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Advantages of constituency: computational perspectives on Samoan word prosody

krisyu
July 27, 2017

Advantages of constituency: computational perspectives on Samoan word prosody

Refereed talk presented at the 22nd Conference on Formal Grammar on July 23, 2017 at the University of Toulouse. http://fg.phil.hhu.de/2017/ Code available at: https://github.com/krismyu/smo-constituency-feet

krisyu

July 27, 2017
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  1. ADVANTAGES OF CONSTITUENCY:
    COMPUTATIONAL PERSPECTIVES
    ON SAMOAN WORD PROSODY
    KRISTINE M. YU
    UNIVERSITY OF MASSACHUSETTS AMHERST
    DEPARTMENT OF LINGUISTICS
    THE 22ND CONFERENCE ON FORMAL GRAMMAR
    UNIVERSITY OF TOULOUSE, JULY 23, 2017

    View full-size slide

  2. PHONOLOGISTS ARE OFTEN EXPLICIT ABOUT
    [OTHER ASSUMPTIONS]. BUT WE TEND TO HELP
    OURSELVES TO PROSODIC DOMAINS WITHOUT
    FURTHER COMMENT.
    Kie Zuraw, 2009
    2
    https://www.mcgill.ca/linguistics/files/ linguistics/Handout_RevisedForMcGill.pdf

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  3. 3
    ▸ Theoretical phonology: referring to phonological constituents
    can capture generalizations in phonological patterns
    ▸ But:
    ▸ Also alternative ways to capture the same generalizations
    ▸ Computational descriptions of phonological patterns have
    revealed strong structural universals without referring to
    constituents at all (Heinz 2009, 2010, et seq.)
    Do constituents make phonological grammars
    more succinct?
    RESEARCH QUESTION

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  4. RESEARCH QUESTION: CASE STUDY
    4
    Do constituents make phonological grammars
    for Samoan word stress more succinct?
    ▸ One way to start to get a grip on whether we get explanatory
    advantage by assuming existence of constituents: succinctness
    ▸ Succinctness as a consequence of constituency has not been
    carefully explored computationally in phonology
    ▸ Case study: Comparison of succinctness of four grammar
    fragments generating Samoan stress patterns in monomorphemic
    words, with and without reference to feet as constituents
    Similar kinds of succinctness comparisons include: Chomsky (1965); Chomsky and Halle (1968); Meyer and
    Fischer (1971); Hartmanis (1980); Stabler (2013); Berwick (2015)

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  5. WHY CASE STUDIES?
    5
    ▸ Question here is if this particular phenomenon motivates
    prosodic constituents
    ▸ cf. case studies to examine where phonological and
    syntactic patterns fall in the Chomsky hierarchy

    View full-size slide

  6. WHY WORK ON SAMOAN STRESS?
    6
    ▸ Recent, detailed phonological analysis using prosodic
    constituents based on a rich set of elicited stress patterns
    (Zuraw, Yu, and Orfitelli 2014)
    ▸ Connections to other computational modeling of grammar
    and parsing at the syntax-phonology interface in Samoan
    (Yu and Stabler to appear, Yu submitted)

    View full-size slide

  7. RESEARCH QUESTION OPERATIONALIZATION
    7
    Do constituents make phonological grammars
    for Samoan word stress more succinct?
    Constituents: Feet

    View full-size slide

  8. WHAT IS A FOOT?
    8
    ▸ Rhythmic unit composed of one or more syllables
    ▸ Organizes syllables into higher-order unit based on regular
    stress patterns
    ▸ Examples:
    ▸ Trochaic foot: Strong-Weak ˈbut.ter
    ▸ Iambic foot: Weak-Strong ba.ˈguette
    ▸ Foot dominated by higher-order-prosodic units all the way
    up to level of utterance

    View full-size slide

  9. DEFINITION OF LANGUAGE “LITTLE SAMOAN”
    9
    ▸ Slightly simplified description of Samoan stress in
    monomorphs presented in Zuraw, Yu, and Orfitelli (2014)
    ▸ Language of strings of light (one vowel only) and heavy (two
    vowels/long vowel) syllables marked for primary, secondary,
    or no stress (“weak”)
    ▸ la(ˈvaː) `energized’ W[L](P[H])
    ▸ (ˈmanu) `animal’ (P[L]W[L])
    ▸ L: light syllable, H: heavy syllable
    ▸ W: weak, P: primary stress, S: secondary stress
    ▸ (…): foot

