,JulieAnneLegate,SamGutmann,BobFrank,Mark
Libermanandtheaudienceatthe2009SchultinkLecture,theUniversityofGroningen,wherethesematerialswere
firstpresented.SpecialthankstoErwinChanandCostasLignosfortheirhelpwithTable3andFigure4.
1
critiqueofthePivotGrammarhypothesis[3],bearsmorethanapassingresemblancewithsome
contemporarytheorizingofchildlanguagereviewedhere.)
This tradition is now challenged by the item or usage-based approach to language [4–7].
This change reflects a current trend in the study of language, as can be seen in these pages
[8–12]), which emphasizes the storage of specific linguistic forms and constructions at the ex-
pense of general combinatorial linguistic principles and overarching points of language varia-
tion[1,13]. Childlanguage,especiallyintheearlystages,isclaimedtoconsistofspecificitem-
basedschemas,ratherthanproductivelinguisticsystemaspreviouslyconceived. Consider,for
instance,threecasestudiesinBox1[5],whichhavebeencitedasevidencefortheitem-based
viewatnumerousplaces.
Box1:ProductionEvidenceforItemBasedApproachtoLanguageLearning
• TheVerbIslandHypothesis[4].Inalongitudinalstudyofearlychildlanguage,itisnoted
that“ofthe162verbsandpredicatetermsused,almosthalfwereusedinoneandonly
oneconstructiontype,andovertwo-thirdswereusedineitheroneortwoconstruction
types ...”. Hence, “the 2-year-old child’s syntactic competence is comprised totally of
verb-specificconstructionswithopennominalslots”,ratherthanabstractandproduc-
tivesyntacticrules.
• Limitedmorphologicalinflection. AstudyofchildItalian[14]findsthat47%ofallverbs
used by 3 children (1;6 to 3;0) were used in 1 person-number agreement form, and an
additional40%wereusedwith2or3forms,wheresixformsarepossible(3person×2
number).Only13%ofallverbsappearedin4ormoreforms.
• Unbalanceddeterminerusage.[15]notesthatwhenchildrenbegantousethedetermin-
ers a and the with nouns, “there was almost no overlap in the sets of nouns used with
the two determiners, suggesting that the children at this age did not have any kind of
abstractcategoryofDeterminersthatincludedbothoftheselexicalitems”. Thisobser-
vationisheldtocontradicttheearlieststudy[16]whichmaintainsthatchilddeterminer
useisadult-likebytheageof2;0.
Sofaraswecantell,however,thesepurportedevidenceforitem-basedlearningisgivenen-
tirelyonthebasisofinformalinspectionsratherthanrigorousempiricaltests. Forallexamples
and observations from child language, not a single statistical test can be found in Tomasello’s
work[4]wheretheVerbIslandHypothesisandrelatedideasaboutitem-basedlearningarefirst
putforward.Ingeneral,theoreticalconclusionsareonlyconvincingwhenthenulloralternative
hypothesiscanbestatisticallyrejected; thatis,theobservationinBox1beshowntobeincon-
sistentwiththeexpectationfromafullyproductivegrammar.Inthisnote,weprovidestatistical
analysisofwhatsuchalternativehypothesiswouldbe.Wedemonstratethatchildren’slanguage
useshowsexactlytheoppositeoftheitem-basedview,andthehypothesisofearlyproductivityis
2
infactsupported. Morebroadly,wedirectcognitivescientiststocertainstatisticalpropertiesof
naturallanguagethatarewidelyknownbutnotwidelyappreciated,andtodiscussthechallenges
theirpropertiesposeforthetheoryoflanguageandlanguagelearning.Ourpointofdepartureis
anamethatoughttostrikefearineverylivingsoul:GeorgeKingsleyZipf.
