Grammar as Choice

Conflict, concord, & optimality
Choice
• Grammar involves Multi-criterion Decision Making
• Similar problems arise in cognitive psychology
(Gigerenzer, Kahneman, Tversky), economics (Arrow),
neural networks (Smolensky), politics, operations
research, and so on.
• Many factors interact to determine the form of words,
phrases, sentences,…
• They need not be remotely in agreement about the best
outcome or course of action.
2
The Three Pillars of Decision
• What are the alternatives?
– from which one must choose.
• What are the criteria?
– which evaluate the alternatives.
• How do the many criteria combine into a single decision?
– given pervasive conflict among them.
3
Alternatives
• The generative stance: the alternatives are actions
• They modify, structure, re-structure, or preserve an input
• As a result, an output is defined.
• The choice is among different (In,Out) pairings.
4
An Example
• The Regular Past Tense of English
Spelled
massed
nabbed
patted
Pronounced
mæst
næbd
pætəd
Observed Suffix
-t
-d
-əd
5
An Example
• The Regular Past Tense of English
Spelled
massed
nabbed
patted
Pronounced
mæst
næbd
pætəd
Observed Suffix
-t
-d
-əd
6
An Example
• The Regular Past Tense of English
Spelled
massed
nabbed
patted
Pronounced
mæst
næbd
pætəd
Observed Suffix
-t
-d
-əd
 No overlap in distribution of suffix variants
7
An Example
• The Regular Past Tense of English
Spelled
massed
nabbed
patted
Pronounced
mæst
næbd
pætəd
Observed Suffix
-t
-d
-əd
 No overlap in distribution of suffix variants
 Suffix variants highly similar phonetically
8
An Example
• The Regular Past Tense of English
Spelled
massed
nabbed
patted
Pronounced
mæst
næbd
pætəd
Observed Suffix
-t
-d
-əd
 No overlap in distribution of suffix variants
 Suffix variants highly similar phonetically
 Choice of variant entirely predictable on general grounds
9
Regular Past Tense Suffix
-t
-ed
-d
10
Regular Past Tense Suffix
-ed
-t
d
-d
Regular Past Tense Suffix
-ed
-t
d
-d
Similarity ← Identity
There is just one suffix: /d/
Lexical Representation
Lexical Representation
• ‘massed’
• ‘nabbed’
• ‘patted’
mæs+d
næb+d
pæt+d
• Relations
Elementary Actions
dd
d t
d  -əd
nil
devoice
insert
13
Dilemmas of Action
• Reluctance
 +voi  –voi
 Ø  ə
doesn’t remove all b,d,g’s from the language
doesn’t spray schwas into every crevice
• Compliance
– Faithful reproduction of input not possible:
• *mæsd, * pætd
 Action is taken only to deal with such problems
• Choices, choices
– Insertion solves all problems. Yet we don’t always do it.
*mæsəd is entirely possible (cf. ‘placid’)
14
The Two Classes of Criteria
.
Markedness. Judging the outcome e.g.
*Diff(voi). (Final) Obstruent clusters may not differ in voicing.
*pd, *bt, *td, *ds, *zt, etc.
*Gem. Adjacent consonants may not be identical.
*tt, *dd, *bb,… [in pronunciation]
This analysis follows Bakovic 2004.
Faithfulness. Judging the action.
Input=Output in a certain property
Every elementary action is individually proscribed: e.g.
NoDevoicing.
NoInsertion.
NoDeletion.
15
The Two Classes of Criteria
.
Markedness. Judging the outcome e.g.
*Diff(voi). (Final) Obstruent clusters may not differ in voicing.
*pd, *bt, *td, *ds, *zt, etc.
*Gem. Adjacent consonants may not be identical.
*tt, *dd, *bb,… [in pronunciation]
This analysis follows Bakovic 2004.
Faithfulness. Judging the action.
Input=Output in a certain property
Every elementary action is individually proscribed: e.g.
NoDevoicing.
NoInsertion.
NoDeletion.
16
The Two Classes of Criteria
.
Markedness. Judging the outcome
Demands compliance with output standards
Faithfulness. Judging the action.
Enforces reluctance to act
17
Penalties
• Constraints assess only penalties
– no rewards for good behavior
• Actions are reluctant because constraints on action
always favor inaction — by penalizing change.
• Actions happen because constraints on outcome force
violation of constraints against action.
18
Conflicts Abound
• The faithfulness constraints disagree among themselves
• And M:*Diff disagrees with F:NoDevoicing.
