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.

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

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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

5

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

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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

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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

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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.

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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

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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

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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!

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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 !

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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 alternativeq = 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

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Simply Bounded

• So a→z cannot possibly be better than a→q– regardless of ranking

– it is better on no constraint, worse on some

Onset NoCoda DepC DepV Max

az ~aq e L L e e

NB: there is no hint that aq is optimal, or even possibly optimal !

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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 ~ L C1 C2 C3 … Cn

az~aq L (L)

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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.

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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 ~ L C1 C2

z ~ a W L

z ~ b L W

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Collective Harmonic Bounding

• An example from Basic Syllable Theory

/bk/ Max DepV Action

bk b a ● 1 1 Ins+Del

ba.ka. 0 2 Ins x 2

● ● 2 0 Del x 2

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Collective Harmonic Bounding

• An example from Basic Syllable Theory

/bk/ Max DepV Action

bk b a ● 1 1 Ins+Del

ba.ka. 0 L 2 W Ins x 2

● ● 2 W 0 L Del x 2

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Collective Harmonic Bounding

• The middle way is no way.

β 0 2

* α 1 1

δ 2 0

A collectively bounded form can easily accumulate fewer total violations that its bounders !

Challenge: construct a realistic example!

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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.

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Ganging Up

• X fuses to L+.

• On any constraint where z~qi shows W, i.e. where zqi

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).

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General Harmonic Bounding

• Def. Candidate z is harmonically bounded

by a nonempty set of candidates B, zB, over a constraint set S iff these conditions are met:

[1] Reciprocity. For every bB, and for every CS,

if C: zb1, then there is a b2B such that C: b2z.

[2] Strictness. Some member of B beats z on some constraint.

- this excludes a candidate violationwise identical to z from bounding it.

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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.

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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

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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.

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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.

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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 !

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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 tableauxC1 C2 C3 C4 C5

a 1 8 5 6 3

b 1 8 5 6 4

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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.

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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.

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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.

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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 ab must also be one in which cd.

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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

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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.

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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).

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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.