2 ZipfianPresence
Undertheso-calledZipf’slaw[17],thedistributionsofwordsfollowacuriouspattern:relatively
fewwordsareusedfrequently—veryfrequently—whilemostwordsarerare,withmanyoccur-
ringonlyonceinevenlargesamplesoftexts.Moreprecisely,thefrequencyofawordtendstobe
approximatelyinverselyproportionaltoitsrankinfrequency.Let f bethefrequencyoftheword
w withtherankofr inasetofN,then:
C
f = whereC issomeconstant
r
IntheBrowncorpus[18],forinstance,thewordwithrank1is“the”,whichhasthefrequency
of about 70,000, and the word with rank 2 is “of”, with the frequency of about 36,000: almost
exactlyasZipf’slawentails.TheZipfiancharacterizationofwordfrequencycanbevisualizedby
plottingthelogofwordfrequencyagainstthelogofwordrank.Bytakingthelogonbothsidesof
theequationabove(logf = constant −logr),aperfectZipfianfitwouldbeastraightlinewith
theslope-1. Indeed, Zipf’slawhasbeenobservedinvocabularystudiesacrosslanguagesand
genres, andthelog-logslopefitisconsistentlyinthecloseneighborhoodof-1.0[19]. Thetop
lineinFigure1plotswordrankandfrequencyonalog-logscalebasedontheBrowncorpus:the
Zipfianfitisexcellent.
Top:words,Bottom:pseudowords
12
10
8
log(freq) 6
4
2
0
0 2 4 6 8 10 12
log(rank)
3
Figure1.ZipfiandistributionofwordsandpseudowordsintheBrowncorpus[18].Thelower
lineisplottedbytaking“words”tobeanysequenceoflettersbetweene’s[23].Thetwostraight
dottedlinesarelinearfunctionswiththeslope-1,whichillustratethegoodnessoftheZipfian
fit.
TherehasbeenagooddealofcontroversyovertheinterpretationofZipf’slaw,whichshows
up not only in the context of words but also other physical and social systems. It is now clear
that the observation of Zipfian distribution alone is of no inherent interest or significance, as
certain random generating processes can produce outcomes that follow Zipf’s law [20–22]. As
notedin[23],ifweredefine“words”asalphabetsbetweenanytwooccurrencesofsomeletter,
say, “e”, ratherthanspaceasinthecaseofwrittentext, theresultingdistributionmayfitZipf’s
lawevenbetter.ThisisillustratedbythelowerlineinFigure1,whichfollowstheZipfianstraight
lineatleastaswellasrealwords.
Itisoftenthecasethatwearenotparticularlyconcernedwiththeactualfrequenciesofwords
buttheirprobabilityofoccurrence. Zipf’slawallowustoestimatetheprobabilityp oftheword
n,whoserankisr amongN wordsinalinguisticsample:
C N
1 (cid:88)1
p = r = whereH istheNthHarmonicNumber (1)
(cid:80)N C rH N i
i=1 i N i=1
Zipf’slawasappliedtothedistributionofwordshasbeenwellknownandstudied. Yetrela-
tivelylittleattentionhasbeengiventothecombinatoricsofwordsunderagrammarandmore
important, how one might draw inference about the grammar given the distribution of word
combinatorics.Weturntothesequestionsimmediately.
3 TheUnbearableLightnessofProductivity
Claimsofitem-basedlearningareestablishedontheassumptionthatlinguisticproductivityen-
tails usage diversity in linguistic production. Take the notion of “overlap” in the case of deter-
mineruseinearlychildlanguage(Box1),followsthelogicoftheVerbIslandhypothesis[4]. If
the child has fully productive use of the syntactic category determiner, then one might expect
her to use determiners with any noun for which they are appropriate. Since the determiners
“the” and “a” have (virtually) identical syntactic distributions, a linguistically productive child
thatuses“a”withanounisexpectedtoautomaticallytransfertheuseofthatnounto“the”.Thus,
determiner-nounoverlapisdefinedasthepercentageofnounsthatappearswithbothdetermin-
ersoutofthosethatappearwitheither. Thelowoverlapsinchildren’sdetermineruse[15]are
takentosupporttheitem-basedviewofchildlanguage. However,usingasimilarmetric,Valian
andcolleagues[24]findthattheoverlapmeasuresforyoungchildrenandtheirmothersarenot
significantly different, and they are both very low. Indeed, when applied to the Brown corpus
(seeBox3formethods),weobtainanoverlap25.2%,whichisactuallylowerthanthoseofsome
childrenreportedin[15]. ItwouldfollowthatthelanguageoftheBrowncorpus, whichdraws
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fromvariousgenresofprofessionalprintmaterials,islessproductiveandmoreitem-basedthan
thatofatoddler—whichseemsabsurd.
