Albright and Hayes (2003): Rules vs. analogy in English past tenses

[AH03] learn the English past tense as a set of stochastic rules. Every stem–past pair is a word-specific rule, and minimal generalization over two rules with the same change keeps the segments they share outward from the change site, reduces the first pair that differ to the class of their shared features, and frees whatever lies beyond ([AH02] gives the full procedure). Each rule is scored by its reliability in the lexicon, hits over scope, discounted for small scope by a lower confidence limit ([Mik97]); the most reliable rules, the islands of reliability, exist for the regular change as well as for irregular ones ([Alb02] for Italian), and a candidate past takes the score of its best rule. Two wug experiments ([Ber58]) on 58 stems that the model chose to cross islands for the regular and for an irregular past find island effects of the same size for both, which the single default rule of the dual mechanism model ([PP88], [PP93]) cannot produce; the irregular effects replicate [BM83]. A generalized context model ([Nos90], with [Bro93a]'s segment similarity) run on the same lexicon misses the regular islands, follows single similar verbs, and misplaces the regular allomorphs, because its similarity is variegated where a rule's is structured: the allomorph depends on the final segment alone.

Implementation notes

Minimal generalization is mg on the substrate's rewrite contexts, in which a leading word boundary is the anchored word-specific rule and its absence the free variable. The paper's own steps (6) and footnote 4, the island (8), and the regular allomorphy of (7b) are computed on the English fragment's segments, whose meet is the featural term and whose natural classes in the English inventory are what a featural term covers. Reliability is Stats; the statistics reported in Tables 1 and 4 and the ratings and production probabilities of Appendix A are rows. Vowel changes, which generalize on both sides of the change, the confidence-limit discount, and the analogical model itself are not modelled.

References

• [AH03]
• [AH02]
• [Alb02]
• [Ber58]
• [PP88]
• [PP93]
• [BM83]
• [Mik97]
• [Nos90]
• [Bro93a]

def AlbrightHayes2003.instDecidableEqContextElem_studies.decEq (x✝ x✝¹ : Subregular.LocalRewrite.ContextElem) : Decidable (x✝ = x✝¹)

Equations

• One or more equations did not get rendered due to their size.
• AlbrightHayes2003.instDecidableEqContextElem_studies.decEq (Subregular.LocalRewrite.ContextElem.seg a) Subregular.LocalRewrite.ContextElem.wordBoundary = isFalse ⋯
• AlbrightHayes2003.instDecidableEqContextElem_studies.decEq Subregular.LocalRewrite.ContextElem.wordBoundary (Subregular.LocalRewrite.ContextElem.seg a) = isFalse ⋯
• AlbrightHayes2003.instDecidableEqContextElem_studies.decEq Subregular.LocalRewrite.ContextElem.wordBoundary Subregular.LocalRewrite.ContextElem.wordBoundary = isTrue ⋯

@[instance_reducible]

instance AlbrightHayes2003.instDecidableEqContextElem_studies : DecidableEq Subregular.LocalRewrite.ContextElem

Equations

• AlbrightHayes2003.instDecidableEqContextElem_studies = AlbrightHayes2003.instDecidableEqContextElem_studies.decEq

Minimal generalization

@[reducible, inline]

abbrev AlbrightHayes2003.Context : Type

The structural description of a past-tense rule: the stem-final material before the change site, read left to right, a leading word boundary anchoring it to a whole stem and its absence standing for the free variable.

Equations

• AlbrightHayes2003.Context = List Subregular.LocalRewrite.ContextElem

def AlbrightHayes2003.wordSpecific (stem : List Phonology.Segment) : Context

The word-specific rule of a stem ((3)).

Equations

• AlbrightHayes2003.wordSpecific stem = Subregular.LocalRewrite.ContextElem.wordBoundary :: List.map Subregular.LocalRewrite.ContextElem.seg stem

def AlbrightHayes2003.Matches (c : Context) (stem : List Phonology.Segment) : Prop

The stem meets the description.

Equations

• AlbrightHayes2003.Matches c stem = (Subregular.LocalRewrite.matchLeftContext c stem = true)

@[instance_reducible]

instance AlbrightHayes2003.instDecidableMatches (c : Context) (stem : List Phonology.Segment) : Decidable (Matches c stem)

Equations

• AlbrightHayes2003.instDecidableMatches c stem = AlbrightHayes2003.instDecidableMatches._aux_1 c stem

def AlbrightHayes2003.mgRev : List Subregular.LocalRewrite.ContextElem → List Subregular.LocalRewrite.ContextElem → List Subregular.LocalRewrite.ContextElem

Generalization proceeds from the change site outward ((5)). Shared segments are kept, the first pair that differ become the class of their shared features, and everything beyond them is freed.

