Harmonic Bounding  Alan Prince,Vieri Samek-Lodovici, Paul Smolensky

2 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!

3 Completeness & Symmetry Perfect System on 3 constraints. C1C2C3 α-1012 α-2 α-2021 α-3102 α-4120 α-5201 α-6210

4 Completeness & Symmetry Perfect System on 3 constraints. C1C2C3 α-1012 α-2 α-2021 α-3102 α-4120 α-5201 α-6210

5 Completeness & Symmetry Perfect System on 3 constraints. C1C2C3 α-1012 α-2 α-2021 α-3102 α-4120 α-5201 α-6210

6 Completeness & Symmetry Perfect System on 3 constraints. C1C2C3 α-1012 α-2 α-2021 α-3102 α-4120 α-5201 α-6210

7 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?

8 Harmonic Bounding Many candidates — ‘almost all’ — can never be optimal

9 Harmonic Bounding Many candidates — ‘almost all’ — can never be optimal What makes it impossible for a certain form to win, ever? Ranking side: no ranking exists that works for it. Candidate side: other candidates are always better –They ‘bound’ how good it can be.

10 Basic Syllable Theory We can use this pattern to derive properties of constraint systems. Consider Basic Syllable Theory (cf.Prince & Smolensky, ch. 6) To make it ‘basic’ assume as part of GEN: –*Complex: no syllable internal C sequences –*Pk/C: no C as syllable nucleus –*Mar/V: no V as syllable margin - Onset or Coda –only C, V in alphabet –all outputs are fully syllabified

11 Basic Syllable Theory Assume as constraints in CON: Markedness: –Onset: every syllable begins with a consonant *(V –NoCoda: no syllable ends on a consonant *C) Faithfulness: –Max: everything in the input has an output correspondent –DepV: every vowel in the output has an input correspondent –DepC: every consonant in the output has an input correspondent

12 Theory of Epenthesis Sites GEN allows any amount of epenthesis in the I,O relation –We place no ad hoc constraints on candidate outputs re epenth. But BST constraints will select only a few sites as realizable in optimal candidates We get a predictive theory of epenthesis without special maneuvers –GEN is quite free –CON says, via the Dep system: never epenthesize!

13 A Typical Restriction CodaProp. Under BST, there is no epenthesis into Coda How can we show conclusively that CP is actually true? Not entirely trivial --- CP says: –For every possible input (and their number is unlimited) –There is no optimal output containing epenth. in Coda And the number of candidates competing for optimality is also unlimited !

14 Harmonic Bounding to the Rescue Consider an output candidate that has Coda epenthesis –they all look like this: z = X (C) COD Y Now consider an alternative q = X Y which is exactly like z except that it lacks the epenth. C. Let a be the input. What marks for each mapping? az: a → z*DepC, *NoCoda in addition to whatever X,Y incur aq: a → q Only the marks in X, Y

15 Simply Bounded So a→z cannot possibly be better than a→q –regardless of ranking –it is better on no constraint, worse on some OnsetNoCodaDepCDepVMax az ~aqeLLee NB: there is no hint that aq is optimal, or even possibly optimal !

16 Harmonic Bounding Generically If there is no constraint on which az  aq, for az  aq — no W in the row — and at least one L — then az can never be optimal. az~aq  L+ aq is always better, so az can’t be the best –Even if aq itself is not optimal, or not even possibly optimal ! e.g. 19 is not the smallest positive number because 18<19. W ~ LC1C2C3…Cn az~aqL(L)

17 Harmonic Bounding Harmonic Bounding is a powerful effect –E.g. Almost all candidates, incl. insertional, are bounded –This gives us a highly predictive theory of insertion But we’re not done. –Simple Harmonic Bounding works without ranking –Any positively weighted combination of violation scores will show the effect. Any system in which you must have something going for you if you want to win.

18 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 b  z –If C2 >>C1 then a  z –The ERC set fuses to L+ a and b cooperate to stifle z W ~ LC1C2 z ~ aWL z ~ bLW

19 Collective Harmonic Bounding An example from Basic Syllable Theory /bk/MaxDepVAction bk  b a ● 11Ins+Del ba.ka.02Ins x 2 ● ●20Del x 2

20 Collective Harmonic Bounding An example from Basic Syllable Theory /bk/MaxDepVAction bk  b a ● 11 Ins+Del ba.ka. 0 L 2 W Ins x 2 ● ● 2 W 0 L Del x 2

21 Collective Harmonic Bounding The middle way is no way. β02 * α * α11 δ 20 A collectively bounded form can easily accumulate fewer total violations that its bounders ! Challenge: construct a realistic example!