    View full-size slide

  10. RESEARCH QUESTION OPERATIONALIZATION
    10
    Do constituents make phonological grammars
    for Samoan word stress more succinct?
    Phonological grammars:
    - ‘Direct’: directly regulate surface patterns
    - ‘Karttunen OT’: Finite state implementation of
    optimality theory (constraint-based)
    Optimality Theory (OT): Prince and Smolensky (1993/2004) ; Karttunen OT implementation: Karttunen (1998)

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  11. OPTIMALITY THEORY 1: (OVER)GENERATE
    11
    Input: LL
    Output:
    P[L]P[L]
    P[L]W[L]
    P[L]S[L]
    W[L]P[L]
    W[L]W[L]
    W[L]S[L]
    S[L]P[L]
    S[L]W[L]
    S[L]S[L]
    GENERATE
    Add stress markup
    ▸ L: light syllable, H: heavy syllable
    ▸ W: weak, P: primary stress, S: secondary stress

    View full-size slide

  12. OPTIMALITY THEORY 2: (OVER)CONSTRAIN
    12
    Candidates:
    P[L]P[L]
    P[L]W[L]
    P[L]S[L]
    W[L]P[L]
    W[L]W[L]
    W[L]S[L]
    S[L]P[L]
    S[L]W[L]
    S[L]S[L]
    CONSTRAIN
    Candidates:
    P[L]P[L]
    P[L]W[L]
    P[L]S[L]
    W[L]P[L]
    W[L]W[L]
    W[L]S[L]
    S[L]P[L]
    S[L]W[L]
    S[L]S[L]
    “Don’t have adjacent
    stressed syllables”
    ▸ L: light syllable, H: heavy syllable
    ▸ W: weak, P: primary stress, S: secondary stress

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  13. OPTIMALITY THEORY 2: (OVER)CONSTRAIN
    13
    (ˌtemo)ka(ˈlasi) ‘democracy’ (S[L]W[L])W[L](P[L]W[L])
    (1) “All syllables must be parsed into feet”
    (2) “The beginning of the word must
    coincide with the beginning of a foot”
    Constraint 1 Constraint 2
    (ˌtemo)ka(ˈlasi) *
    te(ˌmoka)(ˈlasi) * *
    ▸ Both candidates violate at least one of the ordered constraints

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  14. OPTIMALITY THEORY 3: EVALUATE
    14
    Constraint 1 Constraint 2
    (ˌtemo)ka(ˈlasi) *
    te(ˌmoka)(ˈlasi) * *
    ▸ EVAL is not a finite state process! Number of states required
    for EVAL cannot be bounded (Eisner 1997, Karttunen 1998).
    ▸ Karttunen OT: finite state implementation of OT, maps
    underlying forms directly to surface forms rather than
    violation vectors, no EVAL. Equivalent expressive power if
    finite bound on number of violations.
    EVALUATE (ˌtemo)ka(ˈlasi)
    Winner has fewer
    violations of
    higher-ranked
    constraints

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  15. RESEARCH QUESTION OPERATIONALIZATION
    15
    Do constituents make phonological grammars
    for Samoan word stress more succinct?
    Phonological grammars:
    - ‘Direct’: directly regulate surface patterns
    - ‘Karttunen OT’: Finite state implementation of
    optimality theory (constraint-based)
    Optimality Theory (OT): Prince and Smolensky (1993/2004) ; Karttunen OT implementation: Karttunen (1998)

    View full-size slide

  16. 2 X 2 COMPARISON OF GRAMMARS
    16
    Syllables only
    Syllables and
    feet
    Direct
    Direct,
    syllables only
    Direct,
    syll. and feet
    Karttunen OT
    Karttunen OT,
    syllables only
    Karttunen OT,
    syll. and feet

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  17. 17
    ▸ We define the grammars in xfst, a language designed by
    linguists to make it natural to state morphophonological
    grammars (Beesley and Karttunen, 2003)
    ▸ Includes pre-defined operators and capacity for definition of
    own operator and units which allow us to write grammars at
    very high level
    ▸ Compiles our high-level grammars to finite state transducers
    ▸ Provides common formalism in which we can define all four
    grammars and measure grammar size in a controlled
    comparison
    (Beesley and Karttunen, 2003), https://web.stanford.edu/~laurik/fsmbook/home.html
    DEFINING THE GRAMMARS IN xfst

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  18. RESEARCH QUESTION
    18
    Do constituents make phonological grammars
    for Samoan word stress more succinct?