*Gem *Diff
W:
mæs+d  mæst
NoIns NoDev
Action
0
0
0
1
dev
L: mæsəd
0
0
1
0
ins
L:
0
1
0
0
nil
mæsd
19
Conflicts Abound
• The faithfulness constraints disagree among themselves
W:
*Gem *Diff
NoIns NoDev
0
0
0
L: mæsəd
0
0
1
L:
0
1
mæs+d  mæst
mæsd
W
0
1
W
Action
dev
0
L
ins
0
L
nil
 W marks preference for desired winner;  L preference for desired loser
20
Conflicts Abound
• The faithfulness constraints disagree among themselves
• And M:*Diff disagrees with F:NoDev.
W:
*Gem *Diff
NoIns NoDev
0
0
0
L: mæsəd
0
0
1
L:
0
1
mæs+d  mæst
mæsd
W
0
1
W
Action
dev
0
L
ins
0
L
nil
21
All Conflicts Resolved
• Impose a strict priority order ‘>>’ on the set of constraints
– Here: *Gem, *Diff >> NoIns >>NoDel
• In any pairwise comparison of x vs. y
x  y ‘x is better than y’
iff the highest-ranked constraint distinguishing x from y
prefers x.
• Optimal. x is optimal iff x  y for every y
y violationwise distinct from x
22
Lexicographic
• Better Than, ‘’: lexicographic order on the alternatives.
– Sort by the highest ranked constraint
• If it does not decide, on to the next highest.
– And so on.
• Like sorting by first letter (able < baker)
– and then the next, if that doesn’t decide: (aardvark<abacus)
• and then the next (azimuth < azure), and so on.
• Or ordering numerals by place
100 < 200
119 < 130
2235 < 2270
23
Optimality Theory
• Alternatives.
– A set of (input,output) pairs.
– A given input is matched with every possible output.
• Criteria.
– A set of constraints, of two species
• Markedness: judging outcomes
• Faithfulness: judging actions
• Collective judgment.
– Derives from a strict prioritization of the constraint set.
• Imposes lexicographic order on alternatives. Take the best.
24
Universality
To make maximal use of theoretical resources
and minimal commitment to extraneous devices, assume:
• Fixed.
– The set of alternatives is universal.
• Fixed.
– The set of constraints is universal.
• Varying.
– Languages differ freely in the ranking of the constraint set.
25
Harmonic Ascent

Getting better all the time
Beyond Replication
• Faithful mapping: In=Out
‘nabbed’
næb+d  næbd
• What does it take to beat the faithful candidate?
– Moreton 2002, 2004 asks and answers this question.
• Fully Faithful xx satisfies every F constraint.
– Nothing can do better than that on the F’s.
• Nonfaithful xy beats faithful xx iff
– The highest ranked constraint distinguishing them
prefers xy
27
Beyond Replication
• Faithful mapping: In=Out
‘nabbed’
næb+d  næbd
• What does it take to beat the faithful candidate?
– Moreton 2002, 2004 asks and answers this question.
• Fully Faithful xx satisfies every F constraint.
– Nothing can do better than that on the F’s.
• Nonfaithful xy beats faithful xx iff
– The highest ranked constraint distinguishing them
prefers xy
28
Triumph of Markedness
That decisive constraint must be a Markedness constraint.
– Since every F is happy with the faithful candidate.
29
Triumph of Markedness
That decisive constraint must be a Markedness constraint.
– Since every F is happy with the faithful candidate.
M:*Gem M:*Diff F:NoIns
W: pæd+d  pædəd 0
L: pædd 1 W
NoDev
Action
0
1
0
Ins
0
0 L
0
faithful
30
Harmonic Ascent = Markedness Descent
• For a constraint hierarchy H, let H|M be the subhierarchy
of Markedness constraints within it.
• If H:α  φ, for φ fully faithful, then H|M: α  φ
– If things do not stay the same, they must get better.
• Analysis and results due to Moreton 2002, 2004.
31
Markedness Rating by H|M
M: *Diff(voi) >>
M:*Voi
Good
pt, bd (0)
pt
bd
(0)
(2)
bt, pd (1)
bt, pd (1)
Bad
 Note lexicographic refinement of classes
Constraints from Lombardi 1999
32
Markedness-Admissible Mappings
Good
pt
bd
bt
Bad
pd
 Where you stop the ascent, and if you can, depends on H|F.