Equations

• One or more equations did not get rendered due to their size.
• AlbrightHayes2003.mgRev x✝¹ x✝ = []

def AlbrightHayes2003.mg (c₁ c₂ : Context) : Context

The minimal generalization of two descriptions.

Equations

• AlbrightHayes2003.mg c₁ c₂ = (AlbrightHayes2003.mgRev (List.reverse c₁) (List.reverse c₂)).reverse

def AlbrightHayes2003.learned : List (List Phonology.Segment) → Context

The rule learned from a list of stems generalizes each stem's word-specific rule in turn.

Equations

• One or more equations did not get rendered due to their size.
• AlbrightHayes2003.learned [] = []

theorem AlbrightHayes2003.matches_wordSpecific {stem : List Phonology.Segment} : Matches (wordSpecific stem) stem

A stem meets its own word-specific rule.

theorem AlbrightHayes2003.matches_single_iff (k x : Phonology.Segment) : Matches [Subregular.LocalRewrite.ContextElem.seg k] [x] ↔ k ≤ x

A description one segment long is met by exactly the segments it subsumes.

theorem AlbrightHayes2003.mg_comm {c₁ c₂ : Context} : mg c₁ c₂ = mg c₂ c₁

Minimal generalization is symmetric.

theorem AlbrightHayes2003.matches_mg_left {c₁ c₂ : Context} {stem : List Phonology.Segment} (h : Matches c₁ stem) : Matches (mg c₁ c₂) stem

The generalized rule covers everything either rule covered, so generalization only widens.

theorem AlbrightHayes2003.matches_mg_right {c₁ c₂ : Context} {stem : List Phonology.Segment} (h : Matches c₂ stem) : Matches (mg c₁ c₂) stem

theorem AlbrightHayes2003.matches_rtake_one {c : Context} {stem : List Phonology.Segment} (h : List.length c ≤ 1) : Matches c stem ↔ Matches c (stem.rtake 1)

A description one segment long reads the final segment alone, the structured similarity a rule is confined to.

Reliability

def AlbrightHayes2003.scopeOf {C : Type u_1} (c : Context) (lex : List (List Phonology.Segment × C)) : List (List Phonology.Segment × C)

The scope of a rule is the stems of a lexicon meeting its description.

Equations

• AlbrightHayes2003.scopeOf c lex = List.filter (fun (x : List Phonology.Segment × C) => Subregular.LocalRewrite.matchLeftContext c x.1) lex

structure AlbrightHayes2003.Stats : Type

A rule's performance in a lexicon is the forms meeting its description and, among them, those whose past shows its change.

• scope : ℕ
• hits : ℕ

def AlbrightHayes2003.instDecidableEqStats.decEq (x✝ x✝¹ : Stats) : Decidable (x✝ = x✝¹)

Equations

• AlbrightHayes2003.instDecidableEqStats.decEq { scope := a, hits := a_1 } { scope := b, hits := b_1 } = if h : a = b then h ▸ if h : a_1 = b_1 then h ▸ isTrue ⋯ else isFalse ⋯ else isFalse ⋯

@[instance_reducible]

instance AlbrightHayes2003.instDecidableEqStats : DecidableEq Stats

Equations

• AlbrightHayes2003.instDecidableEqStats = AlbrightHayes2003.instDecidableEqStats.decEq

def AlbrightHayes2003.Stats.ofLexicon {C : Type u_1} [DecidableEq C] (ch : C) (c : Context) (lex : List (List Phonology.Segment × C)) : Stats

The statistics of a change in a described context over a lexicon.

Equations

• One or more equations did not get rendered due to their size.

theorem AlbrightHayes2003.hits_le_scope {c : Context} {C : Type u_1} [DecidableEq C] (ch : C) (lex : List (List Phonology.Segment × C)) : (Stats.ofLexicon ch c lex).hits ≤ (Stats.ofLexicon ch c lex).scope

theorem AlbrightHayes2003.scope_mono {c₁ c₂ : Context} {C : Type u_1} [DecidableEq C] (ch : C) (lex : List (List Phonology.Segment × C)) : (Stats.ofLexicon ch c₁ lex).scope ≤ (Stats.ofLexicon ch (mg c₁ c₂) lex).scope

Generalization never loses a form.

def AlbrightHayes2003.Stats.rawConfidence (s : Stats) : ℚ

Raw confidence is hits over scope.

