Booij (2019): The role of schemas in Construction Morphology

This file formalizes [Boo19]'s templatic and second-order schemas as coindexed schemas read through subscriptings of template slots. A templatic schema is a description over root consonants and vowel constants, and a consonant that occupies two slots, the gemination of the Damascus Arabic occupation template C₁aC₂C₂aaC₃ of (2), is a subscripting that is not injective: an item instantiates the schema exactly when its vowels are the template's and the two geminate slots are filled alike (occupation_instantiatesAt_iff). The subscripting is the association of the autosegmental template: the slots it coindexes are the slots associated to one melody element (occupationSub_eq_iff_assoc_eq), and the association spells out the word (xabbaazMatch_spellout).

Second-order schemas relate two templates over one root. Javanese partial reduplication (13) copies the initial consonant into the reduplicant, so the reduplicant repeats one variable and shares it with the base (reduplication_pairs_iff); the Egyptian Arabic comparative (16) pairs an adjective with the comparative template over the same three consonants (comparative_pairs_iff), so kibiir pairs with akbar and not with atxan.

Implementation notes

• The words are the CLDF forms of Data/Forms/Booij2019.json, read as slot-indexed segments on the flat carrier by Data.Forms.Form.slots; slots are Fin n positions. A variable standing for a sequence of sounds, the paper's y, is one segment holding the residue.
• The higher-order schema for Dutch -er and the affixoid discussion are not formalized.

References

• [Boo19]
• [McC81]

The occupation template C₁aC₂C₂aaC₃

inductive Booij2019.OccPart : Type

The parts of the occupation template (2): three root consonants and two vowel constants.

• c1 : OccPart
• c2 : OccPart
• c3 : OccPart
• a : OccPart
• aa : OccPart

@[instance_reducible]

instance Booij2019.instDecidableEqOccPart : DecidableEq OccPart

Equations

• Booij2019.instDecidableEqOccPart x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def Booij2019.occupation : ConstructionMorphology.Schema OccPart (Flat String)

The occupation schema (2): the vowels pinned, the root consonants open.

Equations

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

def Booij2019.occupationSub : Fin 6 → OccPart

The six slots of C₁aC₂C₂aaC₃ subscripted by parts: the second consonant occupies two.

Equations

• Booij2019.occupationSub = ![Booij2019.OccPart.c1, Booij2019.OccPart.a, Booij2019.OccPart.c2, Booij2019.OccPart.c2, Booij2019.OccPart.aa, Booij2019.OccPart.c3]

theorem Booij2019.occupation_instantiatesAt_iff {w : Fin 6 → Flat String} : occupation.InstantiatesAt occupationSub w ↔ w 1 = ↑"a" ∧ w 4 = ↑"aa" ∧ w 2 = w 3

An item instantiates the occupation schema exactly when its vowels are the template's and the two slots of the second consonant are filled alike: gemination is a subscripting that is not injective.

theorem Booij2019.occupation_nouns : occupation.InstantiatesAt occupationSub Forms.xabbaaz.slots ∧ occupation.InstantiatesAt occupationSub Forms.xaddaam.slots ∧ occupation.InstantiatesAt occupationSub Forms.bawwaab.slots ∧ occupation.InstantiatesAt occupationSub Forms.sammaak.slots

The nouns of (1) instantiate the template.

theorem Booij2019.not_occupation_of_ne_medial : ¬occupation.InstantiatesAt occupationSub ![↑"x", ↑"a", ↑"b", ↑"d", ↑"aa", ↑"z"]

A form with distinct medial consonants matches the vowels but not the gemination.

The subscripting as autosegmental association

def Booij2019.occupationTemplate : Morphology.CVTemplate

The template of (2) as a CV skeleton.

Equations

• Booij2019.occupationTemplate = { slots := [Morphology.CVSlot.C, Morphology.CVSlot.V, Morphology.CVSlot.C, Morphology.CVSlot.C, Morphology.CVSlot.V, Morphology.CVSlot.C] }

def Booij2019.xabbaazMatch : Morphology.TemplateMatch String

xabbaaz as the association of the root /x b z/ and the vocalism /a aa/ to the template: the second root consonant is associated to two slots.

Equations

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

def Booij2019.xabbaazAssoc (i : Fin 6) : Option (Morphology.AssocSource × ℕ)

The melody element a slot is associated to.

Equations

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

theorem Booij2019.xabbaazMatch_spellout : xabbaazMatch.spellout = Forms.xabbaaz.segments

The association lines spell out xabbaaz.

theorem Booij2019.occupationSub_eq_iff_assoc_eq (i j : Fin 6) : occupationSub i = occupationSub j ↔ xabbaazAssoc i = xabbaazAssoc j

The subscripting and the association agree: two slots carry the same part exactly when they are associated to the same melody element.