33
Utterly Impossible Mappings
Good
pt
bd
bt
Bad
pd
34
Consequences of Harmonic Ascent
• No Circular Shifts in MF/OT
Shifts that happen
– Western Basque (Kirchner 1995)
a→e
e→i
– Catalan
alaba+a → alabea
seme+e → semie
(Mascaró 1978, Wheeler 1979)
nt → n
n →Ø
kuntent → kunten
plan
→ pla
 Analyzed recently in Moreton & Smolensky 2002
35
 No Circular Shifts
• Harmonic Ascent
– Any such shift must result in betterment vis-à-vis H|M.
– The goodness order imposed on alternatives is
• Asymmetric: NOT[ a b & b a]
• Transitive: [a b & b c]  a b
• Can’t have
• x→y
• y→z
• z→x
• Such a cycle would give: x  x
(contradiction!)
36
Way Up ≠ Way Down
Good
z
y
Bad
x
37
Shift Data
• Large numbers exist
– Moreton & Smolensky collect 35 segmental cases
• 3 doubtful, 4 inferred: 28 robustly evidenced.
• One potential counterexample
– Taiwanese/ Xiamen Tone Circle
– See Yip 2002, Moreton 2002, and many others for discussion.
38
Coastal Taiwanese Tone Shifts
Diagram from Feng-fan Hsieh,
http://www.ling.nthu.edu.tw/teal/TEAL_oral_FengFan_Hsieh.pdf
39
Not the True Article?
• No basis in justifiable Markedness for shifts (Yip).
• “Paradigm Replacement”
– Moreton 2002. Yip 1980, 2002. Chen 2002. Mortensen 2004.
Hsieh 2004. Chen 2000.
40
 No Endless Shifts
NO: x → y →z → … → ……
41
 No Endless Shifts
NO: x → y →z → … → ……
•
E.g: “Add one syllable to input”
42
 No Endless Shifts
NO: x → y →z → … → ……
•
E.g: “Add one syllable to input”
• Because constraints only penalize,
there is an end to getting better.
43
 No Endless Shifts
NO: x → y →z → … → ……
•
E.g: “Add one syllable to input”
• Because constraints only penalize,
there is an end to getting better.
 This is certainly a correct result.
— we can add one syllable to hit a fixed target (e.g. 2 sylls.)
not merely to expand regardless of shape of outcome.
44
Conclusions
• Harmonic Ascent and its consequences nontrivial, since
mod of theory can easily eliminate. E.g. ‘Antifaithfulness.’
• Design of the theory succeeds in taking property of
atomic components (single M constraint) and
propagating it to the aggregate judgment.
• Requires: transitive, asymmetric order, commitment to
penalization, strict limitation to M & F constraints.
45
Concord

Nonconflict in OT
Constraints in conflict
a
C1
0
C2
1
b
1
0
47
Constraints in conflict
a
C1
0
C2
1
b
1
0
ab
48
Constraints in conflict
a
C1
0
C2
1
b
1
0
ab
ba
49
Constraints need not conflict
B1
B2
a
0
0
b
0
1
c
1
1
50
Constraints need not conflict
B1
B2
a
0
0
b
0
1
c
1
1
51
Constraints need not conflict
B1
B2
a
0
0
b
0
1
c
1
1
ac
ac
52
Constraints need not conflict
B1
B2
a
0
0
b
0
1
c
1
1
a?b
ab
53
Constraints need not conflict
B1
B2
a
0
0
b
0
1
c
1
1
ab
54
Constraints need not conflict
B1
B2
a
0
0
b
0
1
c
1
1
55
Constraints need not conflict
B1
B2
a
0
0
b
0
1
c
1
1
bc
56
Constraints need not conflict
B1
B2
a
0
0
b
0
1
c
1
1
ac
bc
ac
ab
57
Constraints need not conflict
B1
B2
a
0
0
b
0
1
c
1
1
abc
 regardless of ranking
58
Constraints and Scales
• Imagine a goodness scale
abcd
59
Abstract Scale
better
a
b
c
d
60
Constraints and Scales
abcd
• Consider every bifurcation:
good  bad
abc  d
B1 = *{d}
ab  cd
B2 = *{c,d}
a  bcd
B3 = *(b,c,d}
61
B1
better
a
b
c
d
62
B2
better
a
b
c
d
63
B3
better
a
b
c
d
64
Binary Constraints in Stringency Relation
B1