    View full-size slide

  19. 19
    ▸ Special case of minimum description length (MDL) relativized to xfst
    notation
    ▸ Common xfst formalism for expressing the grammars
    ▸ Data same across comparisons: stress patterns up to 5 syllables
    ▸ I limit testing empirical coverage of stress patterns to monomorphs
    of 5 syllables because there are no longer words
    ▸ All grammars admit exactly same set of stress patterns up to 5
    syllables
    ▸ Size of encodings of sequences up to 5 syllables is the same, since
    possibilities allowed by grammars in that range is identical
    DEFINING SUCCINCTNESS
    Succinctness: the size of the grammar, i.e. the
    number of symbols it takes to write it down in xfst

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  20. OPERATIONALIZED RESEARCH QUESTION
    20
    Do constituents make phonological grammars
    for Samoan word stress more succinct?
    Does reference to feet reduce the number of
    symbols used in xfst, in defining direct
    approach and Karttunen OT grammars for stress
    patterns in Samoan monomorphs?

    View full-size slide

  21. CODE IS AVAILABLE AT…
    21
    https://github.com/krismyu/smo-constituency-feet

    View full-size slide

  22. IMPLEMENTATION: MARKUP STRESS
    22
    Input:
    .
    .
    .
    LL
    .
    .
    .
    Output:
    .
    .
    .
    P[L]W[L]
    .
    .
    .
    Add stress markup
    Step 0a: Add stress markup
    (all 4 grammars)

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  23. IMPLEMENTATION: MARKUP FEET
    23
    Output:
    .
    .
    .
    (P[L]W[L])
    .
    .
    .
    Add footing markup
    “Wrap parentheses
    around any LL or H”
    Step 0b: Add footing markup
    (foot-based grammars only)
    Input:
    .
    .
    .
    P[L]W[L]
    .
    .
    .

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  24. IMPLEMENTATION: DEFINE FEET/SYLLABLES
    24
    Step 1: Define foot/syllable types
    define Light [ “[“ “L” “]” “;
    define WeakLight [“W” Light ];
    “A weak light is a light immediately preceded by W.”
    “A light syllable is the sequence [L]”
    “A lapse is two adjacent weak lights.”
    define W2 [ WeakLight WeakLight ];

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  25. IMPLEMENTATION: DEFINE FEET/SYLLABLES
    25
    define Foot [“(“ [\[ “(“ | “)” ]]* “)”];
    define PrimaryFoot [ Foot & $[“P”] ];
    “A foot is a string of non-parentheses enclosed in parentheses.”
    “A primary-stressed foot is a foot that contains P.”
    Step 1: Define foot/syllable types

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  26. IMPLEMENTATION: ENFORCE SURFACE RESTRICTIONS
    26
    Step 2: Enforce surface restrictions
    “The end of a word must coincide with the end of a primary-
    stressed foot.” (Edgemost-R)
    define EdgemostR [ \P* PrimaryFoot ];
    “A word may not contain a lapse”
    define No1Lapse ~[$W2]];

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  27. RESULTS: SUCCINCTNESS
    27
    Syllables only
    Syllables and
    feet
    Direct
    Direct,
    syllables only
    145
    Direct,
    syll. and feet
    141
    Karttunen OT
    Karttunen OT,
    syllables only
    >1000
    Karttunen OT,
    syll. and feet
    306
    ‣ Direct accounts: feet don’t increase succinctness
    ‣ OT accounts: feet do increase succinctness
    ‣ OT syllable grammar shows blowup!

