A Lexical Theory of Variation

1
A Lexical Theory of Variation
Andries W. Coetzee Workshop on Variation,
Gradience and Frequency in Phonology Stanford
University, July 2007
2

• Things that are known to influence variation
• Grammar
• Where Where it appears and where not
• Frequency How often does a process apply in some
  context
• Lexical frequency
• Some variable processes affect frequent words
  more, others affect infrequent words more.
• Extra-grammatical factors
• Speech style, speech rate, etc.

3

• Existing theories of variation
• Grammatical
• Variable rule in the Labovian tradition (Labov
  1972 Sankoff 1988)
• Several OT models (Anttila 1997 Boersma and
  Hayes 2000 Coetzee 2006 Reynolds 1994)
• Reasonably successful at accounting for the
  grammatical influence.
• Usage-based/exemplar models (Bybee 2001, 2002
  Pierrehumbert 2001)
• Reasonably successful at accounting for the
  influence of lexical frequency.
• Interaction between the two
• Models that incorporate both are still largely
  absent.

4

• Structure of the presentation
• Usage frequency and variation
• The basics of the proposal
• Phonetically motivated variation
• Analogically motivated variation
• Learning lexical distributions

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Usage Frequency and Variation
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• Phonetically motivated variable process
• Typical phonological process
• Applies more often to lexical items with higher
  usage frequency
• Example t/d deletion
• Pre-C west bank wes bank
• Pre-V west end wes end
• Pre- west wes

Influence of frequency (Bybee 200070)
Chicano English (Santa Ana 1991)
High frequency Low frequency
n 1,650 399
deleted 54.4 34.4
Pre-C Pre-V Pre-
n 3,693 1,574 1,024
deleted 62 45 37
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• Analogically motivated variable process
• Usually some kind of regularization process
  irregular plural/past tense replaced with regular
• Applies more often to lexical items with lower
  usage frequency
• Example Regularization of past tense verbs
• Infrequent verbs are more likely to
  regularize (Hooper 1976100
  Bybee 1985120, 2002269 Bybee Slobin 1982)
• Kucera and Francis frequencies
  (1982) as calculated at www.iphod.com.
• Also many examples from the historical
  literature. (Phillips 1984, 2001 and references
  therein.)

Less likely to regularize Less likely to regularize Less likely to regularize More likely to regularize More likely to regularize More likely to regularize
Present Raw Log Present Raw Log
keep 348 2.54 creep 19 1.28
leave 345 2.54 leap 20 1.30
sleep 106 2.03 weep 22 1.34
drive 174 2.24 dive 32 1.93
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• The challenge
• A formal theory of variation that
• Captures the role of grammar
• Determines what kind of variation is possible
• Influences the frequency of application
• Captures the role of lexical frequency
• Variable process applies differently to different
  lexical items.
• Different kinds of processes are differently
  influenced by lexical frequency.

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The Proposal Variation Through Lexical
Indexation
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• Variable lexical indexation
• Lexically indexed constraints (Pater 1994, 2000
  Itô Mester 1995, 1999)
• Allows a way in for lexical influence
• Yet still keep control in the hands of grammar
• Variation through variable lexical class
  affiliation
• Note that the grammar stays constant what
  varies is the lexical class affiliation of
  lexical items. Variation is hence moved from the
  grammar into the lexicon.

MAX-L2 M MAX-L1
/west/L2 west
wes !
/west/L1 west !
wes
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• Lexical distribution functions
• What determines the lexical class affiliation of
  a lexical item?
• Each lexical item is stored with a probability
  density function.
• Every time a lexical item is submitted to grammar
  for evaluation, a value is chosen randomly along
  the x-axis of the distribution function.
• The x-axis is divided into equally sized adjacent
  regions corresponding to the number of indexed
  versions of the constraint.
• Correlation between frequency and skewness of
  distribution function
• Frequent lexical items left skewed function
• Infrequent lexical items right skewed function

average
low
high
L2 L1
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Example 1 Phonetically Motivated Variation
13
t/d-deletion again
Frequency
Context
High frequency Low frequency
n 1,650 399
deleted 54.4 34.4
Pre-C Pre-V Pre-
n 3,693 1,574 1,024
deleted 62 45 37