B2
B3
a
0
0
0
b
0
0
1
c
0
1
1
d
1
1
1
abc d
ab cd
a bcd
65
Generating Conflations
• From B1, B2, B3 any respectful coarsening of the scale
may be generated
• B1 & B2 = ab  c  d
– i.e., abc d & abcd
• B2 & B3 = a  b  cd
– i.e., abcd & a bcd
• B1 & B2 & B3 = a  b  c  d
and so on…
66
Generating Conflations
• From B1, B2, B3 any respectful coarsening of the scale
may be generated
• B1 & B2 = ab  c  d
– i.e., abc d & abcd
• B2 & B3 = a  b  cd
– i.e., abcd & a bcd
• B1 & B2 & B3 = a  b  c  d
and so on…
67
Generating Conflations
• From B1, B2, B3 any respectful coarsening of the scale
may be generated
• B1 & B2 = ab  c  d
– i.e., abc d & abcd
• B2 & B3 = a  b  cd
– i.e., abcd & abcd
• B1 & B2 & B3 = a  b  c  d
and so on…
68
Generating Conflations
• From B1, B2, B3 any respectful coarsening of the scale
may be generated
• B1 & B2 = ab  c  d
– i.e., abc d & abcd
• B2 & B3 = a  b  cd
– i.e., abcd & a bcd
• B1 & B2 & B3 = a  b  c  d
and so on…
69
B1 & B2
better
a
b
c
d
70
Full DNC on 4 candidates
B1
B2
T12
B3
T13
T23 Q123
a
0
0
0
0
0
0
0
b
0
0
0
1
1
1
1
c
0
1
1
1
1
2
2
d
1
1
2
1
2
2
3
These Do Not Conflict 
71
Full DNC on 4 candidates
B1
B2
T12
B3
T13
T23 Q123
a
0
0
0
0
0
0
0
b
0
0
0
1
1
1
1
c
0
1
1
1
1
2
2
d
1
1
2
1
2
2
3
72
Full DNC on 4 candidates
B1
B2
T12
B3
T13
a
0
0
0
0
0
0
0
b
0
0
0
1
1
1
1
c
0
1
1
1
1
2
2
d
1
1
2
1
2
2
3
 B1 +
T23 Q123
B2 = T12
73
B1 & B2
better
a
b
c
d
74
Linguistic Scales
• Particularly informative is the relation between scales of
relative sonority and placement of stress.
• This allows us to probe the varying behavior of similar
scales across languages.
77
Intrinsic Sonority of vowels
a
eo
iu
schwa
78
Sonority-Sensitive Stress
• Main-stress falls in a certain position
– say, 2nd to last syllable:
xXx
• Except when adjacent vowel has greater sonority
– then the stronger vowel attracts the stress:
Xxx
• This perturbation evidences the fine structure of the
scale.
79
Sonority-Sensitive Stress
Chukchi
(Kenstowicz 1994, Spencer 1999)
• Typically base-final when suffixed:
jará-ŋa
reqokál-gən
welól-gən
piŋé-piŋ
xX+x
migcirét-ək
wiríŋ-ək
ekwét-ək
nuté-nut
• But one syll. back when stronger available:
céri-cer
*cerí-cer
e>i
Xx+x
kéli-kel
wéni-wen
80
Sonority-Sensitive Stress
• Schwa yields to any other
vowel
–
–
–
–
–

ətlá
?əló
ənré
γənín
γənún
a,o, e, i, u > ə
• But behaves normally
with itself
– ə́tləq
– ə́ttəm
– kə́tγət
– cə́mŋə
 ə=ə
NB. stress typically avoids the last syllable of the word.
81
Chukchi Scale
• These considerations motivate a scale like this:
aeo> iu > ə
• In terms of goodness of fit wrt stress:
áéó  íú  ə́
82
Intrinsic Sonority of vowels
a
eo
iu
schwa
83
Flattened Chukchi Scale
better
á
é,ó
í,ú
ə́
84
B1 & B2
a
b
c
d
85
Achieving Chukchi
• How does this relate to the full scale that registers every
level of distinction?
• To coarsen the scale in the Chukchi fashion,
we must disable B3 and activate both B1 and B2.
• Ranking will yield this.
86
Ranking?
• How can the Bi’s be ranked? They don’t conflict!
87
Ranking?
• How can the Bi’s be ranked? They don’t conflict!
• Transitivity. Find a constraint C with which they conflict.
88
Ranking?
• How can the Bi’s be ranked? They don’t conflict!
• Transitivity. Find a constraint C with which they conflict.
{B1, B2} >>
C
89
Ranking?