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  28. KARTTUNEN OT, SYLLABLES ONLY: LAPSE FAMILY
    28
    Penalize 1 lapse: 10 symbols
    Penalize 2 lapses: 25 symbols
    Penalize 3 lapses: 61 symbols
    Penalize 4 lapses: 131 symbols
    Requires violation counting, doing arithmetic
    Finite system can’t make infinitely
    many degrees of well-formedness!
    define No1Lapse ~[$W2]];
    define No2Lapse ~[$[Weak]^3] & ~[[$[W2]]^2];
    define No3Lapse ~[$[Weak]^4] & ~[[$[W2]]^3] & ~[?* [Weak]^3 $[[Weak]^2]] &
    ~[$[[Weak]]^2 [Weak]^3 ?*];
    define No4Lapse ~[$[Weak]^5] & ~[[$[W2]]^4] & ~[?* [Weak]^4 $[[Weak]^2]] &
    ~[$[[Weak]^2] [Weak]^4 ?*] & ~[[$[[Weak]^3]]^2] & ~[?* [Weak]^3 [$[W2]]^2]
    & ~[ [$[W2]]^2 [Weak]^3 ] & ~[$[W2] [Weak]^3 $[W2] ];

    View full-size slide

  29. KARTTUNEN OT, SYLLABLES ONLY: ALIGN-X1-L FAMILY
    29
    Requires counting, doing arithmetic
    Symbol count blow up to
    approximate infinitely many
    degrees of well-formedness!

    View full-size slide

  30. DISCUSSION
    30
    ▸ Surprisingly, except for Karttunen syllable OT account, size of
    grammars in high-level xfst definition very similar.
    ▸ Direct foot: 141 symbols (31 states, 36 arcs)
    ▸ Direct syllable: 145 symbols (27 states, 34 arcs)
    ▸ Karttunen OT foot: 306 symbols (418 states, 535 arcs)
    ▸ Karttunen OT syllable: >1000 symbols (3460 states, 4680 arcs)
    ▸ Within the Karttunen OT formalism, reference to feet does make
    the grammar more succinct
    ▸ But within “direct” approach, reference to feet does not make
    grammar more succinct