The reason for these seemingly paradoxical findings lies in the Zipfian distribution of syn-
tacticcategoriesandthegenerativecapacityofnaturallanguagegrammar.Considerafullypro-
ductiverule“DP→DN”,where“D→ a|the”and“N→ cat|book|desk|...”. Weusethisruleforits
simplicityandforthereadilyavailabledataforempiricaltestsbutonecaneasilysubstitutethe
rulefor“VP→VDP”,“VP→VinConstruction ”,“V →V +Person+Number+Tense”.
x inflection stem
Allsuchcasescanbeanalyzedwiththemethodsprovidedhere.
Suppose a linguistic sample containsS determiner-noun pairs, which consist of D and N
distinctdeterminersandnouns. (InthepresentcaseD =2for“a”and“the”.) Thefullproduc-
tivityoftheDPrule,bydefinition,meansthatthetwocategoriescombineindependently. Two
observationscanbemadeaboutthedistributionsofthetwocategoriesandtheircombinations.
First,nouns(andopenclasswordsingeneral)willfollowzipf’slaw:nounsintheBrowncorpus,
forinstance,showsalog-logslopeof-0.97(seeBox3formethods).Thismeansthatinanygiven
sample, relatively few nouns occur often but most can be expected to occur only once or less
thanonce,whichnecessarilyhavezerooverlapwithdeterminers.
Second,whilethecombinationofD andN issyntacticallyinterchangeable,N’stendtofa-
vor one of the two determiners, a consequence of linguistic pragmatics and conventions. For
instance, we say “the bathroom” more often than “a bathroom” but “a bath” more often than
“the bath”, even though all four DPs are perfectly grammatical. As noted earlier, about 75% of
nounsintheBrowncorpusoccurwitheither“the”or“a”butnotboth. Eventheremaining25%
which do occur with both show strong biases: only a further 25% (297) are used with “a” and
“the”equallyfrequently. Overall,fornounsthatappearwithbothdeterminersasleastonce,the
frequencyratiobetweenthemoreoverthelessfavoreddetermineris2.86:1. Thesegeneralpat-
ternsholdforchildandadultspeechdataaswell. Inthesixchild-adultpairs(thus12samples)
we examined from the CHILDES database (Box 3), the average percentage of balanced nouns
amongthosethatappearwithboth“the”and“a”is22.8%,andthemorefavoredvs. lessfavored
determinerhasanaveragefrequencyratioof2.54:1. Eventhoughtheseratiosdeviatefromthe
perfect2:1ratiounderthestrictversionofZipf’slaw—themorefavoredisevenmoredominant
overthe less—theyclearly pointout theconsiderable asymmetryin categorycombination us-
age. Asaresult,evenwhenanounappearsseveraltimesinasample,thereisstillasignificant
chancethatithasbeenpairedwithasingledeterminerinallinstances.
Together,theseconsequencesofZipf’slawensurethattheaveragedeterminer-nounoverlap
mustberelativelylowunlessthesamplesizeS isverylarge. Box2givesthetheoreticalanalysis
ofoverlapinformallysketchedabove,andFigure2givesanillustration.
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Box2.CalculatingExpectedOverlapinDeterminerNounUsage
LetO(N,S) be the overlap value of N nouns in a sample S pairs of determiner-noun pairs.
Consideranounnwhichhastherankofr outofN.Followingequation(1),ithasaprobability
ofp =1/(rH )ofbeingdrawnatanysingletrialinS;itsexpectedoccurrenceinSisthussimply
N
(Sp). Itsexpectedprobabilityofbeingusedwithmorethan1determinersis1-itsexpected
probability of n being used with exactly 1 determiner in all of the (Sp) trials. Obviously, if a
nounisexpectedtobesampledonceorless,itwillhaveanoverlapofzero. Lettheexpected
overlapofn beO(r,N,S).