• How can the Bi’s be ranked? They don’t conflict!
• Transitivity. Find a constraint C with which they conflict.
{B1, B2} >>
C
>> {B3}
90
Ranking?
• How can the Bi’s be ranked? They don’t conflict!
• Transitivity. Find a constraint C with which they conflict.
{B1, B2} >>
C
>> {B3}
91
Ranking?
• How can the Bi’s be ranked? They don’t conflict!
• Transitivity. Find a constraint C with which they conflict.
{B1, B2} >>
C
>> {B3}
 Here C demands stress in a certain position
92
The Hierarchy
•
B1, B2 >> POS >> B3
93
The Hierarchy
•
B1, B2 >> POS >> B3
– Stress flees from ə to iueoa (B1)
94
The Hierarchy
•
B1, B2 >> POS >> B3
– Stress flees from ə to iueoa (B1)
– Stress flees from əiu to eoa (B2)
95
The Hierarchy
• B1, B2 >> POS >> B3
– Stress flees from ə to iueoa (B1)
– Stress flees from əiu to eoa (B2)
– The distinction eo/a is ignored (B3)
96
The Hierarchy
•
B1, B2 >> POS >> B3
– Stress flees from ə to iueoa (B1)
– Stress flees from əiu to eoa (B2)
– The distinction eo/a is ignored.
• Conjunctivity.
– Because B1 and B2 do not conflict, their demands are
both met.
– see Samek-Lodovici & Prince 1999, 21 ‘Favoring Intersection Lemma’
97
The Optima
•
B1,B2 >> POS >> B3
W~L
B1 = *ə
B2 = *íúə
POS
B3 = *éóíúə
98
The Optima
•
B1,B2 >> POS >> B3
W~L
1.
jará-ŋa
~ jára-ŋa
B1 = *ə
B2 = *íúə
POS
B3 = *éóíúə
W
99
The Optima
•
B1,B2 >> POS >> B3
W~L
1.
jará-ŋa
2. jatjólte
B1 = *ə
B2 = *íúə
POS
~ jára-ŋa
W
~ játjolte
W
B3 = *éóíúə
L
100
The Optima
•
B1,B2 >> POS >> B3
W~L
1.
jará-ŋa
2. jatjólte
3.
kélikel
B1 = *ə
B2 = *íúə
POS
~ jára-ŋa
W
~ játjolte
W
~ kelíkel
W
B3 = *éóíúə
L
L
101
The Optima
•
B1,B2 >> POS >> B3
W~L
B1 = *ə
B2 = *íúə
POS
~ jára-ŋa
W
2. jatjólte
~ játjolte
W
3.
kélikel
~ kelíkel
4.
ətlá
~ ə́tla
1.
jará-ŋa
W
W
B3 = *éóíúə
L
L
L
102
The Optima
•
B1,B2 >> POS >> B3
W~L
B1 = *ə
B2 = *íúə
POS
~ jára-ŋa
W
2. jatjólte
~ játjolte
W
3.
kélikel
~ kelíkel
4.
ətlá
~ ə́tla
5.
ə́tləq
~ ə́tləq
1.
jará-ŋa
W
W
B3 = *éóíúə
L
L
L
W
103
The Ranking
•
B1,B2 >> POS >> B3
W~L
B1 = *ə
B2 = *íúə
POS
~ jára-ŋa
W
2. jatjólte
~ játjolte
W
3.
kélikel
~ kelíkel
4.
ətlá
~ ə́tla
5.
ə́tləq
~ ə́tləq
1.
jará-ŋa
W
W
B3 = *éóíúə
L
L
L
W
104
Currently Known Conflations
ə
i/u
ə
i/u
ə
i/u
e/o
ə
i/u
e/o
ə
i/u
e/o
ə
i/u
ə
i/u
Exemplar
Determining Constraints
Yil
B1
Chukchi
B1, B2
Kobon
B1, B2, B3
Nganasan
B2
a
Kara
B3
a
Gujarati
B1, B3
e/o
e/o
e/o
e/o
a
a
a
a
a
Adapted from de Lacy 2002
106
Conclusion
• All types currently attested except B2+B3
• Assumptions
– Simplest binary interpretation of scale in constraints
– Free ranking of all constraints, as usual
• Result
– All respectful collapses are generated
– Nonconflict automatically provides a theory of scales in OT
107
Optimality

Harmonic bounding
Here Comes Everybody
• Alternatives. Come in multitudes.
• But many rankings produce the same optima.