Equations

• s.rawConfidence = ↑s.hits / ↑s.scope

def AlbrightHayes2003.LessReliable (r s : Stats) : Prop

r is less reliable than s, the ratios cross-multiplied.

Equations

• AlbrightHayes2003.LessReliable r s = (r.hits * s.scope < s.hits * r.scope)

@[instance_reducible]

instance AlbrightHayes2003.instDecidableLessReliable (r s : Stats) : Decidable (LessReliable r s)

Equations

• AlbrightHayes2003.instDecidableLessReliable r s = { decide := (r.hits * s.scope).succ.ble (s.hits * r.scope), reflects_decide := ⋯ }

theorem AlbrightHayes2003.lessReliable_iff {r s : Stats} (hr : 0 < r.scope) (hs : 0 < s.scope) : LessReliable r s ↔ r.rawConfidence < s.rawConfidence

def AlbrightHayes2003.general : Stats

The general suffixation rule (7a) over the learning set.

Equations

• AlbrightHayes2003.general = { scope := 4253, hits := 4034 }

def AlbrightHayes2003.IsIsland (s : Stats) : Prop

An island of reliability is a rule the change works better in than the general rule ((8)).

Equations

• AlbrightHayes2003.IsIsland s = AlbrightHayes2003.LessReliable AlbrightHayes2003.general s

@[instance_reducible]

instance AlbrightHayes2003.instDecidableIsIsland (s : Stats) : Decidable (IsIsland s)

Equations

• AlbrightHayes2003.instDecidableIsIsland s = { decide := (AlbrightHayes2003.general.hits * s.scope).succ.ble (s.hits * AlbrightHayes2003.general.scope), reflects_decide := ⋯ }

The paper's steps on the English fragment

def AlbrightHayes2003.vote : List Phonology.Segment

Equations

• AlbrightHayes2003.vote = [English.Phonology.v, English.Phonology.o, English.Phonology.t]

def AlbrightHayes2003.need : List Phonology.Segment

Equations

• AlbrightHayes2003.need = [English.Phonology.n, English.Phonology.i, English.Phonology.d]

def AlbrightHayes2003.rub : List Phonology.Segment

Equations

• AlbrightHayes2003.rub = [English.Phonology.turnedR, English.Phonology.wedge, English.Phonology.b]

def AlbrightHayes2003.sag : List Phonology.Segment

Equations

• AlbrightHayes2003.sag = [English.Phonology.s, English.Phonology.æ, English.Phonology.g]

def AlbrightHayes2003.plan : List Phonology.Segment

Equations

• AlbrightHayes2003.plan = [English.Phonology.p, English.Phonology.l, English.Phonology.æ, English.Phonology.n]

def AlbrightHayes2003.love : List Phonology.Segment

Equations

• AlbrightHayes2003.love = [English.Phonology.l, English.Phonology.wedge, English.Phonology.v]

def AlbrightHayes2003.flow : List Phonology.Segment

Equations

• AlbrightHayes2003.flow = [English.Phonology.f, English.Phonology.l, English.Phonology.o]

def AlbrightHayes2003.jump : List Phonology.Segment

Equations

• AlbrightHayes2003.jump = [English.Phonology.dezh, English.Phonology.wedge, English.Phonology.m, English.Phonology.p]

def AlbrightHayes2003.miss : List Phonology.Segment

Equations

• AlbrightHayes2003.miss = [English.Phonology.m, English.Phonology.smallCapitalI, English.Phonology.s]

def AlbrightHayes2003.wish : List Phonology.Segment

Equations

• AlbrightHayes2003.wish = [English.Phonology.w, English.Phonology.smallCapitalI, English.Phonology.esh]

def AlbrightHayes2003.laugh : List Phonology.Segment

Equations

• AlbrightHayes2003.laugh = [English.Phonology.l, English.Phonology.æ, English.Phonology.f]

def AlbrightHayes2003.fill : List Phonology.Segment

Equations

• AlbrightHayes2003.fill = [English.Phonology.f, English.Phonology.smallCapitalI, English.Phonology.l]

def AlbrightHayes2003.pick : List Phonology.Segment

Equations

• AlbrightHayes2003.pick = [English.Phonology.p, English.Phonology.smallCapitalI, English.Phonology.k]

theorem AlbrightHayes2003.filter_matches_single (k : Phonology.Segment) (I : Finset Phonology.Segment) : {x ∈ I | Matches [Subregular.LocalRewrite.ContextElem.seg k] [x]} = k.naturalClass I