    View full-size slide

  31. FINITE STATE TRANSDUCERS: FOOTED GRAMMARS
    31
    0
    1
    0:N
    2
    0:X
    3
    0:(
    4
    0:u
    5
    0:[
    7
    0:S
    6
    0:P
    8
    H:l 0:l
    9
    L:l
    10
    L
    12
    0:[
    11
    0:[
    21
    L
    20
    H
    17
    0:]
    13
    H:l
    14
    L:l
    H:l
    15
    L:l
    16
    0:l
    22
    0:X
    23
    0:(
    H:0
    L:0
    H:0
    L:0
    H:0
    L:0
    27
    0:]
    26
    0:]
    32
    0:W
    28
    0:[
    0:P
    29
    0:S
    34
    0:[
    37
    0:[
    33
    H
    L
    38
    0:]
    39
    H
    43
    0:]
    31
    0:)
    0:(
    36
    0:X
    40
    0:[
    45
    L
    44
    H
    50
    0:]
    49
    0:]
    58
    0:(
    57
    0:X
    0:P
    67
    0:S
    66
    0:[
    76
    0:[
    H
    85
    L
    94
    0:]
    55
    0:X
    56
    0:(
    102
    0:W
    48
    0:)
    64
    0:[
    65
    0:S
    H
    74
    L
    H
    L
    75
    0:[
    53
    0:X
    54
    0:(
    111
    0:[
    118
    L
    62
    0:[
    0:P
    63
    0:S
    82
    0:]
    84
    L
    83
    H
    H
    72
    L
    0:X
    91
    0:(
    93
    0:]
    92
    0:]
    125
    0:]
    73
    0:[
    101
    0:W
    100
    0:S
    0:)
    80
    0:]
    H
    81
    L
    89
    0:(
    68
    0:X
    110
    0:[
    90
    0:]
    109
    0:[
    0:P
    98
    0:S
    77
    0:[
    117
    L
    60
    H
    L
    99
    0:W
    124
    0:]
    70
    0:]
    107
    0:[
    108
    0:[
    0:)
    131
    0:W
    115
    H
    116
    L
    L
    122
    0:]
    123
    0:]
    139
    0:[
    130
    0:W
    41
    0:X
    42
    0:(
    46
    0:[
    47
    0:S
    129
    0:)
    146
    L
    H
    51
    L
    136
    0:X
    137
    0:(
    138
    0:[
    153
    0:]
    52
    0:[
    H
    61
    L
    0:)
    59
    0:]
    143
    0:[
    0:P
    144
    0:S
    145
    L
    0:X
    69
    0:(
    71
    0:]
    150
    L
    86
    H
    151
    0:[
    152
    0:]
    78
    0:S
    79
    0:W
    157
    0:]
    95
    0:]
    H
    158
    L
    0:)
    164
    0:]
    163
    0:(
    132
    0:X
    88
    0:[
    87
    0:[
    L
    H
    97
    L
    96
    H
    L
    0:P
    168
    0:S
    140
    0:[
    169
    0:W
    174
    0:[
    106
    0:]
    105
    0:]
    173
    0:[
    0:)
    L
    178
    H
    179
    L
    114
    0:W
    184
    0:]
    185
    0:]
    121
    0:[
    103
    0:X
    104
    0:(
    112
    0:[
    113
    0:S
    190
    0:W
    128
    L
    189
    0:)
    120
    0:[
    196
    0:[
    194
    0:X
    195
    0:(
    H
    119
    L
    135
    0:]
    H
    127
    L
    0:)
    202
    L
    200
    0:[
    0:P
    201
    0:S
    126
    0:]
    134
    0:]
    0:X
    133
    0:(
    208
    0:[
    209
    0:]
    207
    L
    147
    H
    213
    0:]
    154
    0:]
    0:)
    142
    0:W
    141
    0:S
    H
    214
    L
    L
    H
    148
    0:[
    149
    0:[
    218
    0:(
    180
    0:X
    219
    0:]
    156
    L
    155
    H
    224
    0:W
    L
    0:P
    223
    0:S
    186
    0:[
    229
    0:[
    162
    0:]
    161
    0:]
    230
    0:[
    0:)
    234
    H
    235
    L
    L
    167
    0:W
    239
    0:]
    240
    0:]
    172
    0:[
    177
    L
    159
    0:X
    160
    0:(
    165
    0:[
    166
    0:S
    244
    0:)
    245
    0:W
    250
    0:X
    251
    0:(
    252
    0:[
    183
    0:]
    H
    170
    L
    171
    0:[
    256
    0:[
    0:P
    257
    0:S
    175
    0:]
    258
    L
    H
    176
    L
    0:)
    263
    0:[
    264
    0:]
    0:X
    181
    0:(
    182
    0:]
    262
    L
    191
    H
    188
    0:W
    H
    269
    L
    268
    0:]
    197
    0:]
    0:)
    187
    0:S
    L
    H
    274
    0:(
    225
    0:X
    192
    0:[
    193
    0:[
    275
    0:]
    199
    L
    198
    H
    0:P
    279
    0:S
    231
    0:[
    280
    0:W
    L
    285
    0:[
    206
    0:]
    205
    0:]
    284
    0:[
    289
    H
    290
    L
    0:)
    L
    212
    0:W
    217
    0:[
    295
    0:]
    296
    0:]
    300
    0:)
    203
    0:X
    204
    0:(
    210
    0:[
    211
    0:S
    222
    L
    301
    0:W
    228
    0:]
    305
    0:X
    306
    0:(
    307
    0:[
    216
    0:[
    H
    215
    L
    H
    221
    L
    311
    0:[
    0:P
    312
    0:S
    0:)
    220
    0:]
    313
    L
    0:X
    226
    0:(
    320
    0:]
    227
    0:]
    319
    0:[
    318
    L
    236
    H
    H
    325
    L
    324
    0:]
    241
    0:]
    233
    0:W
    0:)
    232
    0:S
    L
    H
    237
    0:[
    238
    0:[
    329
    0:(
    270
    0:X
    330
    0:]
    L
    243
    L
    242
    H
    0:P
    334
    0:S
    276
    0:[
    335
    0:W
    341
    0:[
    340
    0:[
    249
    0:]
    248
    0:]
    345
    H
    346
    L
    0:)
    255
    0:W
    L
    261
    0:[
    350
    0:]
    351
    0:]
    356
    0:W
    267
    L
    355
    0:)
    246
    0:X
    247
    0:(
    253
    0:[
    254
    0:S
    361
    0:X
    362
    0:(
    273
    0:]
    260
    0:[
    H
    259
    L
    363
    0:[
    0:)
    369
    L
    367
    0:[
    0:P
    368
    0:S
    H
    266
    L
    265
    0:]
    374
    0:[
    272
    0:]
    375
    0:]
    0:X
    271
    0:(
    373
    L
    281
    H
    277
    0:S
    H
    380
    L
    379
    0:]
    286
    0:]
    278
    0:W
    0:)
    L
    H
    385
    0:(
    314
    0:X
    282
    0:[
    283
    0:[
    386
    0:]
    0:P
    390
    0:S
    321
    0:[
    391
    0:W
    L
    288
    L
    287
    H
    294
    0:]
    293
    0:]
    394
    0:[
    395
    0:[
    299
    0:W
    L
    398
    H
    399
    L
    0:)
    402
    0:]
    403
    0:]
    304
    0:[
    310
    L
    406
    0:W
    291
    0:X
    292
    0:(
    297
    0:[
    298
    0:S
    405
    0:)
    409
    0:[
    317
    0:]
    303
    0:[
    H 302
    L
    408
    0:(
    336
    0:X
    411
    L
    0:)
    0:P
    410
    0:S
    342
    0:[
    308
    0:]
    H
    309
    L
    412
    0:[
    0:X
    315
    0:(
    413
    0:]
    316
    0:]
    0:)
    322
    0:S
    323
    0:W
    H
    414
    L
    415
    0:]
    L
    326
    H
    327
    0:[
    328
    0:[
    416
    0:W
    L
    333
    L
    332
    H
    331
    0:]
    417
    0:[
    339
    0:]
    338
    0:]
    L
    344
    0:W
    0:)
    349
    0:[
    0:X
    337
    0:(
    343
    0:S
    354
    L
    348
    0:[
    H
    347
    L
    360
    0:]
    352
    0:]
    0:)
    H
    353
    L
    359
    0:]
    357
    0:X
    358
    0:(
    364
    0:[
    0:P
    365
    0:S
    366
    0:W
    372
    0:[
    L
    370
    H
    371
    0:[
    376
    0:]
    378
    L
    377
    H
    L
    384
    0:]
    383
    0:]
    0:)
    389
    0:W
    393
    0:[
    381
    0:X
    382
    0:( 387
    0:[
    0:P
    388
    0:S
    397
    L
    H L
    401
    0:]
    392
    0:[
    19
    L
    18
    H
    0:)
    H
    396
    L
    25
    0:]
    24
    0:]
    400
    0:]
    404
    0:W
    30
    0:W
    35
    0:[
    407
    0:[
    L
    0:)
    L
    0:)
    0
    1
    0:X
    2
    0:(
    3
    0:[
    4
    0:P
    5
    0:S
    6
    L
    9
    0:]
    14
    0:(
    19
    0:S
    0:P
    24
    0:[
    7
    0:[
    12
    H
    17
    0:]
    22
    0:)
    0:(
    26
    0:X
    28
    0:[
    29
    L
    30
    0:] 0:(
    8
    0:[
    11
    L 10
    H
    H 13
    L
    18
    0:]
    16
    0:]
    15
    0:]
    21
    0:W
    23
    0:W
    25
    0:[
    27
    0:[
    L
    L 20
    0:)

