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Grammatical Agent-Based Modeling of Typology Joe Pater Robert Staubs Coral Hughto University of Massachusetts Amherst Modeling Group Meeting, Dec. 11, 2014, LSCP, Paris Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris Grammatical Agent-Based Modeling of Typology 1 / 47

Grammatical Agent-Based Modeling of Typology · 2014. 12. 12. · ways that bear on current debates in linguistic theory { here on the choice between ranked and weighted constraints

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Page 1: Grammatical Agent-Based Modeling of Typology · 2014. 12. 12. · ways that bear on current debates in linguistic theory { here on the choice between ranked and weighted constraints

Grammatical Agent-Based Modeling of Typology

Joe Pater Robert Staubs Coral Hughto

University of Massachusetts Amherst

Modeling Group Meeting, Dec. 11, 2014,LSCP, Paris

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

Grammatical Agent-Based Modeling of Typology 1 / 47

Page 2: Grammatical Agent-Based Modeling of Typology · 2014. 12. 12. · ways that bear on current debates in linguistic theory { here on the choice between ranked and weighted constraints

The standard generative approach to typology, grammar,and learning

A theory of grammar is designed to generate all and onlypossible languages (typological modeling)

A theory of learning is designed to find a correct grammar forall languages in the space defined by the grammatical theory

The learning theory plays no role in the modeling of typology

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

Grammatical Agent-Based Modeling of Typology 2 / 47

Page 3: Grammatical Agent-Based Modeling of Typology · 2014. 12. 12. · ways that bear on current debates in linguistic theory { here on the choice between ranked and weighted constraints

The ‘Iterated Learning’ approach to typology, grammarand learning

Pioneered by Simon Kirby and colleagues (see e.g. Kirby andHurford 2002), developed in a Bayesian framework by Griffithsand Kalish (2007) and Kirby et al. (2007), see Dediu (2009)for useful overview and further extensions, and Wedel (2007,2011), Mackie and Mielke (2011) and Rafferty et al. (2013)for related work in phonology

Uses agent-based modeling (iterated learning) to study therole of learning and transmission in shaping typology

Uses very simple language models, and often highlightsemergence of putatively innate features of language

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

Grammatical Agent-Based Modeling of Typology 3 / 47

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Grammatical Agent-Based Modeling of Typology: a ‘thirdway’?

What is the effect of learning on typology with a relativelyrichly specified theory of grammar?

We take no position on the innateness or even universality ofthe grammatical features, but we do take them as given in thework we discuss here

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

Grammatical Agent-Based Modeling of Typology 4 / 47

Page 5: Grammatical Agent-Based Modeling of Typology · 2014. 12. 12. · ways that bear on current debates in linguistic theory { here on the choice between ranked and weighted constraints

Grammatical Agent-Based Modeling of Typology: a ‘thirdway’?

Today’s talk will be an overview of some results from Pater(2012), Pater and Moreton (2012) and Staubs (2014ab), plusunpublished work by Carroll (2012) and Hughto (2014), andsome more recent joint work (see further Hughto et al. 2014:NECPhon)

See also Boersma and Hamann (2008), as well as Stanton(2014: AMP, NECPhon) for work in this general vein, andHeinz (2009) and colleagues for a quite different approach tolearning and typology (based on formal learning theory, i.e.FSMs) – other references? – and the larger related literatureon modeling linguistic change (overview in Zuraw 2003)

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

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Overview

We’ll show that the incorporation of learning into typologicalexplanation allows standard grammar-based theories oflanguage to deal with phenomena that typically pose deepchallenges: systemic simplicity (Pater 2012), and probabilisticgeneralizations (i.e. typological ‘tendencies’ see esp. Staubs2014ab on stress typology)

We’ll also show that learning can affect predicted typologies inways that bear on current debates in linguistic theory – hereon the choice between ranked and weighted constraints (i.e.Optimality Theory vs. Harmonic Grammar)

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

Grammatical Agent-Based Modeling of Typology 6 / 47

Page 7: Grammatical Agent-Based Modeling of Typology · 2014. 12. 12. · ways that bear on current debates in linguistic theory { here on the choice between ranked and weighted constraints

The challenge of feature economy

A simple example of feature economy(J. Kingston p.c., based on Madiesson and Precoda 1992)

[b] no [b]

[g] 244 11

no [g] 43 153

χ2 = 260, d .f . = 1, p < 0.01

Languages tend to have either both [b] and [g] or neither: it isespecially unlikely for a language to have [g] without [b].