Javanese partial reduplication

inductive Booij2019.RedPart : Type

The parts of the reduplication schema (13): the copied consonant, the schwa constant, and the residue.

• c1 : RedPart
• schwa : RedPart
• y : RedPart

@[instance_reducible]

instance Booij2019.instDecidableEqRedPart : DecidableEq RedPart

Equations

• Booij2019.instDecidableEqRedPart x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def Booij2019.reduplication : ConstructionMorphology.Schema RedPart (Flat String)

The reduplication schema (13): the schwa pinned, the consonant and the residue open.

Equations

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

def Booij2019.redSub : Fin 4 → RedPart

The reduplicant C₁əC₁y: the initial consonant occupies two slots.

Equations

• Booij2019.redSub = ![Booij2019.RedPart.c1, Booij2019.RedPart.schwa, Booij2019.RedPart.c1, Booij2019.RedPart.y]

def Booij2019.baseSub : Fin 2 → RedPart

The base C₁y.

Equations

• Booij2019.baseSub = ![Booij2019.RedPart.c1, Booij2019.RedPart.y]

theorem Booij2019.reduplication_pairs_iff {r : Fin 4 → Flat String} {b : Fin 2 → Flat String} : reduplication.InstantiatesAt (Sum.elim redSub baseSub) (Sum.elim r b) ↔ r 1 = ↑"ə" ∧ r 0 = r 2 ∧ r 0 = b 0 ∧ r 3 = b 1

A reduplicant and a base are a paired instantiation of (13) exactly when the reduplicant has the schwa, copies its initial consonant, and shares consonant and residue with the base.

theorem Booij2019.reduplication_verbs : reduplication.InstantiatesAt (Sum.elim redSub baseSub) (Sum.elim Forms.tetamu.slots Forms.tamu.slots) ∧ reduplication.InstantiatesAt (Sum.elim redSub baseSub) (Sum.elim Forms.jejawah.slots Forms.jawah.slots) ∧ reduplication.InstantiatesAt (Sum.elim redSub baseSub) (Sum.elim Forms.gegeni.slots Forms.geni.slots)

The verbs of (12) with their bases.

The Egyptian Arabic comparative

inductive Booij2019.CmpPart : Type

The parts of the comparative schema (16): three root consonants, the adjective's two open vowels, and the comparative's constant a.

• c1 : CmpPart
• c2 : CmpPart
• c3 : CmpPart
• v1 : CmpPart
• v2 : CmpPart
• a : CmpPart

@[instance_reducible]

instance Booij2019.instDecidableEqCmpPart : DecidableEq CmpPart

Equations

• Booij2019.instDecidableEqCmpPart x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def Booij2019.comparative : ConstructionMorphology.Schema CmpPart (Flat String)

The comparative schema (16): the comparative vowel pinned, all else open.

Equations

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

def Booij2019.adjSub : Fin 5 → CmpPart

The adjective template C₁VC₂VC₃.

Equations

• Booij2019.adjSub = ![Booij2019.CmpPart.c1, Booij2019.CmpPart.v1, Booij2019.CmpPart.c2, Booij2019.CmpPart.v2, Booij2019.CmpPart.c3]

def Booij2019.cmpSub : Fin 5 → CmpPart

The comparative template aC₁C₂aC₃.

Equations

• Booij2019.cmpSub = ![Booij2019.CmpPart.a, Booij2019.CmpPart.c1, Booij2019.CmpPart.c2, Booij2019.CmpPart.a, Booij2019.CmpPart.c3]

theorem Booij2019.comparative_pairs_iff {p q : Fin 5 → Flat String} : comparative.InstantiatesAt (Sum.elim adjSub cmpSub) (Sum.elim p q) ↔ q 0 = ↑"a" ∧ q 3 = ↑"a" ∧ p 0 = q 1 ∧ p 2 = q 2 ∧ p 4 = q 4

An adjective and a comparative are a paired instantiation of (16) exactly when the comparative has its vowels and the two share their three consonants.

theorem Booij2019.comparative_pairs : comparative.InstantiatesAt (Sum.elim adjSub cmpSub) (Sum.elim Forms.kibiir.slots Forms.akbar.slots) ∧ comparative.InstantiatesAt (Sum.elim adjSub cmpSub) (Sum.elim Forms.tixiin.slots Forms.atxan.slots) ∧ ¬comparative.InstantiatesAt (Sum.elim adjSub cmpSub) (Sum.elim Forms.kibiir.slots Forms.atxan.slots)

kibiir pairs with akbar and tixiin with atxan, and kibiir not with atxan.