Grammar as Choice

Grammar as Choice

Conflict, concord, & optimality

2

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.

3

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.

4

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.

5

An Example

• The Regular Past Tense of English

Spelled Pronounced Observed Suffixmassed mæst -tnabbed næbd -dpatted pætəd -əd

6

An Example



7

An Example



No overlap in distribution of suffix variants

8

An Example




Suffix variants highly similar phonetically

9

An Example




Suffix variants highly similar phonetically

Choice of variant entirely predictable on general grounds

10

Regular Past Tense Suffix

-t-ed

-d

-t -ed

-d


d

-t -ed

-d


d

Similarity ← Identity There is just one suffix: /d/

13

Lexical Representation

Lexical Representation• ‘massed’ mæs+d• ‘nabbed’ næb+d• ‘patted’ pæt+d

• Relations Elementary Actions d d nild t devoiced -əd insert

14

Dilemmas of Action

• Reluctance +voi –voi doesn’t remove all b,d,g’s from the language Ø ə 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’)

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


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.

17


Markedness. Judging the outcome.Demands compliance with output standards

Faithfulness. Judging the action.

Enforces reluctance to act

18

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.

19

Conflicts Abound

• The faithfulness constraints disagree among themselves

• And M:*Diff disagrees with F:NoDevoicing.

*Gem *Diff NoIns NoDev Action

W: mæs+d mæst 0 0 0 1 dev

L: mæsəd 0 0 1 0 ins

L: mæsd 0 1 0 0 nil

20

Conflicts Abound




L: mæsəd 0 0

1 W 0 L ins

L: mæsd 0 1 W 0 0 L nil

W marks preference for desired winner; L preference for desired loser

21

Conflicts Abound


• And M:*Diff disagrees with F:NoDev.



L: mæsəd 0 0

1 W 0 L ins

L: mæsd 0 1 W 0 0 L nil

22

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. yx 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

23

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

24

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.

25

Universality

To make maximal use of theoretical resourcesand 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.

Harmonic Ascent

Getting better all the time

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 xx satisfies every F constraint.– Nothing can do better than that on the F’s.

• Nonfaithful xy beats faithful xx iff– The highest ranked constraint distinguishing them

prefers xy

28

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 xx satisfies every F constraint.– Nothing can do better than that on the F’s.

• Nonfaithful xy beats faithful xx iff– The highest ranked constraint distinguishing them

prefers xy

29

Triumph of Markedness

That decisive constraint must be a Markedness constraint.– Since every F is happy with the faithful candidate.

30

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 NoDev Action

W: pæd+d pædəd 0 0 1 0 Ins

L: pædd 1 W 0 0 L 0 faithful

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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.

32

Markedness Rating by H|M

M: *Diff(voi) >> M:*Voi

pt, bd (0) pt (0)bd (2)

bt, pd (1) bt, pd (1)

Good

Bad

Constraints from Lombardi 1999

Note lexicographic refinement of classes

33

Markedness-Admissible Mappings

pt

bd

bt pd

Good

Bad

Where you stop the ascent, and if you can, depends on H|F.

34

Utterly Impossible Mappings

pt

bd

bt pd

Good

Bad

35

Consequences of Harmonic Ascent

• No Circular Shifts in MF/OTShifts that happen– Western Basque (Kirchner 1995)

a → e alaba+a → alabeae → i seme+e → semie

– Catalan (Mascaró 1978, Wheeler 1979)nt → n kuntent → kuntenn → Ø plan → pla

Analyzed recently in Moreton & Smolensky 2002

36

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!)

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Way Up ≠ Way Down

z

y

x

Good

Bad

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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.

39

Coastal Taiwanese Tone Shifts

Diagram from Feng-fan Hsieh, http://www.ling.nthu.edu.tw/teal/TEAL_oral_FengFan_Hsieh.pdf

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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.

