Ranking and Necessity

From RCD to FRed

Adrian Brasoveanu & Alan Prince

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RCD Overview

• Given a set of desired optima, and a bunch of competitors, RCD finds– At least one ranking that works, if there is at least one– That there is no ranking that works, if there is none

• It does this by stratifying constraints as soon as they are rankable, and dismissing the ERCs they solve, thereby creating a new, smaller ranking problem of the same type.

• Rankability is detectable by fusion

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The Successful Tableau

• Since the fate of a competition is decided by the highest ranking constraint that goes one way or the other (W,L), a successful tableau shows W as the first polar member of each row.

• Fusion detects W’s like these: because constraints are rankable when they do not present a leading L, which would force an L in the fusion.

W

W

W

W

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RCD is Sufficient

• But often loses sight of necessity.

• Consider the stratified hierarchy

C1 C2 | C3

• It can come from several sources

• Single ERC:

• (W, W, L) – either one of C1, C2 dominates C3

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What a Stratified Hierarchy Represents

• Single ERC:

(W, e, L) – C1 MUST dominate C3– Example provided by Hunter Brooks, member of this class

• Two ERCs:

(W, e, L)

(e, W, L) – C1 and C2 MUST BOTH dominate C3

• Looking at the Stratified Hierarchy, we can’t tell which!

C1 C2 | C3

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What a Stratified Hierarchy Represents

• Single ERC:

(W, e, L) – C1 MUST dominate C3– Example provided by Hunter Brooks, member of this class

• Two ERCs:

(W, e, L)

(e, W, L) – C1 and C2 MUST BOTH dominate C3

• Looking at the Stratified Hierarchy, we can’t tell which!

C1 C2 | C3

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What a Stratified Hierarchy Represents

• Single ERC:

(W, e, L) – C1 MUST dominate C3– Example provided by Hunter Brooks, member of this class

• Two ERCs:

(W, e, L)

(e, W, L) – C1 and C2 MUST BOTH dominate C3

• Looking at the Stratified Hierarchy, we can’t tell which!

C1 C2 | C3

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No Big Deal ?

• For many users, this will not be an issue– The learner, the parser, the speaker, the hearer, the modeler– These merely need a grammar that works– They need no meta-appreciation of the interactional delicacies

• But the analyst cannot escape the need to know– The necessary conditions tell us what interactions are crucial– E.g. If a constraint is posited because it is locally “phonetically

motivated”, one might like to know that it in the global ensemble, it is the one doing the relevant work.

– In general, the patterns of explanation emerge from the structure of the necessary domination relations

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Source of Loss of Necessity

• Stratal lumping hides details of interaction– As just seen

• The stratum is treated like a single mega-constraint

• Thus, in multi-step cases of RCD, we continue recursively with the remaining ERCs that have e in every constraint of the current stratum. – These are the ERCs that are not ‘solved’ by any constraint in the

stratum.

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RCD Example

C1 C2 C3 C4

α W e L e

β e W e L

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RCD Example

C1 C2 C3 C4

α W e L e

β e W e L

fαβ W W L L

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RCD Example

C1 C2 C3 C4

α W e L e

β e W e L

fαβ W W L L

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RCD Example

C1 C2 C3 C4

α W e L e

β e W e L

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As If

C1C2 C3C4α W L

β W L

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But in reality

C1C2 C3C4α W L

β W L

But α = (W,e, L, e) = C1 >> C3 β= (e, W,e, L) = C2 >> C4

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Where Stratified Constraints Differ

C1 C2 C3 C4

α W e L e

β e W e L

Each constraint fusing to W has a Residue of unsolved ERCs, those to which it awards e.

R1 = β R2 = α

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Where Stratified Constraints Differ

C1 C2 C3 C4

α W e L e

β e W e L

Each constraint fusing to W has a Residue of unsolved ERCs, those to which it awards e.

R1 = β R2 = α

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Where Stratified Constraints Differ

C1 C2 C3 C4

α W e L e

β e W e L

Each constraint fusing to W has a Residue of unsolved ERCs, those to which it awards e.

