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 dd 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 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 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 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 ab 48 Constraints in conflict a C1 0 C2 1 b 1 0 ab ba 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 ac ac 52 Constraints need not conflict B1 B2 a 0 0 b 0 1 c 1 1 a?b ab 53 Constraints need not conflict B1 B2 a 0 0 b 0 1 c 1 1 ab 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 bc 56 Constraints need not conflict B1 B2 a 0 0 b 0 1 c 1 1 ac bc ac ab 57 Constraints need not conflict B1 B2 a 0 0 b 0 1 c 1 1 abc regardless of ranking 58 Constraints and Scales • Imagine a goodness scale abcd 59 Abstract Scale better a b c d 60 Constraints and Scales abcd • 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 & abcd • B2 & B3 = a b cd – i.e., abcd & 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 & abcd • B2 & B3 = a b cd – i.e., abcd & 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 & abcd • B2 & B3 = a b cd – i.e., abcd & abcd • 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 & abcd • B2 & B3 = a b cd – i.e., abcd & 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, 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. 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 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. 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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