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No Endless Shifts

NO: x → y →z → … → ……

42

No Endless Shifts

NO: x → y →z → … → ……

• E.g: “Add one syllable to input”

43

No Endless Shifts

NO: x → y →z → … → ……


• Because constraints only penalize, there is an end to getting better.

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No Endless Shifts

NO: x → y →z → … → ……


• 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.

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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.

Concord

Nonconflict in OT

47

Constraints in conflict

C1 C2 a 0 1 b 1 0

48


C1 C2 a 0 1 b 1 0 ab

49


C1 C2 a 0 1 b 1 0 ab ba

50

Constraints need not conflict

B1 B2

a 0 0 b 0 1 c 1 1

51


B1 B2

a 0 0 b 0 1 c 1 1

52


B1 B2

a 0 0 b 0 1 c 1 1

ac ac

53


B1 B2

a 0 0 b 0 1 c 1 1

a ? b ab

54


B1 B2

a 0 0 b 0 1 c 1 1

ab

55


B1 B2

a 0 0 b 0 1 c 1 1

56


B1 B2

a 0 0 b 0 1 c 1 1

bc

57


B1 B2

a 0 0 b 0 1 c 1 1

acbc

acab

58


B1 B2

a 0 0 b 0 1 c 1 1

a b c

regardless of ranking

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Constraints and Scales

• Imagine a goodness scale a b c d

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a b c d

Abstract Scale

better

61

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}

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a b c d

B1

better

63

a b c d

B2

better

64

a b c d

B3

better

65

Binary Constraints in Stringency Relation

B1 B2 B3

a 0 0 0

b 0 0 1

c 0 1 1

d 1 1 1abc d ab cd a bcd

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Generating Conflations

• From B1, B2, B3 any respectful coarsening of the scalemay be generated

• B1 & B2 = ab c d– i.e., abc d & abcd

• B2 & B3 = a b cd– i.e., abcd & a bcd

• B1 & B2 & B3 = a b c d and so on…

67






68




• B2 & B3 = a b cd– i.e., abcd & abcd


69






70

a b c d

B1 & B2

better

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

These Do Not Conflict

72


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

73


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

B1 + B2 = T12

74

a b c d

B1 & B2

better

77

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.

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a eo iu schwa

Intrinsic Sonority of vowels

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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.

80


Chukchi (Kenstowicz 1994, Spencer 1999)

• Typically base-final when suffixed: xX+x jará-ŋa migcirét-əkreqokál-gən wiríŋ-ək welól-gən ekwét-ək piŋé-piŋ nuté-nut

• But one syll. back when stronger available: Xx+x céri-cer *cerí-cer e>i kéli-kel wéni-wen

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• 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.

82

Chukchi Scale

• These considerations motivate a scale like this:

aeo> iu > ə

• In terms of goodness of fit wrt stress:

áéó íú ə́

83

a eo iu schwa

Intrinsic Sonority of vowels

84

á é,ó í,ú ə́

Flattened Chukchi Scale

better

85

a b c d

B1 & B2

86

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.

87

Ranking?

• How can the Bi’s be ranked? They don’t conflict!

88

Ranking?


• Transitivity. Find a constraint C with which they conflict.

89

Ranking?


• Transitivity. Find a constraint C with which they conflict.

{B1, B2} >> C

90

Ranking?


• Transitivity. Find a constraint C with which they conflict.{B1, B2} >> C >> {B3}

91

Ranking?



92

Ranking?



