when does variation lead to change? a dynamical...

When does variation lead to change? A dynamical systemsaccount of an English stress shift

Partha Niyogi Morgan Sonderegger{niyogi, morgan}@cs.uchicago.edu

Department of Computer Science, University of Chicago

Introduction

I Phonetic variation extensive, yetusually does not lead to change. Howdo V & C coexist?

Actuation problem (restated):1. Why does language change occur?2. Why does it arise from variation?3. What determines whether variation is

stable (→ change) or not?Two approaches to language change:

1. Diachronic datasets: Historicallinguists (Wang), sociolinguists(Labov)

2. Math. models of linguisticpopulations: Dynamical systems forlanguage [4, 3]

Here use appropriate models (2) tounderstand empirically observedtrends (1).

Claim: Bifurcations in linguistic systems are possible explanation for theactuation problem: Long-term stability and sudden change coexist when dyn.

sys. models contain bifurcations, correspond to learners with “ambiguity”.

Background: Dynamical systems

I Describe system state α at t + 1 asfunction of state at t .

I Fixed points: αt+1 = αt, can bestable or unstable (when systemperturbed).

I Ex: Pendulum has 1 of each.

I Bifurcations: stability of f.p.(s)changes suddenly as systemparameter changes continuously.

I e.g. phase transitions in physics: astemp passes 100◦ C, stable state ofwater liquid→ gas.

Stress in English N/V pairs

I English 2-syllable noun/verb pairshave variable stress:

N V(1, 1) σ́σ σ́σ (elbow, fracture)(1, 2) σ́σ σσ́ (consort, protest)(2, 2) σσ́ σσ́ (police, review)

I Ongoing variation: perfume,research, ally...

Variation within individuals: NationalPublic Radio (US):

Word 1 only 2 only Var Spkrsresearch (N) 0.53 0.12 0.35 17perfume (N) 0.22 0.44 0.33 9address (N) 0.4 0.4 0.2 5

Data collection

I (1, 1), (1, 2), (2, 2) are stable states:In random subset (List 2), most donot change over time.

I But some do (Sherman [5]): ≈ 149pairs (List 1) changed since 1600,pron. data from dictionaries =⇒lexical diffusion to (1,2)?

Dataset:I 149 words (List 1), 76 dictionaries,

1550–2007 (most 1700 on).I Sherman (1550-1800), MS

(1800-2007) collected.I British and American, only British

used here.

Pronunciation trajectories

I Plot moving average of N, V prons ofeach pair, 50 year window.

I Complete changes observed:I (1, 1)→ (1, 2) (A)I (1, 2)→ (1, 1) (B)I (2, 2)→ (1, 2) (C)I (1, 2)→ (2, 2) (D)

I Short-term variation common (E),long-term variation rare (F).

I (2, 1) never occurs.I Multidirectional diffusionI What causes sudden loss of stability?

A B C

D E F

What is driving change?

Mishearing [2]:I (English) N occur more often than V

in trochaic-biasing (σ́σ) contextsI Biases perception:

N→ σ́σ, V→ σσ́

Prefix similarity:I Similar trajectories for words sharing

a prefix, e.g. con- .I Effect stronger for larger classes.I Low-frequency words in a class

change first, not true cross-class.

Model notation

Dynamical systems models, assume:I Infinite populationI Discretized generations: gen t + 1

learns from gen t .I For each N/V pair, each speaker

keeps:

α̃, β̃ ∈ [0, 1] = prob of producing the 2form.I Let αt = probability N example at t

produced as 2, βt same for V.I Mishearing probs a1, b1, a2, b2:

a1 = P(N heard as 1 | 2 intended)

Model 1: No ambiguity

I Batch learner: at t hears N1 noun examples, N2 verb examplesI Hears K1 N examples as 2, K2 V examples as 2, then sets

α̃ =K1

N1, β̃ =

K2

N2

I Then take expectations: αt+1 = E(α̃), βt+1 = E(β̃)I Find fixed points: (αt+1, βt+1) = (αt, βt), get

α∗ =b1

a1 + b1, β∗ =

b2

a2 + b2I Unique stable N,V freqs, depend on ai/bi ratios. Long-term stability, but

doesn’t explain sudden change.

