Harmonic Bounding 

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

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!

Completeness & Symmetry Perfect System on 3 constraints.

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?

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

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.

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

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

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!

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 !

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, *NoCodain addition to whatever X,Y incur aq: a → q Only the marks in X, Y

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

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 — thenaz 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.

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.

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+ • aand b cooperate to stifle z

Collective Harmonic Bounding • An example from Basic Syllable Theory

Collective Harmonic Bounding • The middle way is no way. A collectively bounded form can easily accumulate fewer total violations that its bounders ! Challenge: construct a realistic example!

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.

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).

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.

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.

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

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.

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.

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 !

Lexicographic Order • An even closer analogy: order of numbers in decimal notation (padding with initial 0’s). • 18593 < 20000 • 18563 < 19211 • 18563 < 18700 • 18563 < 18564 • This applies directly to the reading of violation tableaux C1 C2 C3 C4 C5 a 1 8 5 6 3 b 1 8 5 6 4

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. • 1000 is an upper bound for the set of ages that human beings have reached.

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.

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.

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.

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

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.

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).

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.

Harmonic Bounding 