The stems of an inventory that a description one segment long covers are the natural class of its featural term.

theorem AlbrightHayes2003.learned_vote_need : learned [vote, need] = [Subregular.LocalRewrite.ContextElem.seg (English.Phonology.t ⊓ English.Phonology.d)] ∧ (English.Phonology.t ⊓ English.Phonology.d).naturalClass English.Phonology.inventory = {English.Phonology.t, English.Phonology.d}

In (6), vote and need differ first in their final segments, so the learned -əd rule keeps what [t] and [d] share, and that featural term characterizes the class [t, d] uniquely.

theorem AlbrightHayes2003.learned_rub_sag_plan : learned [rub, sag, plan] = [Subregular.LocalRewrite.ContextElem.seg (English.Phonology.b ⊓ English.Phonology.g ⊓ English.Phonology.n)] ∧ English.Phonology.d ∈ (English.Phonology.b ⊓ English.Phonology.g ⊓ English.Phonology.n).naturalClass English.Phonology.inventory

As footnote 4 observes, whatever [b], [g] and [n] share, [d] has, so the -d rule learned from rub, sag and plan reaches need, and only the phonology keeps needd out.

def AlbrightHayes2003.voiced : Phonology.Segment

The voiced segments, as a description.

Equations

• AlbrightHayes2003.voiced = Phonology.Segment.ofSpecs [(Phonology.Feature.voice, true)]

def AlbrightHayes2003.voiceless : Phonology.Segment

The voiceless segments, as a description.

Equations

• AlbrightHayes2003.voiceless = Phonology.Segment.ofSpecs [(Phonology.Feature.voice, false)]

theorem AlbrightHayes2003.learned_voiced : learned [rub, sag, plan, love, flow] = [Subregular.LocalRewrite.ContextElem.seg (English.Phonology.b ⊓ English.Phonology.g ⊓ English.Phonology.n ⊓ English.Phonology.v ⊓ English.Phonology.o)] ∧ (English.Phonology.b ⊓ English.Phonology.g ⊓ English.Phonology.n ⊓ English.Phonology.v ⊓ English.Phonology.o).naturalClass English.Phonology.inventory = voiced.naturalClass English.Phonology.inventory \ {English.Phonology.l} ∧ (English.Phonology.b ⊓ English.Phonology.g ⊓ English.Phonology.n ⊓ English.Phonology.v ⊓ English.Phonology.o ⊓ English.Phonology.l).naturalClass English.Phonology.inventory = voiced.naturalClass English.Phonology.inventory

In (7b) the -d rule generalizes to [+voice] as the stems come to span the voiced segments. From rub, sag, plan, love and flow, whose final segments are all non-lateral, it reaches every voiced segment but /l/, and a stem in /l/ completes it.

theorem AlbrightHayes2003.learned_voiceless : learned [jump, miss, laugh] = [Subregular.LocalRewrite.ContextElem.seg (English.Phonology.p ⊓ English.Phonology.s ⊓ English.Phonology.f)] ∧ (English.Phonology.p ⊓ English.Phonology.s ⊓ English.Phonology.f).naturalClass English.Phonology.inventory = voiceless.naturalClass English.Phonology.inventory \ {English.Phonology.k} ∧ (English.Phonology.p ⊓ English.Phonology.s ⊓ English.Phonology.f ⊓ English.Phonology.k).naturalClass English.Phonology.inventory = voiceless.naturalClass English.Phonology.inventory

In (7b) the -t rule does likewise. From jump, miss and laugh, whose final segments are all non-dorsal, it reaches every voiceless segment but /k/, and a stem in /k/ completes it.

def AlbrightHayes2003.voicelessFricative : Phonology.Segment

The island (8) is -t after a voiceless fricative.

Equations

• AlbrightHayes2003.voicelessFricative = Phonology.Segment.ofSpecs [(Phonology.Feature.sonorant, false), (Phonology.Feature.continuant, true), (Phonology.Feature.voice, false)]

theorem AlbrightHayes2003.naturalClass_voicelessFricative : voicelessFricative.naturalClass English.Phonology.inventory = {English.Phonology.f, English.Phonology.θ, English.Phonology.s, English.Phonology.esh}

(8) is met by the four voiceless fricatives and nothing else.

theorem AlbrightHayes2003.learned_le_voicelessFricative : learned [miss, wish, laugh] = [Subregular.LocalRewrite.ContextElem.seg (English.Phonology.s ⊓ English.Phonology.esh ⊓ English.Phonology.f)] ∧ voicelessFricative ≤ English.Phonology.s ⊓ English.Phonology.esh ⊓ English.Phonology.f

The rule learned from miss, wish and laugh lies inside the island (8), and further fricative-final forms widen it to the island.