N
1 (cid:88)
O(N,S)= O(r,N,S) (2)
N
r=1
(cid:40)1−(cid:80)D d(Sp) ifSp >1whered = 1 ,p = 1
O(r,N,S)= i=1 i i iHD rHN (3)
0 otherwise
Theprobabilityofdeterminersd (i =1,2)in(3)alsofollowsZipf’slaw(1), i.e., d =2/3and
i 1
d =1/3.a
2
aAlthoughtheempiricalfrequenciesofdeterminersdeviatesomewhatfromthestrictZipfianratioof2:1,numer-
icalresultsshowthatthe2:1ratioisaveryaccuratesurrogateforawiderangeofactualrationsinthecalculationof
(2)and(3). Thisisbecausemostofaverageoverlapvaluecomesfromtherelativelyfewandhighfrequentnouns;
seeFigure2.
1 ××
×
0.9 ×
×
0.8
×
0.7 ×
×
0.6
×
Expected 0.5 ×
Overlap ×
0.4 ×
×
0.3 ×
×
0.2 ×
×
×
0.1 ×
×
×
0 ×××××××××××××××××××××××××××××
0 5 10 15 20 25 30 35 40 45 50
Rank
Figure2.Expectedoverlapvaluesfornounsorderedbyrank,forN =50nounsinasamplesizeof
6
S=100.WordfrequenciesareassumedtofollowtheZipfiandistribution.Ascanbeseen,fewofnouns
havehighprobabilitiesofoccurringwithbothdeterminers,butmostare(far)belowchance.Theaverage
overlapis20.6%.
Wenowstudydeterminer-nounoverlapinchildlanguageandcomparetheresultswiththe
theoretical expectations. The empirical methods for data analysis are given in Box 3, and the
resultsaresummarizedinTable1.
Box3.EmpiricalStudiesofOverlapinLanguageProduction
a. WeconsiderthedatafromAdam,Eve,Sarah,Naomi,Nina,andPeter[25]. Thesearetheall
andonlychildrenintheCHILDESdatabasewithsubstantiallongitudinaldatathatstartsat
theverybeginningofsyntacticdevelopment(i.e,oneortwowordstage)sothattheitem-
basedstage,ifexists,couldbeobserved.
b. We first removed the extraneous annotations from the child text and then applied
an open source implementation of a rule-based part-of-speech tagger [26] (available
http://gposttl.sourceforge.net/): wordsarenowassociatedwiththeirpart-of-speech((e.g.,
preposition,singularnoun,pasttenseverbetc.). ForlanguagessuchasEnglish,whichhas
relativelysalientcuesforpart-of-speech(e.g.,rigidwordorder,lowdegreeofmorphologi-
calsyncretism),suchtaggerscanachievehighaccuracyatover97%,whichissufficientfor
our purposes. For comparison, we also consider the Brown corpus [18], which has been
manuallytagged.
c. WithPOStaggeddatasets,weextractedadjacentdeterminer-nounpairssuchasD iseither
“a”or“the”,andN hasbeentaggedasasingularnoun. Wordsthataremarkedasunknown
asdiscarded.Asisstandardinchildlanguageresearch,repetitionscountsonlyoncetoward
the tally. For instance, when the child says “I made a queen. I made a queen. I made a
queen”,“aqueen”iscountedonceforthesample.
d. Foranadditionaltest,wehavepooledtogetherthefirst100,300,and500determiner-noun
tokens of the six children and created three hypothetical children from the very earliest
ageoflanguageacquisition,whichwouldpresumablybetheleastproductiveknowledgeof
determinerusage.
e. Foreachlearner,thetheoreticalexpectationofoverlapiscalculatedbasedonequationsin
Box2,thatis,onlywiththesamplesizeSandthenumberofdistinctnounsN indeterminer-
nounpairs.