More generally, a segment is more likely if its feature valuesare shared by other segments.

In other words, languages tend toward feature economy(Martinet 1968; Clements 2003)

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

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The challenge of feature economy

First difficulty - feature economy is a property of systems, notof individual representations or derivations

How do we express the dependency of [b] on [g] and viceversa?

Standard phonological theories, be they rule- orconstraint-based, do not provide a formal mechanism toexpress such systemic dependencies

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

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Page 9: Grammatical Agent-Based Modeling of Typology · 2014. 12. 12. · ways that bear on current debates in linguistic theory { here on the choice between ranked and weighted constraints

The challenge of feature economy

Second difficulty - feature economy is a tendency

Languages with [p k g] or [p k b] are rare, not unattested

Standard phonological theories deal only with typologicalabsolutes, not probabilities

These problems for an analysis of feature economy seem tohave not been discussed prior to Pater (2012), but the issuesare the same as those discussed for the largely parallel case ofcross-linguistic tendencies toward uniform phrasal headednessacross syntactic categories (see e.g. Aristar 1991, Dryer 1992).

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

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General constraints, tendencies to generality andcategoricity

Could we get tendencies toward systemic simplicity by havingconstraints or rules that state a general prohibition against, orpreference for, a structure across contexts?

For example, could we get [voice] economy by includingconstraints on voicing that generalize across place ofarticulation?

And could we get a tendency toward uniform syntacticheadedness with constraints on headedness that generalizeacross category?

Yes, if we include learning in our model of typology (Pater2012, see Pater and Staubs 2013: MFM for more on featureeconomy)

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

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General constraints, tendencies to generality andcategoricity

Whether we adopt a parametric framework, or ranked (OptimalityTheory) or weighted constraints (Harmonic Grammar), thegrammar model can represent arbitrarily non-uniform headedness

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

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General constraints, tendencies to generality andcategoricity

The interaction of the general Head-R/L and the category-specificconstraints is an example of a vacuous gang effect: the HGtypology is the same as OT, or a parametric theory. In all of theseframeworks, in terms of the set of systems that our theory canrepresent, the general constraint is doing no work for us.

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

Grammatical Agent-Based Modeling of Typology 12 / 47

Page 13: Grammatical Agent-Based Modeling of Typology · 2014. 12. 12. · ways that bear on current debates in linguistic theory { here on the choice between ranked and weighted constraints

General constraints, tendencies to generality andcategoricity

One approach: predicted probability is proportional to thenumber of rankings generating it (Coetzee 2002, Bane andRiggle 2008)

In HG, randomly sampling constraint weights from a uniformdistribution from 0 to 1 gives 0.4 probability consistentheadedness with the general constraint, 0.125 without it

We get a similar effect by constructing a model of the effectof learning on typology, though as we will see, our modelpredictions can diverge considerably from random sampling

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

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General constraints, tendencies to generality andcategoricity

Another approach: relate typological attestation to ease oflearning

This will be basically the approach we take, althoughagent-based modeling goes beyond the case of a single fixedteacher-learner relationship

See Pater (2012) for some discussion of issues that arise inattempting to equate simple ease of learning with typologicalattestation for the case of feature economy; see moregenerally Rafferty et al. (2013)

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

Grammatical Agent-Based Modeling of Typology 14 / 47

Page 15: Grammatical Agent-Based Modeling of Typology · 2014. 12. 12. · ways that bear on current debates in linguistic theory { here on the choice between ranked and weighted constraints

General constraints, tendencies to generality andcategoricity

We assume a probabilistic version of HG in which the probability ofa candidate is proportional to the exponential of its Harmony(MaxEnt grammar, Goldwater and Johnson 2003)

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

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General constraints, tendencies to generality andcategoricity

Learning: difference between datum produced by teacher andlearner’s expectation scaled by a learning rate (below always 0.1),added to current weights

Gradient ascent: training data given in batch.Stochastic gradient ascent (SGA): one piece of training data at atime.Sampling SGA: learner expectation also sampled from distributiondefined by grammar

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

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Page 17: Grammatical Agent-Based Modeling of Typology · 2014. 12. 12. · ways that bear on current debates in linguistic theory { here on the choice between ranked and weighted constraints

0.5  

0.55  

0.6  

0.65  

0.7  

0.75  

0.8  

0.85  

0.9  

0.95  

1  

0   10   20   30   40   50   60   70   80   90  100  110  120  130  140  150  160  170  180  190  200  210  220  230  240  250  

