@cite{zuraw-2010}: Factorial Typology of Nasal Substitution

Formalizes the factorial typology of Tagalog-style nasal substitution from @cite{zuraw-2010} (NLLT 28: 417–472). When a nasal-final prefix (e.g. maŋ-) is concatenated with an obstruent-initial stem, the nasal and the obstruent may coalesce into a single nasal retaining the place of the latter:

• maŋ+bigáj → mamigáj 'to distribute' (nasal substitution = YES)
• paŋ+tabój → pantabój 'to goad' (faithful cluster = NO)

Constraint set

Six constraints drive the typology:

• *NC (markedness): penalizes all nasal + obstruent clusters
• *NC̥ (markedness): penalizes nasal + voiceless obstruent clusters
• *ASSOC (markedness): penalizes multiple association (coalescence)
• *[ŋ, *[n, *[m (markedness, stringent hierarchy): penalizes stem-initial nasals resulting from coalescence, with *[m most inclusive (penalizes all nasals) and *[ŋ least inclusive (penalizes velar only)

The stringent [N hierarchy is defined following Prince's (1997) stringency theory: "[n" means "a stem must not begin with n or a backer nasal", so *[n penalizes both n and ŋ but not m. Similarly, *[m penalizes m, n, and ŋ (all nasals).

Factorial typology

With 6 constraints, there are 720 rankings generating exactly 10 distinct substitution patterns over 6 stem-initial consonants {p, t, k, b, d, g}. The 10 patterns satisfy two implicational universals:

1. Voicing effect: if a voiced C undergoes substitution, the voiceless C at the same place also does (e.g., b→YES implies p→YES).
2. Place effect: if a backer C undergoes substitution, all fronter Cs also do (within voicing class: k→t→p and g→d→b). Labials are easiest to substitute (least *[N cost), velars hardest (most *[N cost).

Dictionary data

@cite{zuraw-2010}'s Tagalog dictionary counts confirm the voicing effect: voiceless stems show higher substitution rates than voiced stems at the labial place (p: 253/263 vs b: 177/277).

def Phenomena.PhonologicalAlternation.Studies.Zuraw2010.starNC : Core.OT.NamedConstraint Fragments.Tagalog.Phonology.NSCand

*NC: penalizes all nasal + obstruent sequences. Violated by NO (the faithful output preserves the NC cluster).

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def Phenomena.PhonologicalAlternation.Studies.Zuraw2010.starNCvoiceless : Core.OT.NamedConstraint Fragments.Tagalog.Phonology.NSCand

*NC̥: penalizes nasal + voiceless obstruent sequences. Violated by NO for voiceless stems only.

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def Phenomena.PhonologicalAlternation.Studies.Zuraw2010.starAssoc : Core.OT.NamedConstraint Fragments.Tagalog.Phonology.NSCand

*ASSOC: penalizes multiple association (coalescence). Violated by YES for all stems. On this 6-consonant domain, *ASSOC has the same violation profile as *[m (both assign 1 to every YES candidate). They remain distinct constraints because they encode different linguistic generalizations and their co-presence affects factorial percentages.

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def Phenomena.PhonologicalAlternation.Studies.Zuraw2010.starInitVelar : Core.OT.NamedConstraint Fragments.Tagalog.Phonology.NSCand

*[ŋ: "A stem must not begin with ŋ or a backer nasal." Least inclusive member of the stringent *[N hierarchy. Violated by YES for velar stems only (coalescence yields ŋ).

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def Phenomena.PhonologicalAlternation.Studies.Zuraw2010.starInitCorVel : Core.OT.NamedConstraint Fragments.Tagalog.Phonology.NSCand

*[n: "A stem must not begin with n or a backer nasal." Middle member of the stringent *[N hierarchy. Violated by YES for coronal and velar stems (coalescence yields n or ŋ, both at or backer than n).

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def Phenomena.PhonologicalAlternation.Studies.Zuraw2010.starInitAll : Core.OT.NamedConstraint Fragments.Tagalog.Phonology.NSCand

*[m: "A stem must not begin with m or a backer nasal." Most inclusive member of the stringent *[N hierarchy. Violated by YES for all stems (coalescence always yields a nasal, and every nasal is m or backer).