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  32. DISCUSSION
    32
    ▸ Introduction of feet doesn’t make direct grammars more succinct.
    So why might we still want to use feet?
    ▸ Clearly more structure in the grammar referring to feet
    ▸ Without feet, require case-by-case stipulations in grammar
    ▸ Direct accounts more succinct than Karttunen OT accounts.
    So why use OT?
    ▸ Certain stress patterns very difficult to describe under usual
    constraints, OT framework captures universals that
    phonologists have noticed
    ▸ Any regular stress pattern could be captured in direct
    approach (unless we introduced further restrictions)

    View full-size slide

  33. CONCLUSION
    33
    ▸ Answer: not necessarily! Depends on grammar formalism.
    ▸ Here, preliminary “proof-of-concept” exploration
    ▸ Not clear that counting symbols right way to assess how well
    grammar is capturing generalizations
    ▸ Not clear that results here would generalize for other phonological
    phenomena
    ▸ Shows a way we can study concrete, specific linguistic proposals and
    engage closely with linguistic data and practice while maintaining a
    rigorous approach
    Do constituents make phonological grammars
    for Samoan word stress more succinct?

    View full-size slide

  34. FOR MORE INFORMATION…
    34
    ▸ Paper and code
    ▸ https://github.com/krismyu/smo-constituency-feet
    ▸ Related work at my academic website
    ▸ www.krisyu.org
    Merci beaucoup!

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  35. APPENDIX: CONVENTIONS FOR SYMBOL COUNTING
    35

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