Appendix A

structure AlbrightHayes2003.IORCategory : Type

A cell of Table 3 records whether the stem occupies an island for the regular past and for some irregular past.

• iorForRegular : Bool
• iorForIrregular : Bool

def AlbrightHayes2003.instDecidableEqIORCategory.decEq (x✝ x✝¹ : IORCategory) : Decidable (x✝ = x✝¹)

Equations

• One or more equations did not get rendered due to their size.

@[instance_reducible]

instance AlbrightHayes2003.instDecidableEqIORCategory : DecidableEq IORCategory

Equations

• AlbrightHayes2003.instDecidableEqIORCategory = AlbrightHayes2003.instDecidableEqIORCategory.decEq

def AlbrightHayes2003.IORCategory.ofString : String → Option IORCategory

Equations

• AlbrightHayes2003.IORCategory.ofString "both" = some { iorForRegular := true, iorForIrregular := true }
• AlbrightHayes2003.IORCategory.ofString "regOnly" = some { iorForRegular := true, iorForIrregular := false }
• AlbrightHayes2003.IORCategory.ofString "irregOnly" = some { iorForRegular := false, iorForIrregular := true }
• AlbrightHayes2003.IORCategory.ofString "neither" = some { iorForRegular := false, iorForIrregular := false }
• AlbrightHayes2003.IORCategory.ofString x✝ = none

def AlbrightHayes2003.digits (s : String) : ℕ

A printed decimal is read as the integer of its digits, ratings in hundredths and production probabilities in thousandths.

Equations

• AlbrightHayes2003.digits s = List.foldl (fun (n : ℕ) (c : Char) => if c.isDigit = true then 10 * n + (c.toNat - '0'.toNat) else n) 0 s.toList

def AlbrightHayes2003.value (key : String) (r : Datum) : ℕ

A row's numeric feature.

Equations

• AlbrightHayes2003.value key r = AlbrightHayes2003.digits ((r.feature? key).getD "0")

def AlbrightHayes2003.statsOf (r : Datum) : Option Stats

A row's reported rule statistics (Tables 1 and 4).

Equations

• AlbrightHayes2003.statsOf r = (r.nat? "ruleScope").bind fun (s : ℕ) => Option.map (fun (x : ℕ) => { scope := s, hits := x }) (r.nat? "ruleHits")

theorem AlbrightHayes2003.table4_islands (r : Datum) : r ∈ Examples.all → (r.feature? "island").isSome = true → ∀ s ∈ statsOf r, IsIsland s

In Table 4 the twelve regular islands all outscore the general rule.

theorem AlbrightHayes2003.gleed_ranking (s₁ : Stats) : s₁ ∈ statsOf Examples.a1_25_gleed → ∀ s₂ ∈ statsOf Examples.a1_25_gled, ∀ s₃ ∈ statsOf Examples.a1_25_gleeded, LessReliable s₁ s₂ ∧ LessReliable s₂ s₃

Table 1 ranks gleed's pasts by raw confidence, gleed below gled below gleeded.

def AlbrightHayes2003.rows (regular : Bool) (p : IORCategory → Bool) : List Datum

The Appendix A rows of one past type in the cells satisfying p (the Peripheral stems of Table A2 have no cell).

Equations

• One or more equations did not get rendered due to their size.

def AlbrightHayes2003.total (key : String) (rs : List Datum) : ℕ

The sum of a numeric feature over rows.

Equations

• AlbrightHayes2003.total key rs = (List.map (AlbrightHayes2003.value key) rs).sum

def AlbrightHayes2003.MeanGT (key : String) (A B : List Datum) : Prop

The mean of key over A exceeds its mean over B, cross-multiplied.