4R  

3R,  1L  

2R,  2L  

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

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Page 18: Grammatical Agent-Based Modeling of Typology · 2014. 12. 12. · ways that bear on current debates in linguistic theory { here on the choice between ranked and weighted constraints

General constraints, tendencies to generality andcategoricity

Interactive learning: a simple way to generate typologies

Two ‘agents’ are started with zero weight on the constraints,and hence equal probability for the two candidates in eachtableau

In each trial, an agent is picked at random as the teacher, atableau is picked at random, and learning datum is sampledfrom the distribution defined by the teacher’s grammar(sampling SGA)

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

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General constraints, tendencies to generality andcategoricity

This is done 10,000 times to generate a language; I’ll reportresults over 100 languages

There are 4 phrase types each with a binary choice, so thereare 24 = 16 possible headedness combinations across them.Of these, only 2/16 have consistent headedness: 0.125 is abaseline probability of consistent headedness.

Taking the higher probability candidate from each tableau asthe choice of headedness, 74/100 = 0.74 have consistentheadedness

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

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General constraints, tendencies to generality andcategoricity

100 languages generated without the general constraints produced7 cases of consistent headedness (0.07, similar to 0.125 baseline)

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Page 21: Grammatical Agent-Based Modeling of Typology · 2014. 12. 12. · ways that bear on current debates in linguistic theory { here on the choice between ranked and weighted constraints

General constraints, tendencies to generality andcategoricity

A tendency toward categoricity can be seen by looking at theprobabilities granted to the candidates at the end of simulations

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

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Page 22: Grammatical Agent-Based Modeling of Typology · 2014. 12. 12. · ways that bear on current debates in linguistic theory { here on the choice between ranked and weighted constraints

General constraints, tendencies to generality andcategoricity

Why are we getting categoricity?

1 Weight changes can push the agents towards either more orless categorical states.

2 As the agents drift into categorical grammars1 They change less and less.2 The effective change to probability from a change in weights

shrinks.

3 The system spends most of its time in categorical states.

(cf. Wedel 2007 on models where a positive feedback loop createssimilar pressures)

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

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Page 23: Grammatical Agent-Based Modeling of Typology · 2014. 12. 12. · ways that bear on current debates in linguistic theory { here on the choice between ranked and weighted constraints

General constraints, tendencies to generality andcategoricity

The network structure of this agent-based model is the polaropposite of those in Kirby and Hurford (2002) and Griffithsand Kalish (2007), besides the fact that there are only twoagents (at a time) in both

Their models are purely generational: a designated learnerlearns for some number of trials from a designated teacher,and then becomes the teacher for the next generation

This model is purely interactive: agents always have the sameprobability of being the teacher or learner

Reality lies between these two poles; in any case the purelygenerational model also produces the categoricity tendency(Dediu 2009: 555, Hughto et al. 2014)

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

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Page 24: Grammatical Agent-Based Modeling of Typology · 2014. 12. 12. · ways that bear on current debates in linguistic theory { here on the choice between ranked and weighted constraints

General constraints, tendencies to generality andcategoricity

Take-home message on categoricity: a probabilistic grammar,embedded in a model of learning and transmission, can predictthat languages should tend to be categorical

Aside: why do generative grammarians tend to shunprobabilistic models? Is it partially due to observed tendenciesto categoricity? If so, these results are relevant.

Take-home message on generality: a general constraint can yieldtendencies toward generality if we incorporate learning intotypological modeling

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

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Page 25: Grammatical Agent-Based Modeling of Typology · 2014. 12. 12. · ways that bear on current debates in linguistic theory { here on the choice between ranked and weighted constraints

HG and typology: background

The predecessor to Optimality Theory (Prince and Smolensky1993) was a weighted constraint theory of grammar calledHarmonic Grammar (Legendre, Miyaya, and Smolensky 1990,Smolensky and Legendre 2006)

There has been a recent re-emergence of interest in HG, interms of learning (e.g. Wilson 2006, Jaeger 2007, Jesney andTessier 2011, Boersma and Pater to appear, though see Magri2013), and also in terms of typology (e.g. Pater 2009, Pottset al. 2010, Jesney 2012)

We’ll look at perhaps the best case of an attested HG-specificpattern, and two potential problems for HG in light ofGrammatical Agent-Based Modeling

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

Grammatical Agent-Based Modeling of Typology 25 / 47

Page 26: Grammatical Agent-Based Modeling of Typology · 2014. 12. 12. · ways that bear on current debates in linguistic theory { here on the choice between ranked and weighted constraints