22 The General Picture Ranking side: candidate z will fail to be optimal iff there’s no ranking that works for it as desired optimum. From ERC theory, we know that the set of ERCs A taking z to be the desired optimum will be inconsistent, unsatisfiable by any ranking. Therefore, A contains a subset X that fuses to L+. (We can easily find this subset using RID.) Candidate side: from this we can deduce how the candidates must be arrayed against z.

23 Ganging Up X fuses to L+. On any constraint where z~qi shows W, i.e. where z  qi there must be another ERC z~qk showing L, i.e qk  z. Whenever z betters some member of X, there’s another member that is better than z. On no constraint is z better than all of X, though it may equal all of X on some (fusing to e).

24 General Harmonic Bounding Def. Candidate z is harmonically bounded by a nonempty set of candidates B, z  B, over a constraint set S iff these conditions are met: [1] Reciprocity. For every b  B, and for every C  S, if C: z  b1, then there is a b2  B such that C: b2  z. [2] Strictness. Some member of B beats z on some constraint. - this excludes a candidate violationwise identical to z from bounding it.

25 Summary By reciprocity (the heart of the matter) 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.

26 Some Stats Tesar 1999 studies a system of 10 prosodic constraints. –with a quite 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)  Reported in Samek-Lodovici & Prince 1999

27 Bounding and Order Bounding is result of the order structure of OT A constraint hierarchy chooses an optimum, but it also imposes an order on the entire candidate set, including all of its darker regions. The order between any two candidates may be determined by consider a comparison between them, i.e. by thinking of a 2 candidate set featuring just them.

28 Order from a Constraint The order imposed by a single constraint is a ‘stratified partial order’ or ‘rank order’. Every candidate incurring k violations is better than any candidate incurring more. But among the k-violators, no order is determined. These are the ranks or strata of the order. –Members of the same violation stratum share all order properties with respect to the other candidates.

29 Lexicographic Order The order imposed by a Ranking, amalgamated from the orders of the individual constraints is a lexicographic order. In alphabetic [lexicographic, dictionary] order, in comparing two words, we try the first letter. –If it decides the order, we are done. adze < zap –If not, we go on to the second. adze < apple –and so, until we reach a difference This is exactly the way constraints filter the candidates !

30 Lexicographic Order An even closer analogy: order of numbers in decimal notation (padding with initial 0’s). –18593 < –18563 < –18563 < –18563 < This applies directly to the reading of violation tableaux C1C2C3C4C5 a18563 b18564

31 Bounding as Lingo We therefore speak of ‘bounding’ because order theory recognizes the notion of ‘upper’ and ‘lower’ bounds. An entity x is an upper bound for the elements in a set S, if no element of S is greater than x is an upper bound for the set of ages that human beings have reached.

32 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). 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.

33 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, 2002 for discussion.

34 Bounding and ERC Entailment The sense of entailment: If ERC [a~b] entails ERC [c~d], then whenever the first holds, the second must also hold. Any hierarchy in which a  b must also be one in which c  d.

35 Bounding and Entailment Suppose a bounds z. Then [q~a] entails [q~z] –Whenever q  a, it must be that q  z, because a  z. Bounding produces entailment. The opposite is not guaranteed. –Challenge: produce an example that shows this

36 Independence = No Bounding Lack of entailment --- logical independence – in an ERC set therefore implies lack of bounding among its members. This gives a taste of the relations between bounding and entailment. For more, see ERA, ch. 6.

37 Generality of ERC Entailment Bounding and Entailment are not mutually reducible, though related. Entailment is perhaps a more widely applicable notion, since it allows us to compare across candidate sets. –Thus we can ask not just if a single form is possible, but whether an aggregate of forms & mappings can possibly belong to the same system. Bounding plays a central role in eliminating candidate structures from consideration. –In analysis, and even in learning (Riggle 2002).

38 Challenge We argue with limited candidate sets and limited constraint sets. What relations of optimality and/or bounding are preserved as we [1] enlarge the candidate set while keeping the constraints constant [2] enlarge the constraint set while keeping the candidate set constant.