Model 2: With ambiguity

I Try another error type: no mishearing, but each example can be heard as 1, 2or ambiguous =⇒ discarded.

I Let ri = P(heard as ambiguous | i intended) (i = 1, 2)I Learner hears K1, K2, N − K1 − K2 as 1,2, ambig, sets

α̃ =K2

K1 + K2if K1 + K2 > 0

(and 1/2 if K1 + K2 = 0)I For large N , find

αt+1 =α(1− r2)

(1− r1) + α(r1 − r2)I Get fixed points

x+ = 1 stable for r1 > r2 x− = 0 stable for r1 < r2

I Bifurcation at r1 = r2, sudden change explained as loss of stability of a f.p.

Variations

I Frequency effects: Make N finite, getbifurcation-like behavior,low-frequency words change first.

I Prefix effects: Couple α, β variablesfor words with same prefix.

I S-shaped curves: Mixture ofambiguity and mishearing, R=%errors mishearing.

I Determines how bifurcation-like curveis.

Conclusions

I Observe sudden change in N/Vstress between multiple statesfollowing long-term stability.

I Model as bifurcations in nonlineardynamics of linguistic populations.

I Models have bifurcations ⇐⇒ haveambiguity.

I Bifurcations possible explanation foractuation.

List 1: Sherman’s word list [5]Color is first reported pronunciation: (1,1), (2,2), (1,2)

abstractaccentaddictaddressaffectaffixalloyallyannexassaybombardcementcollectcombatcommunecompactcompoundcompressconcert

concreteconductconfectconfineconflictconscriptconserveconsortcontentcontestcontractcontrastconverseconvertconvictconvoydecoydecreasedefect

defiledescantdesertdetaildictatedigestdiscarddischargediscorddiscountdiscourseegressejectescortessayexcerptexciseexile

exploitexportextractfermentimpactimportimpressimprintincenseinclineincreaseindentinfixinflowinlayinletinsertinsetinsult

invertlegatemisprintobjectoutcastoutcryoutgooutlawoutleapoutlookoutpouroutspreadoutstretchoutworkperfumepermitpervertpostdateprefix

preludepremisepresagepresentproduceprogressprojectprotestpurportrampagerebaterebelreboundrecallrecastrecessrecoilrecordrecount

redraftredressrefillrefitrefundrefuseregressrehashrejectrelapserelayrepeatreprintresearchresetsojournsubjectsublease

subletsurchargesurveysuspecttormenttransfertransplanttransporttransversetraverseundressupcastupgradeupliftuprightupriseuprushupset

List 2: Words in use 1700–2007Color is 1700 pronunciation [1], *=changed by 2007.

abuseaccentadvanceaffrontally*anchorarrestassaultassayattackbellowblunderbottombreakfastbucklebundle

buttercement*challengechannelcommandconcernconductconsortcontestcontractconvictcoverdecrease*decreedietdigest*

dispatchdissentdistressdoubleenvyexile*expressfavourferretflourishforecast*forwardgallopgloryhammerhandle

harbourhollowimport*increase*interestironjourneylevellevylicencelicensemattermeasurementionmeritmotion

murdermusterorderoutlawpepperplasterpremise*presentproceed*protest*purchasepuzzlequarryreasonredressreform

regardrelapse*relishremarkrepealreposereservereviewrivalsaddlesecondshivershouldersquabblestablestomach

tabletallythundertitletormenttraveltrebletriumphtroublevaluevisitvomitwhistlewitness

References[1] Abel Boyer. The royal dictionary: in two parts... London, 1700.

[2] M.H. Kelly. Rhythm and language change in English. Journal of Memory and Language, 28:690–710, 1989.

[3] P. Niyogi. The Computational Nature of Language Learning and Evolution. MIT Press, Cambridge, 2006.

[4] P. Niyogi and R. Berwick. The logical problem of language change. AI Memo 1516, MIT, 1995.

[5] D. Sherman. Noun-verb stress alternation: An example of the lexical diffusion of sound change in English. Linguistics, 159:43–71, 1975.