HG and typology: background

Japanese loanword devoicing (Nishimura 2003, 2006), Kawahara(2006), Pater (2009)

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HG and typology: background

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

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HG and typology: background

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

Grammatical Agent-Based Modeling of Typology 28 / 47

Page 29: Grammatical Agent-Based Modeling of Typology · 2014. 12. 12. · ways that bear on current debates in linguistic theory { here on the choice between ranked and weighted constraints

HG and typology: background

This can be analyzed in OT with a larger constraint set, but is onlypredicted by HG

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

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Page 30: Grammatical Agent-Based Modeling of Typology · 2014. 12. 12. · ways that bear on current debates in linguistic theory { here on the choice between ranked and weighted constraints

Stress windows

We can also get gang effects between just two constraints

Two constraint gang effect

3 2X Y H

→A -1 -2B -1 -3

→C -1 -3D -2 -4

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

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Page 31: Grammatical Agent-Based Modeling of Typology · 2014. 12. 12. · ways that bear on current debates in linguistic theory { here on the choice between ranked and weighted constraints

Stress windows

Potential problem for HG: stress windows (Legendre, Sorace andSmolensky 2006, Pater 2009)

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

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Page 32: Grammatical Agent-Based Modeling of Typology · 2014. 12. 12. · ways that bear on current debates in linguistic theory { here on the choice between ranked and weighted constraints

Stress windows

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

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Page 33: Grammatical Agent-Based Modeling of Typology · 2014. 12. 12. · ways that bear on current debates in linguistic theory { here on the choice between ranked and weighted constraints

Stress windows

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

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Page 34: Grammatical Agent-Based Modeling of Typology · 2014. 12. 12. · ways that bear on current debates in linguistic theory { here on the choice between ranked and weighted constraints

HG vs. OT

Only 3-syllable windows exist, not 4-syllable or larger, and3-syllable windows are much rarer than 2 (Staubs 2014b: 89)

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

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Page 35: Grammatical Agent-Based Modeling of Typology · 2014. 12. 12. · ways that bear on current debates in linguistic theory { here on the choice between ranked and weighted constraints

Stress windows

Prince (1993/2007) points out the potential relevance oflearning to the unattestedness of larger windows, insofar as theweight ratios needed for larger windows are harder to acquire

Staubs (2014b): if we use a generational version of GABM tomodel the relative probabilities of 2- and 3-syllable windows,what do we predict for 4 syllable windows?

‘Although the fitted model predicts a value of one as most likely(32.69% probability), the observed zero count is by no meansunlikely (20.60% probability).’

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

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Page 36: Grammatical Agent-Based Modeling of Typology · 2014. 12. 12. · ways that bear on current debates in linguistic theory { here on the choice between ranked and weighted constraints

Neutralization in the ‘elsewhere’ environment

Our second case of an apparent problem for HG relates to anobservation first made by Prince (1998) about OT

Allophony with a marked feature in a specific environment:e.g. ‘vowels are predictably [+nasal] before nasal consonants,[–nasal] elsewhere’

In some versions of OT (Pater 1999) faithfulness constraintscan specify the direction of change (i.e. from ‘+’ to ‘–’ andfrom ‘–’ to ‘+’)

Predicts neutralization in the elsewhere environment: e.g.‘vowels contrast for [+/–nasal] before nasal consonants, andare [–nasal] elsewhere’

Though Prince (1998) doesn’t talk about the attestedtypology, the standard OT prediction seems basically correct:that neutralization in the elsewhere environment does not exist

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

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Page 37: Grammatical Agent-Based Modeling of Typology · 2014. 12. 12. · ways that bear on current debates in linguistic theory { here on the choice between ranked and weighted constraints

Neutralization in the ‘elsewhere’ environment

Carroll (2012) points out standard OT constraints used in HGalso predict ‘neutralization elsewhere’ (gang effect betweenFaith and specific Mark)

He looks at the typology of consonant palatalization, wheremarked palatals are often found before front vowels

There is in fact one language with neutralization elsewhere:Gujarati ‘[s] and [S] before front vowels, [s] elsewhere’

There are many more languages with the expected patterns ofallophony (23 languages, 11 families) and of neutralization topalatalized before front vowels (15, 10)

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

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Page 38: Grammatical Agent-Based Modeling of Typology · 2014. 12. 12. · ways that bear on current debates in linguistic theory { here on the choice between ranked and weighted constraints

Neutralization in the ‘elsewhere’ environment

With standard OT constraints, uniformly sampling the weightsindependently from a bounded uniform distribution yieldsneutralization elsewhere three times more often thanallophony, or regular neutralization.

