Bybee (1995): Regular morphology and the lexicon

This file formalizes [Byb95]'s network model of productivity, in which schemas emerge over stored forms and the productivity of a schema is determined by two things: its type frequency, the number of distinct forms instantiating it, and the openness of its variables. Token frequency does the opposite work, strengthening an individual form's own entry. Over a corpus of tokens, a schema's type frequency counts the distinct forms it relates and its token frequency counts every occurrence (typeFrequency_le_tokenFrequency), and a more general schema has the larger type frequency (typeFrequency_mono). What a schema generates depends on the corpus only through its types, since the lexicon of its roles is the corpus's set of types, so repetition of a stored form leaves its generative capacity unchanged. Openness is the other determinant: a schema generates every instance whatever is stored exactly when all of its variables are open (Schema.isProductive_iff_forall_generates_iff), which is what makes the English past in -ed fully productive (edSchema_isProductive).

Schemas are product-oriented: generalizations over the derived forms themselves, not over pairs of base and derived form. The class of strung is the shape of its past tenses, and the dialectal new members struck, snuck and drug fit that shape (struck_instantiates) while their bases lack the vowel a source-oriented pairing from string would demand (not_sourceOriented_pairs), the finding of [BM83].

Implementation notes

• A corpus is a multiset of forms; the lexicon of a schema's roles is its set of types.
• The counts of Table 8.1, the French conjugations children overgeneralize by type rather than token frequency, and the German participle and plural cases are not formalized.

References

• [Byb95]
• [Byb07]
• [BM83]

Type frequency and token frequency

def Bybee1995.typeFrequency {P : Type u_1} {α : Type u_2} [PartialOrder α] [DecidableEq (P → α)] (s : ConstructionMorphology.Schema P α) [DecidablePred s.Instantiates] (c : Multiset (P → α)) : ℕ

The type frequency of a schema over a corpus: the number of distinct forms instantiating it.

Equations

• Bybee1995.typeFrequency s c = Multiset.countP s.Instantiates c.dedup

def Bybee1995.tokenFrequency {P : Type u_1} {α : Type u_2} [PartialOrder α] (s : ConstructionMorphology.Schema P α) [DecidablePred s.Instantiates] (c : Multiset (P → α)) : ℕ

The token frequency of a schema over a corpus: the number of occurrences of forms instantiating it.

Equations

• Bybee1995.tokenFrequency s c = Multiset.countP s.Instantiates c

theorem Bybee1995.typeFrequency_le_tokenFrequency {P : Type u_1} {α : Type u_2} [PartialOrder α] [DecidableEq (P → α)] (s : ConstructionMorphology.Schema P α) [DecidablePred s.Instantiates] (c : Multiset (P → α)) : typeFrequency s c ≤ tokenFrequency s c

Types are at most tokens.

theorem Bybee1995.typeFrequency_mono {P : Type u_1} {α : Type u_2} [PartialOrder α] [DecidableEq (P → α)] {s t : ConstructionMorphology.Schema P α} [DecidablePred s.Instantiates] [DecidablePred t.Instantiates] (h : t.body ≤ s.body) (c : Multiset (P → α)) : typeFrequency s c ≤ typeFrequency t c

A more general schema has the larger type frequency.

Openness

def Bybee1995.edSchema : ConstructionMorphology.Schema (Fin 2) (Flat String)

The English past in -ed, over a base slot and an affix slot: the base open, so totally open in the sense of Schema.isProductive_iff_forall_generates_iff.

Equations

• Bybee1995.edSchema = ConstructionMorphology.Schema.productive ![⊥, ↑"ed"]

theorem Bybee1995.edSchema_isProductive : edSchema.IsProductive

The product-oriented class of strung

def Bybee1995.strungSchema : ConstructionMorphology.Schema (Fin 3) (Flat String)

The product-oriented schema of the strung class over onset, nucleus and coda: the nucleus pinned to /ʌ/, the onset open, the coda a closed variable filled from the class.

Equations

• Bybee1995.strungSchema = { body := ![⊥, ↑"ʌ", ⊥], opens := {0} }

def Bybee1995.strungClass : Set (Fin 3 → Flat String)

The stored members of the class.

Equations

• Bybee1995.strungClass = {Bybee1995.Forms.strung.slots, Bybee1995.Forms.stung.slots, Bybee1995.Forms.flung.slots, Bybee1995.Forms.hung.slots}

theorem Bybee1995.struck_instantiates : strungSchema.Instantiates Forms.struck.slots ∧ strungSchema.Instantiates Forms.snuck.slots ∧ strungSchema.Instantiates Forms.drug.slots

The dialectal new members fit the shape of the class.

inductive Bybee1995.PairVar : Type

The variables of a source-oriented pairing of base and past: the shared onset and coda, and the two nuclei.

• onset : PairVar
• coda : PairVar
• base : PairVar
• past : PairVar

@[instance_reducible]

instance Bybee1995.instDecidableEqPairVar : DecidableEq PairVar

Equations

• Bybee1995.instDecidableEqPairVar x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def Bybee1995.sourceOriented : ConstructionMorphology.Schema PairVar (Flat String)

A source-oriented schema pairing a base in /ɪ/ with a past in /ʌ/ over a shared onset and coda: the generalization over pairs such as string, strung.

Equations

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

def Bybee1995.baseSub : Fin 3 → PairVar

The subscripting of the base's positions.

Equations

• Bybee1995.baseSub = ![Bybee1995.PairVar.onset, Bybee1995.PairVar.base, Bybee1995.PairVar.coda]

def Bybee1995.pastSub : Fin 3 → PairVar

The subscripting of the past's positions.

Equations

• Bybee1995.pastSub = ![Bybee1995.PairVar.onset, Bybee1995.PairVar.past, Bybee1995.PairVar.coda]

theorem Bybee1995.string_strung_pairs : sourceOriented.InstantiatesAt (Sum.elim baseSub pastSub) (Sum.elim Forms.string.slots Forms.strung.slots)

string and strung are a paired instantiation of the source-oriented schema.

theorem Bybee1995.not_sourceOriented_pairs : ¬sourceOriented.InstantiatesAt (Sum.elim baseSub pastSub) (Sum.elim Forms.strike.slots Forms.struck.slots) ∧ ¬sourceOriented.InstantiatesAt (Sum.elim baseSub pastSub) (Sum.elim Forms.sneak.slots Forms.snuck.slots) ∧ ¬sourceOriented.InstantiatesAt (Sum.elim baseSub pastSub) (Sum.elim Forms.drag.slots Forms.drug.slots)

The new members are not: their bases lack /ɪ/, so no source-oriented schema pairs them, while the product-oriented schema admits them.