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Sample a&theNoun aortheNoun Overlap Overlap S
Child
Size(S) types types(N) (expected) (empirical) N
Naomi(1;1-5;1) 884 60 349 19.0 19.8 2.53
Eve(1;6-2;3) 831 61 283 22.7 21.6 2.94
Sarah(2;3-5;1) 2453 187 640 26.4 29.2 3.83
Adam(2;3-4;10) 3729 252 780 32.0 32.3 4.78
Peter(1;4-2;10) 2873 194 480 43.0 40.4 5.99
Nina(1;11-3;11) 4542 308 660 47.2 46.7 6.88
First100 600 53 243 19.6 21.8 2.47
First300 1800 141 483 26.7 29.1 3.73
First500 3000 219 640 32.3 34.2 4.68
Browncorpus 20650 1175 4664 23.8 25.2 4.43
Table1.Empiricalandexpecteddeterminer-nounoverlapsinchildspeech.TheBrowncorpus
isincludedforcomparison.Resultsincludethedatafromsixindividualchildrenandthefirst
100,300,500determiner-nounpairsfromallchildrenpooledtogether,whichreflecttheearliest
stagesoflanguageacquisition.Theexpectedvaluesincolumn5arecalculatedusingonlythe
samplesizeSandthenumberofnounsN (column2and4respectively),followingtheanalytic
resultsinBox2.
Thetheoreticalexpectationsandtheempiricalmeasuresofoverlapagreeextremelywell(col-
umn5and6inTable1). Neitherpairedt-testnorpairedWilcoxontestshowsignificantdiffer-
encebetweenthetwosetsofvalues. Perhapsamorerevealingtestislinearregression:aperfect
agreement between the two sets of value would have the slope of 1.0, and the actual slope is
1.08(adjustedR2 =0.9705). Inotherwords, thedeterminerusagedatafromchildlanguageis
consistentwiththeproductiverule“DP→DN”.
Theempiricalstudiesalsorevealconsiderableindividualvariationintheoverlapvalues,and
itisinstructivetounderstandwhy. AstheBrowncorpusresultsshow(Table1lastrow),sample
sizeS,thenumberofnounsN,orthelanguageuser’sagealoneisnotpredictiveoftheoverlap
value. Thevariationcanbeformallyanalyzed. GivenN nounsinasampleofS,greateroverlap
valuewillbeobtainedwhenmorenounsareexpectedtooccurmorethanonce,orSp >1. That
is,wordswhoseoccurrenceprobabilitiesthataregreaterthan1/S cancontributetotheoverlap
value; Zipf’s law allows us to express this probability cutoff line in terms with ranks, following
equation (1). The approximation below uses a well-known result from Euler’s summation for-
mula.
1
S =1
rH
N
S S
r = ≈ (4)
H lnN
N
Thatis,onlynounswhoseranksarelowerthanS/(lnN)canbeexpectedtobenon-zerooverlaps.
ThetotaloverlapisthusamonotonicallyincreasingfunctionofS/(NlnN)which,giventheslow
growth of lnN, is approximately S/N, a term that must be positively correlated with overlap
8
measures.Thisisconfirmedinstrongestterms:S/N isanearperfectpredictorfortheempirical
valuesofoverlap(lasttwocolumnsofTable1):r =0.986,p <0.00001.
Wenowbrieflyexplorethequestionwhetherthedeterminerusagedatabychildrencanbe
accounted for by the item based approach to language learning. Our effort is hampered by
thelackofconcretemodelsfortheitem-basedlearningapproach,apointthatTomasellocon-
cedes [4, p274]. Analytical results like those in Box 2 cannot be similarly obtained. A plausi-
ble approach can be construed based on a central tenet of item-based learning, that the child
doesnotformgrammaticalgeneralizationsbutrathermemorizesspecificanditemizedcombi-
nations. Similarapproachessuchasconstructiongrammar[10],usage[27]andexemplarbased
models [28] make similar commitment to the role of verbatim memory. To this end, we con-
sider a type of learning model that memorizes determiner-noun pairs in the input, and these
pairs are then sampled jointly, following the commitment of item-based learning, rather than
independently(whichwouldbetheproductiverule-basedview).