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def Phenomena.PhonologicalAlternation.Studies.Zuraw2010.allConstraints : List (Core.OT.NamedConstraint Fragments.Tagalog.Phonology.NSCand)

The six constraints forming the factorial typology.

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theorem Phenomena.PhonologicalAlternation.Studies.Zuraw2010.stringency_violations : starInitAll.eval (Fragments.Tagalog.Phonology.StemC.p, Fragments.Tagalog.Phonology.SubSt.yes) + starInitCorVel.eval (Fragments.Tagalog.Phonology.StemC.p, Fragments.Tagalog.Phonology.SubSt.yes) + starInitVelar.eval (Fragments.Tagalog.Phonology.StemC.p, Fragments.Tagalog.Phonology.SubSt.yes) = 1 ∧ starInitAll.eval (Fragments.Tagalog.Phonology.StemC.t, Fragments.Tagalog.Phonology.SubSt.yes) + starInitCorVel.eval (Fragments.Tagalog.Phonology.StemC.t, Fragments.Tagalog.Phonology.SubSt.yes) + starInitVelar.eval (Fragments.Tagalog.Phonology.StemC.t, Fragments.Tagalog.Phonology.SubSt.yes) = 2 ∧ starInitAll.eval (Fragments.Tagalog.Phonology.StemC.k, Fragments.Tagalog.Phonology.SubSt.yes) + starInitCorVel.eval (Fragments.Tagalog.Phonology.StemC.k, Fragments.Tagalog.Phonology.SubSt.yes) + starInitVelar.eval (Fragments.Tagalog.Phonology.StemC.k, Fragments.Tagalog.Phonology.SubSt.yes) = 3

The stringent [N hierarchy assigns increasing violation counts to nasals at backer places: labial m incurs 1 violation ([m only), coronal n incurs 2 (*[n + [m), velar ŋ incurs 3 ([ŋ + *[n + *[m). This makes labials easiest to substitute and velars hardest.

theorem Phenomena.PhonologicalAlternation.Studies.Zuraw2010.assoc_eq_initAll (c : Fragments.Tagalog.Phonology.StemC) (s : Fragments.Tagalog.Phonology.SubSt) : starAssoc.eval (c, s) = starInitAll.eval (c, s)

*ASSOC and *[m have identical violation profiles on the 6-consonant domain (both penalize every YES candidate).

structure Phenomena.PhonologicalAlternation.Studies.Zuraw2010.SubPattern : Type

A substitution pattern: for each of the 6 consonants, whether substitution is optimal (true = YES wins).

• sub_p : Bool
• sub_t : Bool
• sub_k : Bool
• sub_b : Bool
• sub_d : Bool
• sub_g : Bool

def Phenomena.PhonologicalAlternation.Studies.Zuraw2010.instDecidableEqSubPattern.decEq (x✝ x✝¹ : SubPattern) : Decidable (x✝ = x✝¹)

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instance Phenomena.PhonologicalAlternation.Studies.Zuraw2010.instDecidableEqSubPattern : DecidableEq SubPattern

Equations

• Phenomena.PhonologicalAlternation.Studies.Zuraw2010.instDecidableEqSubPattern = Phenomena.PhonologicalAlternation.Studies.Zuraw2010.instDecidableEqSubPattern.decEq

instance Phenomena.PhonologicalAlternation.Studies.Zuraw2010.instBEqSubPattern : BEq SubPattern

Equations

• Phenomena.PhonologicalAlternation.Studies.Zuraw2010.instBEqSubPattern = { beq := Phenomena.PhonologicalAlternation.Studies.Zuraw2010.instBEqSubPattern.beq }

def Phenomena.PhonologicalAlternation.Studies.Zuraw2010.instBEqSubPattern.beq : SubPattern → SubPattern → Bool