Equations

• AlbrightHayes2003.MeanGT key A B = (AlbrightHayes2003.total key A * B.length > AlbrightHayes2003.total key B * A.length)

@[instance_reducible]

instance AlbrightHayes2003.instDecidableMeanGT (key : String) (A B : List Datum) : Decidable (MeanGT key A B)

Equations

• AlbrightHayes2003.instDecidableMeanGT key A B = { decide := (AlbrightHayes2003.total key B * A.length).succ.ble (AlbrightHayes2003.total key A * B.length), reflects_decide := ⋯ }

theorem AlbrightHayes2003.regulars_ior : MeanGT "adjustedRating" (rows true fun (x : IORCategory) => x.iorForRegular) (rows true fun (x : IORCategory) => !x.iorForRegular) ∧ MeanGT "production" (rows true fun (x : IORCategory) => x.iorForRegular) (rows true fun (x : IORCategory) => !x.iorForRegular)

Regulars show islands of reliability (Fig. 2), in that novel regular pasts are rated higher, and volunteered more often, when the stem occupies an island for the regular change.

theorem AlbrightHayes2003.irregulars_ior : MeanGT "adjustedRating" (rows false fun (x : IORCategory) => x.iorForIrregular) (rows false fun (x : IORCategory) => !x.iorForIrregular) ∧ MeanGT "production" (rows false fun (x : IORCategory) => x.iorForIrregular) (rows false fun (x : IORCategory) => !x.iorForIrregular)

Islands of reliability for irregulars, likewise.

theorem AlbrightHayes2003.regulars_not_cell_invariant : total "adjustedRating" (rows true fun (x : IORCategory) => decide (x = { iorForRegular := true, iorForIrregular := false })) * (rows true fun (x : IORCategory) => decide (x = { iorForRegular := false, iorForIrregular := true })).length ≠ total "adjustedRating" (rows true fun (x : IORCategory) => decide (x = { iorForRegular := false, iorForIrregular := true })) * (rows true fun (x : IORCategory) => decide (x = { iorForRegular := true, iorForIrregular := false })).length

The single-default-rule prediction, that novel regular ratings do not vary with the stem's cell, fails on the Core data: the regulars-only and irregulars-only cells differ.

theorem AlbrightHayes2003.tradeoff : MeanGT "adjustedRating" (rows true fun (x : IORCategory) => decide (x = { iorForRegular := true, iorForIrregular := false })) (rows true fun (x : IORCategory) => decide (x = { iorForRegular := true, iorForIrregular := true })) ∧ MeanGT "adjustedRating" (rows true fun (x : IORCategory) => decide (x = { iorForRegular := false, iorForIrregular := false })) (rows true fun (x : IORCategory) => decide (x = { iorForRegular := false, iorForIrregular := true })) ∧ MeanGT "adjustedRating" (rows false fun (x : IORCategory) => decide (x = { iorForRegular := false, iorForIrregular := true })) (rows false fun (x : IORCategory) => decide (x = { iorForRegular := true, iorForIrregular := true })) ∧ MeanGT "adjustedRating" (rows false fun (x : IORCategory) => decide (x = { iorForRegular := false, iorForIrregular := false })) (rows false fun (x : IORCategory) => decide (x = { iorForRegular := true, iorForIrregular := false }))

Trade-off (Figs. 3–4): a past is rated higher when its rival is not in an island, for regulars and irregulars alike.

theorem AlbrightHayes2003.ruleBased_islands : MeanGT "ruleBased" (rows true fun (x : IORCategory) => x.iorForRegular) (rows true fun (x : IORCategory) => !x.iorForRegular)

The rule-based model's predicted regular ratings are higher in the regular islands: the stimuli were chosen by the model.

theorem AlbrightHayes2003.analogical_misses_islands (r : Datum) : r ∈ Examples.all → (r.feature? "island").isSome = true → value "analogical" r < value "adjustedRating" r ∧ value "analogical" r < value "ruleBased" r

In Table 4, on the twelve regular pasts in the best islands the analogical model, unable to locate structured similarity, scores below both the participants and the rule-based model.

def AlbrightHayes2003.burnt : List Datum

The pseudo-burnt irregulars of (15).

Equations

• AlbrightHayes2003.burnt = List.filter (fun (x : Datum) => decide (x.feature? "set" = some "burnt")) AlbrightHayes2003.Examples.all

theorem AlbrightHayes2003.burnt_underestimated : total "ruleBased" burnt < total "adjustedRating" burnt ∧ total "adjustedRating" burnt < total "analogical" burnt

The rule-based model's one systematic error: it underrates the burnt-class forms, which the analogical model overrates.

theorem AlbrightHayes2003.regulars_preferred : MeanGT "rating" (List.filter (fun (r : Datum) => decide (r.feature? "pastType" = some "regular")) Examples.all) (List.filter (fun (r : Datum) => decide (r.feature? "pastType" = some "irregular")) Examples.all)

Participants preferred regular pasts overall.