Sample Overlap Overlap Overlap
Child
Size(S) (BIGlearner) (smalllearner) (empirical)
Eve 831 16.0 17.8 21.6
Naomi 884 16.6 18.9 19.8
Sarah 2453 24.5 27.0 29.2
Peter 2873 25.6 28.8 40.4
Adam 3729 27.5 28.5 32.3
Nina 4542 28.6 41.1 46.7
First100 600 13.7 17.2 21.8
First300 1800 22.1 25.6 29.1
First500 3000 25.9 30.2 34.2
Table2.Isthefullproductivitydatainchildlanguageconsistentwithitem-basedlearning?Two
variantsoflearnersareconsidered.Onetype,theBIGlearner,isdesignedtomimicthelong
termcommitmenttomemory;itstoresalargesetofdeterminer-nounpairs,whichconsistsof
asampleof1.1millionchilddirectedutterancesfromtheCHILDESdatabase(methodsas
describedinBox3).Theothervariant–thesmalllearner–onlymemorizestheadultutterances
presentineachchild’stranscript.Forbothlearners,wedrawanindependentandrandom
samplefromthesestoredD-Npairswithrespecttotheirjointempiricalfrequencies;thisis
contrastedwiththerule-basedmodelinwhichDandNaredrawnindependently.Thesample
sizematchesthoseineachchild’sproduction(Table1,column2).Theoverlapvaluesarethen
calculatedasthepercentageofnounsthatappearwithboth“a”and“the”overthosethat
appearwitheither.TheresultsaregiveninTable2,averagedover1000trialsperchild.
Bothsetsofoverlapvaluesfromitem-basedlearning(column3and4)aresignificantlydif-
ferent from the empirical measures (column 5): p < 0.005 for both paired t-test and paired
Wilcoxon test. This suggests that children’s use of determiners do not follow the predictions
oftheitem-basedlearningapproach.Naturally,ourevaluationhereistentativesincetheproper
test can be carried out only when the theoretical predictions of item-based learning are made
9
clear.Andthatisexactlythepoint:theadvocatesofitem-basedlearningnotonlyrejectedtheal-
ternativehypothesiswithoutadequatestatisticaltests,butalsoacceptedthefavoredhypothesis
withoutadequatestatisticaltests. Intuitionisnosubstitutefortheoreticalanalysisorstatistical
validation.
4 AnItemizedLookatVerbs
Theformalanalysisinsection3canbegeneralizedtothestudyofchildverbsyntaxandmor-
phology(Box1).Unfortunately,theacquisitiondatainsupportoftheVerbIslandHypothesis[4]
andtheitem-basednatureofearlymorphology[14]isnotavailableinthepublicdomain.
ButthereisnoescapefromtheZipfiangrasp:thecombinatoricsofverbsandtheirmorpho-
logicalandsyntacticassociatesaresimilarlylopsidedintheirusagedistributionasinthecaseof
determiners. ConsiderfirstthekindofverbalsyntaxdistributionsattributedtotheVerbIsland
Hypothesis.Wefocusonconstructionsthatinvolveatransitiveverbanditsnominalobjects,in-
cludingpronounsandnounphrases.Followingthedefinitionof“sentenceframe”inTomasello’s
original Verb Island study [4, p242], each unique lexical item in the object position counts as
a unique construction for the verb. Figure 3 shows the construction frequencies of the top 15
transitiveverbsin1.1millionchilddirectedutterances.
8
×
7.5
7
×
6.5
×
log(freq) 6
×
×
5.5
×
5 ×
×
4.5 ×
×
4
0 0.5 1 1.5 2 2.5
log(rank)
Figure3.Rankandfrequencyofverb-objectconstructionsbasedon1.1millionchild-directed
utterances.ProcessingmethodsareasdescribedinBox3exceptherewefocusonadjacentverb-
nominalpairsinpart-of-speechtaggedtexts. Theverbsarethetop15mostfrequenttransitive
verbs: put,tell,see,want,let,give,take,show,got,ask,makeeat,like,bring andhear. Foreach
verb,wecounteditstop10mostfrequentconstructions,whicharedefinedastheverbfollowed
auniquelexicalitemintheobjectposition(e.g.,“askhim”and“askJohn”aredifferentconstruc-
tions.) For each of the 10 ranks, we tallied the construction frequencies for all 15 verbs: the
10