R1 = β

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Where Stratified Constraints Differ

C1 C2 C3 C4

α W e L e

β e W e L

Each constraint fusing to W has a Residue of unsolved ERCs, those to which it awards e.

R2 = α

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Toward Necessity

To avoid the loss of detail ---

• There can be no lumping of constraints into strata– We cannot proceed with the coarse Stratal Residue

• We must continue recursively, based on each relevant constraint, pursuing the ERCs it alone does not ‘solve’.– These are the separate Residues of each individual constraint

• The goal: a set of derived ERCs that display the necessary and sufficient conditions with maximal perspicuity.

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The Fusional Reduction Algorithm

The basic structure of FRed

[1] Fuse All

[2] Save the fusion (to be refined)

[3] Reapply the algorithm to the Residue Rk of each constraint Ck that fuses to W in [1]

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A Simple Example

C1 C2 C3 C4

α W L W W

β e W L W

γ e e W L

NB: α, β highly disjunctive

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FRed [1] - Fuse All

C1 C2 C3 C4

α W L W W

β e W L W

γ e e W L

fαβγ W L L L

NB: loss of disjunctivity of α in fusion

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FRed [2] - Save the fusion

C1 C2 C3 C4

α W L W W

β e W L W

γ e e W L

fαβγ W L L L

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FRed [3] - Proceed with the Residues

C1 C2 C3 C4

α W L W W

β e W L W

γ e e W L

fαβγ W L L L

R1 = { β,γ }. There are no other residues.

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FRed Again [1] - Fuse All

R1 C1 C2 C3 C4

β e W L W

γ e e W L

fβγ e W L L

NB: loss of disjunctivity of β in fusion

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FRed Again [2] - Save the fusion

R1 C1 C2 C3 C4

β e W L W

γ e e W L

fβγ e W L L

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FRed Again [3] - Proceed with Residues

R1 C1 C2 C3 C4

β e W L W

γ e e W L

fβγ e W L L

Note that γ is the only Residue.

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FRed – Third Time Around

R12 C1 C2 C3 C4

γ e e W L

fγ e W L

Save fγ and we’re done

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The Result

MIB C1 C2 C3 C4

fαβγ W L L L

fβγ e W L L

fγ e e W L

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Entailment Check

• The fusion of the whole is not always informative

• It may be entailed by the collection of all Residues

• Recall our first example:

α W e L e

β e W e L

fαβ W W L L

The fusion is entailed by the set of residues! It’s useless.

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Entailment Check – How to do it

• Instead of blithely saving the fusion of the whole, first check it against the collection of all Residues

• Form the fusion of the entire Residue collectionfk Rk

• If fkRk entails the fusion of the whole, do not save that fusion.

• Otherwise, save it – it is informative

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The Fusional Reduction Algorithm

[1] Fuse All

[2a] Check to see if the fusion of the whole is entailed by the fusion of the Residues.

[2b] If unentailed, save it. If entailed, discard it.

[3] Reapply the algorithm to the Residue Rk of each constraint Ck that fuses to W

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An Example of Nontrivial Entailment Check

• See paper non-slide handout

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What FRed Gives You

• A set of derived ERCs, the Most Informative Basis

• Free of Entailments – “logically independent”

• Maximally informative with respect to the ranking of each individual constraint

• Therefore, maximally condensed in size

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Different Representations of a Ranking

• Suppose we have C1 >> C2 >> C3

• This can arise from a number of ERC sets, e.g.

S1 S2 S3• W L e W L W W L L• e W L e W L e W L

• Given any of these as input, FRed outputs S3.

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FRed Condenses

• Consider the ERC set

α W L W

β W W L

• FRed outputs a single ERC to represent it:

fαβ = W L L• In just this case, where no W is ever matched to e in a

column fusing to W --- fusion is equivalent to conjunction, and the fusion retains all the information in the original ERCs. (No residues !)

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FRed in Summary

• FRed extracts all the ranking information from any ERC set

• That information is presented in a unique form, as a logically independent, maximally condensed, maximally informative set of ERCs

• The result is the Most Informative Basis --- the unique set of ERCS, equivalent to the original, which has all these properties.