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• Phenomena.PhonologicalAlternation.Studies.Zuraw2010.instBEqSubPattern.beq x✝¹ x✝ = false

instance Phenomena.PhonologicalAlternation.Studies.Zuraw2010.instReprSubPattern : Repr SubPattern

Equations

• Phenomena.PhonologicalAlternation.Studies.Zuraw2010.instReprSubPattern = { reprPrec := Phenomena.PhonologicalAlternation.Studies.Zuraw2010.instReprSubPattern.repr }

def Phenomena.PhonologicalAlternation.Studies.Zuraw2010.instReprSubPattern.repr : SubPattern → ℕ → Std.Format

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def Phenomena.PhonologicalAlternation.Studies.Zuraw2010.subWins (ranking : List (Core.OT.NamedConstraint Fragments.Tagalog.Phonology.NSCand)) (c : Fragments.Tagalog.Phonology.StemC) : Bool

Determine whether YES wins over NO for a given consonant under a ranking.

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def Phenomena.PhonologicalAlternation.Studies.Zuraw2010.patternOf (ranking : List (Core.OT.NamedConstraint Fragments.Tagalog.Phonology.NSCand)) : SubPattern

Extract the substitution pattern for a given ranking.

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def Phenomena.PhonologicalAlternation.Studies.Zuraw2010.factorialPatterns : List SubPattern

All distinct substitution patterns generated by the factorial typology.

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theorem Phenomena.PhonologicalAlternation.Studies.Zuraw2010.ten_language_types : factorialPatterns.length = 10

The factorial typology generates exactly 10 language types (Table 5).

def Phenomena.PhonologicalAlternation.Studies.Zuraw2010.pat_a : SubPattern

Pattern a: no substitution for any consonant.

Equations

• Phenomena.PhonologicalAlternation.Studies.Zuraw2010.pat_a = { sub_p := false, sub_t := false, sub_k := false, sub_b := false, sub_d := false, sub_g := false }

def Phenomena.PhonologicalAlternation.Studies.Zuraw2010.pat_b : SubPattern

Pattern b: substitution for labial voiceless only (p). Attested in Da'a (@cite{zuraw-2010} Table 5).

Equations

• Phenomena.PhonologicalAlternation.Studies.Zuraw2010.pat_b = { sub_p := true, sub_t := false, sub_k := false, sub_b := false, sub_d := false, sub_g := false }

def Phenomena.PhonologicalAlternation.Studies.Zuraw2010.pat_c : SubPattern

Pattern c: substitution for both labials (p, b). Attested in Wolio (@cite{zuraw-2010} Table 5).

Equations

• Phenomena.PhonologicalAlternation.Studies.Zuraw2010.pat_c = { sub_p := true, sub_t := false, sub_k := false, sub_b := true, sub_d := false, sub_g := false }

def Phenomena.PhonologicalAlternation.Studies.Zuraw2010.pat_d : SubPattern

Pattern d: substitution for voiceless labial + coronal (p, t). Attested in Balantak (@cite{zuraw-2010} Table 5).

Equations

• Phenomena.PhonologicalAlternation.Studies.Zuraw2010.pat_d = { sub_p := true, sub_t := true, sub_k := false, sub_b := false, sub_d := false, sub_g := false }

def Phenomena.PhonologicalAlternation.Studies.Zuraw2010.pat_e : SubPattern

Pattern e: substitution for p, t, b. Attested in Toba Batak (@cite{zuraw-2010} Table 5).

Equations

• Phenomena.PhonologicalAlternation.Studies.Zuraw2010.pat_e = { sub_p := true, sub_t := true, sub_k := false, sub_b := true, sub_d := false, sub_g := false }

def Phenomena.PhonologicalAlternation.Studies.Zuraw2010.pat_f : SubPattern

Pattern f: substitution for p, t, b, d. Attested in Cebuano (@cite{zuraw-2010} Table 5).

Equations

• Phenomena.PhonologicalAlternation.Studies.Zuraw2010.pat_f = { sub_p := true, sub_t := true, sub_k := false, sub_b := true, sub_d := true, sub_g := false }

def Phenomena.PhonologicalAlternation.Studies.Zuraw2010.pat_g : SubPattern

Pattern g: substitution for all voiceless (p, t, k). Attested in Bolaang Mongondow (@cite{zuraw-2010} Table 5).

