Morphological Paradigm Substrate

@cite{ackerman-malouf-2013} @cite{rathi-hahn-futrell-2026}

Cross-paper substrate for paradigm-cell-level analyses: inflection classes, paradigm systems, cell distributions, and entropy-based measures over them.

Core.Morphology.ParadigmCell (in MorphRule.lean) is the type for one cell of a paradigm with its features and form; ParadigmSystem here is the whole table with frequency weights, organized by inflection class.

Form parameterization

InflectionClass n Form and ParadigmSystem n Form are parameterized over the surface-form type. Most consumers use Form := String (the natural representation for actual morphological forms in @cite{ackerman-malouf-2013}'s A&M language sample). The Rathi 2026 toy paradigms use a small [Fintype] form alphabet so that PMF-based entropy operators (Entropy.lean) apply to cell distributions; that integration lives in a sibling PMFCellDistribution.lean file added in a later phase of the unification refactor.

Public API

• InflectionClass n Form — total function Fin n → Form from cell index to form
• ParadigmSystem n Form — list of inflection classes with frequency weights
• cellDistribution, jointCellDistribution — empirical distributions
• cellEntropy, conditionalCellEntropy — Shannon entropies via Core.InformationTheory
• isImplicative, isTransparent — structural predicates on paradigm shape
• fromStems — derive a ParadigmSystem n String from a list of Stem σ
• eComplexity — count of inflection classes (Ackerman-Malouf E-complexity)

iComplexity (the paper-specific average conditional entropy) and LCECHolds (the LCEC threshold predicate) live in AckermanMalouf2013.lean because they are particular aggregations the paper defines, not substrate primitives.

structure Core.Morphology.InflectionClass (numCells : ℕ) (Form : Type u_1) : Type u_1

An inflection class: function from cell index to surface realization of type Form.

• realize : Fin numCells → Form

@[implicit_reducible]

instance Core.Morphology.instBEqInflectionClass {n : ℕ} {Form : Type} [BEq Form] : BEq (InflectionClass n Form)

Equations

• Core.Morphology.instBEqInflectionClass = { beq := fun (a b : Core.Morphology.InflectionClass n Form) => (List.finRange n).all fun (i : Fin n) => a.realize i == b.realize i }

structure Core.Morphology.ParadigmSystem (numCells : ℕ) (Form : Type u_1) : Type u_1

A paradigm system: inflection classes paired with frequency weights.

• entries : List (InflectionClass numCells Form × ℚ)

def Core.Morphology.groupBySum {α : Type} [BEq α] (tagged : List (α × ℚ)) : List (α × ℚ)

Group a tagged list by key, summing associated ℚ values.

Equations

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

def Core.Morphology.ParadigmSystem.cellDistribution {n : ℕ} {Form : Type} [BEq Form] (ps : ParadigmSystem n Form) (c : Fin n) : List (Form × ℚ)

Empirical distribution of forms at cell c: pairs each surface form with the total frequency of inflection classes realizing it at c.

Equations

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

def Core.Morphology.ParadigmSystem.jointCellDistribution {n : ℕ} {Form : Type} [BEq Form] (ps : ParadigmSystem n Form) (ci cj : Fin n) : List ((Form × Form) × ℚ)

Joint empirical distribution of forms at cell pair (ci, cj).

Equations

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

def Core.Morphology.ParadigmSystem.eComplexity {n : ℕ} {Form : Type u_1} (ps : ParadigmSystem n Form) : ℕ

E-complexity (@cite{ackerman-malouf-2013}): the number of inflection classes in the paradigm system.

Equations

• ps.eComplexity = ps.entries.length

noncomputable def Core.Morphology.ParadigmSystem.cellEntropy {n : ℕ} {Form : Type} [BEq Form] [DecidableEq Form] (ps : ParadigmSystem n Form) (c : Fin n) : ℝ

Shannon entropy of the empirical distribution at cell c (in nats).

Equations

• ps.cellEntropy c = match Core.Morphology.distOfList✝ (ps.cellDistribution c) with | (support, prob) => Core.Morphology.entropy✝ support prob

noncomputable def Core.Morphology.ParadigmSystem.conditionalCellEntropy {n : ℕ} {Form : Type} [BEq Form] [DecidableEq Form] (ps : ParadigmSystem n Form) (ci cj : Fin n) : ℝ

Conditional entropy H(C_i | C_j) of cell ci given cell cj (in nats). The integrand of @cite{ackerman-malouf-2013}'s i-complexity.

Equations

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

def Core.Morphology.ParadigmSystem.isImplicative {n : ℕ} {Form : Type} [BEq Form] [DecidableEq Form] (ps : ParadigmSystem n Form) (ci cj : Fin n) : Prop

A cell pair (ci, cj) is implicative iff knowing the form at cj perfectly determines the form at ci (zero conditional entropy).

Equations

• ps.isImplicative ci cj = (ps.conditionalCellEntropy ci cj = 0)

def Core.Morphology.ParadigmSystem.isTransparent {n : ℕ} {Form : Type} [BEq Form] [DecidableEq Form] (ps : ParadigmSystem n Form) : Prop

A paradigm is transparent iff every off-diagonal cell pair is implicative — every cell perfectly predicts every other.

Equations

• ps.isTransparent = ∀ (ci cj : Fin n), ci ≠ cj → ps.isImplicative ci cj

def Core.Morphology.ParadigmSystem.fromStems {σ : Type} (stems : List (Stem σ)) (baseMeaning : σ) (numCells : ℕ) (cellExtractor : List (String × Features × σ) → Fin numCells → String) : ParadigmSystem numCells String

Extract a ParadigmSystem n String from a list of Stems. The cellExtractor function maps each stem's generated forms (with their features and base meaning) to a per-cell surface realization. Specialized to Form := String since Stem's allForms returns String-typed surface forms.

Equations

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