• Grammar
• Markedness constraints
• PRE-C No t/d in the context C_C
• PRE-V No t/d in the context C_V
• PRE- No t/d in the context C_
• Contextual licensing constraints a la Steriade
  (1997)
• Four indexed versions of MAX.
• Ranking
• MAX-L4 ? PRE-C ? MAX-L3 ? PRE-V ? MAX-L2 ?
  PRE- ? MAX-L1

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• The grammar in Pre-C condition
• Preservation if MAX-L4, deletion if MAX-L3,
  MAX-L2, MAX-L1

MAX-L4 PRE-C MAX-L3 PRE-V MAX-L2 PRE- MAX-L1
/westL4 bank/ ? west bank
wes bank !
/westL3 bank/ west bank !
? wes bank
/westL2 bank/ west bank !
? wes bank
/westL1 bank/ west bank !
? wes bank
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• The grammar in Pre-V condition
• Preservation if MAX-L4, MAX-L3, deletion if
  MAX-L2, MAX-L1

MAX-L4 PRE-C MAX-L3 PRE-V MAX-L2 PRE- MAX-L1
/westL4 end/ ? west end
wes end !
/westL3 end/ ? west end
wes end !
/westL2 end/ west end !
? wes end
/westL1 end/ west end !
? wes end
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• The grammar in Pre-Pause condition
• Preservation if MAX-L4, MAX-L3, MAX-L2, deletion
  if MAX-L1

MAX-L4 PRE-C MAX-L3 PRE-V MAX-L2 PRE- MAX-L1
/westL4/ ? west
wes !
/westL3/ ? west
wes !
/westL2/ ? west
wes !
/westL1/ west !
? wes
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• Likelihood of deletion based on grammar alone
• Grammar MAX-L4 ? PRE-C ? MAX-L3 ? PRE-V
  ? MAX-L2 ? PRE- ? MAX-L1
• Note that grammar determines
• What variation is observed only a process that
  reduces markedness, only a process that is
  grammatically motivated.
• How frequently process applies in which context.
• But we still need to give the lexicon its due.

Context Example Indexation resulting in retention Indexation resulting in deletion indexations resulting in deletion
Pre-C west side L4 L3, L2, L1 75
Pre-V west end L4, L3 L2, L1 50
Pre-Pause west L4, L3, L2 L1 25
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• The influence of lexical frequency
• Frequencies from
  Francis Kucera (1982), calculated at
  www.iphod.com.

Raw frequency Log frequency Expected deletion
Infrequent vest 6 0.60 Low
Intermediate modest 29 1.46 Medium
Frequent best 361 2.56 High
Mean 29.8 1.47
best
modest
vest
MAX-L4 MAX-L3 MAX-L2
MAX-L1 PRE-C PRE-V
PRE-
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Example 2 Analogically Motivated Variation
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• Regularization of the strong past tense in
  English
• Specific examples from Kucera and Francis (1982)
  (www.iphod.com)
• Irregular morphology/suppletion as allomorphy
• Two morphological options for formation of the
  past tense.
• Both options are input to grammar, so that choice
  of the one allomorph does not violate
  faithfulness relative to the other. (Anttila
  1997, Bonet 2004, Itô and Mester 2006, Kager
  1996, Mascaró 1996, etc.)
• Constraints
• OO-FAITH Some kind of paradigm uniformity (Benua
  2000, Kenstowicz 1996,etc.)
• USELISTED The input of a candidate must be a
  single lexical entry (Zuraw 2000)

Base Base
Raw Log Regular past Strong past regular
speed 91 1.96 3 9 25
dive 32 1.51 5 4 56
leap 20 1.2 20 2 91
mean 29.8 1.47
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• The grammar

OO-FAITH-L2 USELISTED OO-FAITH-L1
/leapL1 ed/, /leaptL1/ leaped !
OO-Base leap ? leapt
/leapL2 ed/, /leaptL2/ ? leaped
OO-Base leap leapt !

• And the influence from the lexicon

dive
leap
speed
OO-FAITH-L2
OO-FAITHL1 USELISTED
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Lexical Distribution Functions
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• What needs to be learned?
• Grammar Ranking between constraints
• Lexicon Lexical items, with their
  probabilistic distribution functions.
• These are two separate learning problems, each
  with their own solution.
• Learning the grammar
• Well developed learnability literature in OT.
  (Tesar and Smolensky 1998, 2000, etc.)
• And specifically on learning an indexed grammar.
  (Pater 2006, to appear).
• I will therefore not dwell on this aspect here.
• Learning the lexicon
• Focus here on how the lexical distribution
  functions might be acquired.