Here C demands stress in a certain position

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The Hierarchy

• B1, B2 >> POS >> B3

94

The Hierarchy

• B1, B2 >> POS >> B3– Stress flees from ə to iueoa (B1)

95

The Hierarchy

• B1, B2 >> POS >> B3– Stress flees from ə to iueoa (B1)– Stress flees from əiu to eoa (B2)

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 (B3)

97

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’

98

The Optima

• B1,B2 >> POS >> B3

W ~ L B1 = *ə B2 = *íúə POS B3 = *éóíúə

99

The Optima

• B1,B2 >> POS >> B3


1. jará-ŋa ~ jára-ŋa W

100

The Optima

• B1,B2 >> POS >> B3



2. jatjólte ~ játjolte W L

101

The Optima

• B1,B2 >> POS >> B3




3. kélikel ~ kelíkel W L

102

The Optima

• B1,B2 >> POS >> B3





4. ətlá ~ ə́tla W L

103

The Optima

• B1,B2 >> POS >> B3





4. ətlá ~ ə́tla W L

5. ə́tləq ~ ə́tləq W

104

The Ranking

• B1,B2 >> POS >> B3



2. jatjólte ~ játjolte W L3. kélikel ~ kelíkel W L4. ətlá ~ ə́tla W L5. ə́tləq ~ ə́tləq W

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Currently Known Conflations

ə i/u e/o a Exemplar Determining Constraints

ə i/u e/o a Yil B1

ə i/u e/o a Chukchi B1, B2

ə i/u e/o a Kobon B1, B2, B3

ə i/u e/o a Nganasan B2

ə i/u e/o a Kara B3

ə i/u e/o a Gujarati B1, B3

Adapted from de Lacy 2002

107

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

Optimality

Harmonic bounding

109

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

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



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



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



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

114

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?

115

Harmonic Bounding

• Many candidates — ‘almost all’ — can never be optimal

116

Harmonic Bounding


• Example: Profuse insertion


a. pæd+d pædəd 0 0 1 0 Ins

b. əpædəd 0 0 2 0 Ins x 2

117

Harmonic Bounding


• Example: Profuse insertion


a. pæd+d pædəd 0 0 1 0 Ins

b. əpædəd 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

118

Harmonic Bounding

• Attempt the overinserted candidate as desired optimum

• It can’t win this competition: – no constraint prefers it, – and one prefers its competitor !

W ~ L *Gem *Diff NoIns NoDev

pæd+d əpædəd ~ pædəd L

119

Harmonic Bounding

• Generically

• 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.

W ~ L C1 C2 C3 … Cn

α~β L (L)

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

121

Harmonic Bounding


• Even though there are no restrictions on insertions at all in defining the set of possible alternatives!

122

Harmonic Bounding


• 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.

123

Collective Harmonic Bounding

• A ranking will not exist unless all competitions can be won simultaneously

• 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

W ~ L C1 C2

α ~ β W L

α ~ δ L W

124


• An example from Basic Syllable Theory

/bk/ No-Del No-Ins Action

bk ba 1 1 Ins+Del

ba.ka. 0 2 Ins x 2

ØØ 2 0 Del x 2

125


• An example from Basic Syllable Theory

/bk/ No-Del No-Ins Action

bk ba 1 1 Ins+Del

ba.ka. 0 L 2 W Ins x 2

ØØ 2 W 0 L Del x 2

126


• The middle way is no way.

β 0 2

* α 1 1

δ 2 0

127

General Harmonic Bounding

• Def. Candidate α is harmonically bounded by a nonempty set of candidates B, xB, over a constraint set S iff for every xB, and for every CS,

if C: αx, then there is a yB such that C: yα.

• 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


• Def. Candidate α is harmonically bounded by a nonempty set of candidates B, xB, over a constraint set S iff for every xB, and for every

CS, if C: αx, then there is a yB such that C: yα.



129


• Def. Candidate α is harmonically bounded by a nonempty set of candidates B, xB, over a constraint set S iff for every xB, and for every

CS, if C: αx, then there is a yB such that C: yα.



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

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

132

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.

133

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).

134

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.

135

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.

Grammar as Choice

Conclusion, retrospect, & overview

137

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

138

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.

139

Gigerenzer &Goldstein 1996

140

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

141

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.

142

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.

143

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.

http://roa.rutgers.edu/

144

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

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Grammar as Choice

Conflict, concord, & optimality

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Grammar as Choice