This case, in which a pattern predicted only by HG exists, butis rarer than predicted, may well fit with many phonologists’‘seat of the pants’ assessment of the general situation

Carroll (2012) also shows that usual adjustments to the priorof a MaxEnt model don’t change the probabilities in thedesired way (a high-variance log-normal prior could work, butthis essentially just builds in ranking-like behavior)

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

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Page 39: Grammatical Agent-Based Modeling of Typology · 2014. 12. 12. · ways that bear on current debates in linguistic theory { here on the choice between ranked and weighted constraints

Neutralization in the ‘elsewhere’ environment

Hughto’s (2014) attack on this problem is based on Carroll’s(2012) observation that the space of HG-specific patternsoverlaps greatly with the space of patterns yielding(noticeable) variation in a MaxEnt model

Relatedly, Flemming (p.c.) reports that attested gang effectstend to be variable, like Japanese loanword devoicing

Given that the results of agent-based modeling tend towardscategoricity, perhaps they would also tend away fromcumulativity / gang effects? (Can also be harder to learn)

They do – what follows is a demonstration with a simpletwo-constraint case

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

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Page 40: Grammatical Agent-Based Modeling of Typology · 2014. 12. 12. · ways that bear on current debates in linguistic theory { here on the choice between ranked and weighted constraints

Cumulativity tableaux

3 2X Y H

→A -1 -2B -1 -3

→C -1 -3D -2 -4

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Page 41: Grammatical Agent-Based Modeling of Typology · 2014. 12. 12. · ways that bear on current debates in linguistic theory { here on the choice between ranked and weighted constraints

0.5  

0.55  

0.6  

0.65  

0.7  

0.75  

0.8  

0.85  

0.9  

0.95  

1  

0   50   100   150   200   250   300   350   400   450   500   550   600   650   700   750   800   850   900   950  1000  

AC  

AD  

BC  

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Cyan and orange: no cumulativity effect. Black: cumulativity.Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

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Cumulativity

A simulation like our syntax one, except that the stopping criterionis 0.95 probability on some candidate in each tableau

X Y

A -1B -1

C -1D -2

Language Count

A, D 312B, C 688A, C 0

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0 500 1000 1500

0.0

0.4

0.8

Avoidance of cumulative pattern, starting at 88% probability

Learning Step

Pro

babi

lity

of [a

]

0 500 1000 1500

0.0

0.4

0.8

Learning Step

Pro

babi

lity

of [c

]

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

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0 1000 2000 3000 4000 5000

0.0

0.4

0.8

Maintaining cumulative pattern, starting at very high probabilities

Learning Step

Pro

babi

lity

of [a

]

0 1000 2000 3000 4000 5000

0.0

0.4

0.8

Learning Step

Pro

babi

lity

of [c

]

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

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HG vs. OT

Take-home message on ranking vs. weighting: a weightedconstraint grammar, in a model of typology that includes learning,predicts that languages should tend to avoid gang effects

Much remains to be done in terms of studying the generalityof this result, but the potential consequences are striking,since the grammatical model alone might overgenerate, withthe full model remaining sufficiently restrictive

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

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Page 47: Grammatical Agent-Based Modeling of Typology · 2014. 12. 12. · ways that bear on current debates in linguistic theory { here on the choice between ranked and weighted constraints

Conclusions

In generative linguistics, we typically make relatively directinferences about the structure of UG from typology

These results suggest that abstracting from learning may notbe entirely safe (categoricity, cumulativity) and may result inmissed opportunities (generality from general constraints,accounts of tendencies)

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

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Page 48: Grammatical Agent-Based Modeling of Typology · 2014. 12. 12. · ways that bear on current debates in linguistic theory { here on the choice between ranked and weighted constraints

Thank you!

This material is based upon work supported by the NationalScience Foundation under Grant No. S121000000211 to thesecond author, Grants BCS-0813829 and BCS-424077 to theUniversity of Massachusetts Amherst, and by the city of Parisunder a Research in Paris fellowship to the first author.

We would also like to thank Lucien Carroll, Giorgio Magri,Joaquim Brandao de Carvalho, and Elliott Moreton for discussion,as well as John McCarthy and other members of the phonologygrant group at UMass.

Joe Pater, Robert Staubs, Coral Hughto UMass Amherst LSCP Paris

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