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• General properties of lexical distribution
  functions
• MAX L1
• DEP L1 L2 L3
• IDENTF L1 L2
  L3 L4

average
frequent
infrequent
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• General properties of lexical distribution
  functions
• Basic requirements
• Minimum and maximum value.
• Shape parameters that determine skewness
• Beta-distribution (Evans, Hastings Peacock
  2000)
• ? ? ? symmetric
• ? lt ? ? right skewed
• ? gt ? ? left skewed

frequent
average
infrequent
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• A small scale simulation
• IPhOD 1.3 (www.iphod.com)
• 33,432 words, with CMU transcriptions and
  KuceraFrancis frequencies
• Multiple KF by 10 to avoid having to work with
  log(1)
• Calculated the following
• Mean frequency of all words in IPhOD 297.89.
  Log(?) 2.47.
• Collected all words that end -Ct or -Cd,
  excluding past tense verbs, and took the log of
  the frequency for each of these.
• Distribution functions

Frequency Frequency Skewness Skewness ? ?
frequent (f gt ?) left (? gt ?) log(f) log(?)
infrequent (f lt ?) right (? lt ?) log(?) log(f)
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• A small scale simulation

aghast vest modest best most ?
Frequency 10 40 290 3610 11610 297.89
Log 1 1.60 2.46 3.56 4.07 2.47
aghast
most
modest
best
vest
MAX-L4 MAX-L3 MAX-L2
MAX-L1 PRE-C PRE-V
PRE-
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• How well do the predictions line up with reality?
• Once the values of ? and ? for a word are known,
  it is easy to calculate the likelihood of an
  x-value falling in a specific range along the
  x-axis, and hence the likelihood of deletion in
  each of the three contexts for each word.
• Using this, I ran a simulation, feeding each
  -Ct and -Cd word through the grammar,
  according to its frequency in IPhOD.

Phonological context (value in brackets is
ratio to Pre-C) (Santa Ana 1991)
Pre-C Pre-V Pre-
Chicano English 62 45 (.73) 37 (.60)
Predictions of LTV 90 62 (.69) 27 (.30)
Frequency (value in brackets is ratio to gt
35/million) (Bybee 2000)
gt 35/million lt 35/million
Chicano English 54 34 (.63)
Predictions of LTV 65 43 (.66)
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• How can this be refined further?
• Currently, the lexical distribution functions are
  determined purely based on lexical frequency. But
  we know that different dialects show different
  deletion rates.
• Either different dialects have different lexical
  frequencies.
• Or there are other parameters that can be set
  independently from lexical frequency.
• Maybe some constant is added/subtracted from the
  mean?
• Added more words become infrequent more
  conservative dialect.
• Subtracted more words become frequent more
  deletion.
• Maybe the lexical space can be warped i.e. the
  regions along the x-axis that correspond to
  lexical classes are not of equal size.
• Maybe lexical distribution functions are best-fit
  functions i.e. learn a function that would
  result in the correct deletion rate but then we
  lose the connection between usage frequency and
  deletion rates.

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Conclusion
31

• Conclusion
• Existing grammatical models of variation do not
  allow the lexicon enough opportunity to play a
  role. (Pierrehumbert 2001)
• p. 138
• p. 148
• Purely usage-based models probably does not allow
  the grammar enough say.
• Bybee (200073)
• Bybee (2002268)
• LTV is an attempt to do both. Does it succeed?

A second challenge arises from the fact that the
differential phonetic outcomes relate
specifically to word frequency. Standard
generative models do not encode word frequency.
They treat the word frequency effects as
matters of linguistic performance rather than
linguistic competence. Thus the intrusion of word
frequency into a traditional area of linguistics,
namely to conditioning of allophony, is not
readily accommodated in the classical generative
viewpoint. The exemplar model is the only current
model which has these properties.
it does mean that there is no variable rule of
t/d-deletion. Rather there is a gradual process
of shortening or reducing the lingual gesture
If we take linguistic behavior to be highly
practiced neuromotor activity then we can view
reductive sound changes as the result of the
automation of linguistic production. It is well
known that repeated neuromotor patterns become
more efficient as they are practiced transitions
are smoothed by the anticipatory overlap of
gestures, and unnecessary or extreme gestures
decrease in magnitude or are omitted.
32
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Die einde