– Not all constraints conflict
• Extreme formal symmetry to produce all possible optima
– Not often encountered ecologically
109
Completeness & Symmetry
Perfect System on 3 constraints.
C1
C2
C3
α-1
0
1
2
α-2
0
2
1
α-3
1
0
2
α-4
1
2
0
α-5
2
0
1
α-6
2
1
0
110
Completeness & Symmetry
Perfect System on 3 constraints.
C1
C2
C3
α-1
0
1
2
α-2
0
2
1
α-3
1
0
2
α-4
1
2
0
α-5
2
0
1
α-6
2
1
0
111
Completeness & Symmetry
Perfect System on 3 constraints.
C1
C2
C3
α-1
0
1
2
α-2
0
2
1
α-3
1
0
2
α-4
1
2
0
α-5
2
0
1
α-6
2
1
0
112
Completeness & Symmetry
Perfect System on 3 constraints.
C1
C2
C3
α-1
0
1
2
α-2
0
2
1
α-3
1
0
2
α-4
1
2
0
α-5
2
0
1
α-6
2
1
0
113
Optima and Alternatives
• Limited range of possible optima
– Much, much less than n! for n constraint system
• But there are Alternatives Without Limit.
– Augmenting actions (insertion, adjunction, etc.) increase size
and number of alternatives, no end in sight.
• Where is everybody?
114
Harmonic Bounding
• Many candidates — ‘almost all’ — can never be optimal
115
Harmonic Bounding
• Many candidates — ‘almost all’ — can never be optimal
• Example: Profuse insertion
a. pæd+d  pædəd
b.
əpædəd
*Gem
*Diff
NoIns
NoDev
Action
0
0
1
0
Ins
0
0
2
0
Ins x 2
116
Harmonic Bounding
• Many candidates — ‘almost all’ — can never be optimal
• Example: Profuse insertion
a. pæd+d  pædəd
b.
əpædəd
*Gem
*Diff
NoIns
NoDev
Action
0
0
1
0
Ins
0
0
2
0
Ins x 2
Candidate (b) has nothing going for it.
It is equal to (a) — or worse than it — on every constraint
117
Harmonic Bounding
• Attempt the overinserted candidate as desired optimum
W~L
*Gem
pæd+d  əpædəd ~ pædəd
*Diff
NoIns
NoDev
L
• It can’t win this competition:
– no constraint prefers it,
– and one prefers its competitor !
118
Harmonic Bounding
• Generically
W~L
α~β
C1
C2
L
C3
…
Cn
(L)
• If there is no constraint on which α  β, for α  β violationwise,
— no W in the row — and at least one L —
then α can never be optimal.
• β is always better, so α can’t be the best
– Even if β itself is not optimal, or not possibly optimal !
• e.g. 19 is not the smallest positive number because 18<19.
119
Harmonic Bounding
• Harmonic Bounding is a powerful effect
– E.g. Almost all insertional candidates are bounded
– This gives us a highly predictive theory of insertion
120
Harmonic Bounding
• Harmonic Bounding is a powerful effect
– E.g. Almost all insertional candidates are bounded
– This gives us a highly predictive theory of insertion
• Even though there are no restrictions on insertions at all in
defining the set of possible alternatives!
121
Harmonic Bounding
• Harmonic Bounding is a powerful effect
– E.g. Almost all insertional candidates are bounded
– This gives us a highly predictive theory of insertion
• Even though there are restrictions on insertion at all in
defining the set of possible alternatives!
• But we’re not done.
– Simple Harmonic Bounding works without ranking
– Any positively weighted combination of violation scores will show
the effect.
122
Collective Harmonic Bounding
• A ranking will not exist unless all competitions
can be won simultaneously
W~L
C1
C2
α~β
W
L
α~δ
L
W
• Neither C1 nor C2 may be ranked above the other
– If C1>>C2, then δ  α
– If C2 >>C1 then β  α
•
β and δ cooperate to stifle α
123
Collective Harmonic Bounding
• An example from Basic Syllable Theory
/bk/
bk 
No-Del
No-Ins
Action
ba
1
1
Ins+Del
ba.ka.
0
2
Ins x 2
ØØ
2
0
Del x 2
124
Collective Harmonic Bounding
• An example from Basic Syllable Theory
/bk/
bk 
No-Del
No-Ins
Action
ba
1
1
Ins+Del
ba.ka.
0
L
2
W
Ins x 2
ØØ
2
W
0
L
Del x 2
125
Collective Harmonic Bounding
• The middle way is no way.
β
0
2
*α
1
1
δ
2
0
126
General Harmonic Bounding
• Def. Candidate α is harmonically bounded
by a nonempty set of candidates B, xB, over a
constraint set S iff for every xB, and for every CS,
if C: αx, then there is a yB such that C: yα.
• If any member of B is beaten by α on a constraint C,
another member of B comes to the rescue, beating α.
– If any α~x earns W, then some α~y earns L.
– If B has only one member, then α can never beat it.
• No harmonically bounded candidate can be optimal.
127
General Harmonic Bounding
• Def. Candidate α is harmonically bounded
by a nonempty set of candidates B, xB, over a
constraint set S iff for every xB, and for every CS,
if C: αx, then there is a yB such that C: yα.
• If any member of B is beaten by α on a constraint C,
another member of B comes to the rescue, beating α.
– If any α~x earns W, then some α~y earns L.
– If B has only one member, then α can never beat it.
• No harmonically bounded candidate can be optimal.
128
General Harmonic Bounding
• Def. Candidate α is harmonically bounded
by a nonempty set of candidates B, xB, over a
constraint set S iff for every xB, and for every CS,
if C: αx, then there is a yB such that C: yα.
• If any member of B is beaten by α on a constraint C,
another member of B comes to the rescue, beating α.
– If any α~x earns W, then some α~y earns L.
– If B has only one member, then α can never beat it.
• No harmonically bounded candidate can be optimal.
129
Some Stats
• Tesar 1999 studies a system of 10 prosodic constraints.
– with a large number of prosodic systems generated
• Among the 4 syllable alternatives
– ca. 75% are bounded on average
– ca. 16% are collectively bounded (approx. 1/5 of bounding cases)
• Among the 5 syllable alternatives
– ca. 62% are bounded
– ca. 20% are collectively bounded (approx. 1/3 of bounding cases)
 Calculated in Samek-Lodovici & Prince 1999
130
Some Stats
• Tesar 1999 studies a system of 10 prosodic constraints.
– with a large number of prosodic systems generated
• Among the 4 syllable alternatives
– ca. 75% are bounded on average
– ca. 16% are collectively bounded (approx. 1/5 of bounding cases)
• Among the 5 syllable alternatives
– ca. 62% are bounded
– ca. 20% are collectively bounded (approx. 1/3 of bounding cases)
 Calculated in Samek-Lodovici & Prince 1999
131
Bounding in the Large
• Simple Harmonic Bounding is ‘Pareto optimality’
– An assignment of goods is Pareto optimal or ‘efficient’ if there’s
no way of increasing one individual’s holdings without
decreasing somebody else’s.
– Likewise, it is non-efficient if someone’s holdings can be
increased without decreasing anybody else’s.
– A simply bounded alternative is non-Pareto-optimal. We can
better its performance on some constraint(s) without worsening it
on any constraint.
• Collective Harmonic Bounding is the creature of freely
permutable lexicographic order.
– See Samek-Lodovici & Prince 1999 for discussion.
132
Intuitive Force of Bounding
• Simple Bounding relates to the need for individual
constraints to be minimally violated.
•
If we can get (0,0,1,0) we don’t care about (0,0,2,0).
133
Intuitive Force of Bounding
• Collective Bounding reflects the taste of lexicographic
ordering for extreme solutions.
• If a constraint is dominated, it will accept any number of
violations to improve the performance of a dominator.
• There is no compensation for a high-ranking violation
• If (1,1) meets (0,k), the value of k is irrelevant.
134
Explanation from Bounding
• Bounded alternatives are linguistically impossible.
• Yet their impossibility is not due to a direct restriction on
linguistic structure.
• Impossibility follows from the interaction of constraints
under ranking.
• Explanation emerges from the architecture of the theory.
135
Grammar as Choice

Conclusion, retrospect, & overview
Among the Cognitive Sciences
• Perspectives on cognitive theory tend to bifurcate
discrete math
continuous math
logic
probability
symbolic
featural
rule, constraint
association
ordinal preference
utility function
innate
nihil in intellectu
See esp. Smolensky’s work for analysis
137
Among the Cognitive Sciences
• OT sits on the left side of every opposition
• But in every case there is currently an active technical
interchange between advocates and critics leading to
new understanding of the relations between apparent
dichotomies.
• In psychology of reasoning, e.g., Gigerenzer and
colleagues argue for the use of criteria under
lexicographic order.
138
Gigerenzer &Goldstein 1996
139
Fast and Frugal
• For Gigerenzer et al. the main contrast is with Bayesian
probabilistic calculation over alternatives.
• Lexicographic choice is ‘one reason’ decision making
– i.e. at the level of deciding between 2 alternatives
– Therefore, fast and frugal.
• OT aims for neither speed nor frugality, but deploys the
same mechanism of lexicographic decision-making
140
Looking Both Ways
• OT seeks to explain the basic properties of human
language through a formal theory of the linguistic faculty.
• OT, as a lexicographic theory of ordinal preference,
points toward new kinds of connections with the
cognitive apparatus that acquires and uses grammatical
knowledge.

141
Thanks
• Thanks to Vieri Samek-Lodovici, Paul Smolensky,
John McCarthy, Jane Grimshaw, Paul de Lacy, Alison
Prince, Adrian Brasoveanu, Naz Merchant, Bruce
Tesar, Moira Yip.
142
Where to learn more about OT
• http://roa.rutgers.edu
• Many researchers have made their work freely available
at the Rutgers Optimality Archive.
• Thanks to the Faculty of Arts & Sciences, Rutgers
University for support.
143
References
ROA = http://roa.rutgers.edu
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Alderete, J. 1999. Morphologically governed accent in Optimality Theory. ROA-393.
Arrow, K. 1951. Social choice and Individual Values. Yale.
Bakovic, E. 2004. Partial Identity Avoidance as Cooperative Interaction. ROA-698.
Chen, M. 2000. Tone Sandhi. CUP.
de Lacy, Paul. 2002. The Formal Expression of Markedness. ROA-542.
Gigerenzer, G., P. Todd, and the ABC Research Group. Simple Heuristics that Make us Smart.
OUP.
Gigerenzer, G. and D. Goldstein. 1996. Reasoning the fast and frugal way: Models of bounded
rationality. Psych. Rev. 103, 650-669.
Hsieh, Feng-fan. 2004. Tonal Chain-shifts as Anti-neutralization-induced Tone Sandhi. In
Proceedings of the 28th Penn Linguistics Colloquium. http://web.mit.edu/ffhsieh/www/ANTS.pdf
Kager, R. Optimality Theory. [Textbook]. CUP.
Kirchner, 1995. Going the distance: synchronic chain shifts in OT. ROA-66.
Kirchner, Robert. 1996. Synchronic chain shifts in optimality theory. LI 27:2: 341-350.
Lombardi, L. 1999. Positional Faithfulness and Voicing Assimilation in Optimality Theory. NLLT 17,
267-302.
Lubowicz, A. 2002. Contrast Preservation in Phonological Mappings. ROA-554
Mascaró, J. 1978. Catalan Phonology and the Phonological Cycle. Ph. D.
144
dissertation, MIT. Distributed by Indiana University Linguistics Club.
References
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
McCarthy, J. 2002. A Thematic Guide to Optimality Theory. CUP.
McCarthy, J., ed. 2004. Optimality Theory in Phonology. Blackwell.
Moreton, E. 2002, 2004. Non-Computable Functions in Optimality Theory. ROA-364. Revised, in
McCarthy 2004, pp.141-163.
Moreton, E. and P. Smolensky. 2002. Typological consequences of local constraint conjunction.
ROA-525.
Mortensen, D. 2004. Abstract Scales in Phonology. ROA-667.
Prince, A. 1997ff. Paninian Relations. http://ling.rutgers.edu/faculty/prince.html
Prince, A.2002. Entailed Ranking Arguments. ROA-500
Prince, A. 2002. Arguing Optimality. ROA-562.
Prince, A. and P. Smolensky, 1993/2004. Optimality Theory: Constraint Interaction in Generative
Grammar. Blackwell. ROA-537.
Samek-Lodovici, V. and A. Prince. 1999. Optima. ROA-363.
Samek-Lodovici, V. and A. Prince. Fundamental Properties of Harmonic Bounding. RuCCS-TR-71.
http://ruccs.rutgers.edu/tech_rpt/harmonicbounding.pdf
Smolensky, P and G. Legendre. To appear 2005. The Harmonic Mind. MIT.
Spencer, A. 1999. Chukchee.
http://privatewww.essex.ac.uk/~spena/Chukchee/chapter2.html#stress
Wheeler, Max. 1979. Phonology of Catalan. Blackwell.
Yip, M. 2002. Tone. CUP.
145
Grammar as Choice

Conflict, concord, & optimality
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