Equations

• Phenomena.PhonologicalAlternation.Studies.Zuraw2010.pat_g = { sub_p := true, sub_t := true, sub_k := true, sub_b := false, sub_d := false, sub_g := false }

def Phenomena.PhonologicalAlternation.Studies.Zuraw2010.pat_h : SubPattern

Pattern h: substitution for p, t, k, b. Attested in Ratahan (@cite{zuraw-2010} Table 5).

Equations

• Phenomena.PhonologicalAlternation.Studies.Zuraw2010.pat_h = { sub_p := true, sub_t := true, sub_k := true, sub_b := true, sub_d := false, sub_g := false }

def Phenomena.PhonologicalAlternation.Studies.Zuraw2010.pat_i : SubPattern

Pattern i: substitution for p, t, k, b, d. Attested in Pendau (@cite{zuraw-2010} Table 5).

Equations

• Phenomena.PhonologicalAlternation.Studies.Zuraw2010.pat_i = { sub_p := true, sub_t := true, sub_k := true, sub_b := true, sub_d := true, sub_g := false }

def Phenomena.PhonologicalAlternation.Studies.Zuraw2010.pat_j : SubPattern

Pattern j: substitution for all consonants. Attested in Tagalog, Indonesian, Malay (@cite{zuraw-2010} Table 5).

Equations

• Phenomena.PhonologicalAlternation.Studies.Zuraw2010.pat_j = { sub_p := true, sub_t := true, sub_k := true, sub_b := true, sub_d := true, sub_g := true }

theorem Phenomena.PhonologicalAlternation.Studies.Zuraw2010.all_patterns_attested : [pat_j, pat_g, pat_a, pat_i, pat_f, pat_d, pat_h, pat_e, pat_c, pat_b] = factorialPatterns

All 10 named patterns appear in the factorial typology.

theorem Phenomena.PhonologicalAlternation.Studies.Zuraw2010.voicing_effect (pat : SubPattern) : pat ∈ factorialPatterns → (pat.sub_b = true → pat.sub_p = true) ∧ (pat.sub_d = true → pat.sub_t = true) ∧ (pat.sub_g = true → pat.sub_k = true)

Voicing effect: for every pattern in the typology, if a voiced consonant undergoes substitution then its voiceless counterpart does too. E.g., b→YES implies p→YES.

theorem Phenomena.PhonologicalAlternation.Studies.Zuraw2010.place_effect_voiceless (pat : SubPattern) : pat ∈ factorialPatterns → (pat.sub_k = true → pat.sub_t = true) ∧ (pat.sub_t = true → pat.sub_p = true)

Place effect (voiceless): among voiceless consonants, if a backer place undergoes substitution then all fronter places do too. k→YES implies t→YES implies p→YES.

theorem Phenomena.PhonologicalAlternation.Studies.Zuraw2010.place_effect_voiced (pat : SubPattern) : pat ∈ factorialPatterns → (pat.sub_g = true → pat.sub_d = true) ∧ (pat.sub_d = true → pat.sub_b = true)

Place effect (voiced): among voiced consonants, if a backer place undergoes substitution then all fronter places do too. g→YES implies d→YES implies b→YES.

theorem Phenomena.PhonologicalAlternation.Studies.Zuraw2010.g_implies_all (pat : SubPattern) : pat ∈ factorialPatterns → pat.sub_g = true → pat.sub_p = true ∧ pat.sub_t = true ∧ pat.sub_k = true ∧ pat.sub_b = true ∧ pat.sub_d = true ∧ pat.sub_g = true

The voicing and place effects together imply that if g undergoes substitution, every other consonant does too (pattern j is the only pattern where g = YES).

def Phenomena.PhonologicalAlternation.Studies.Zuraw2010.subCount (c : Fragments.Tagalog.Phonology.StemC) : ℕ

Count how many of the 720 rankings produce substitution for consonant c.

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theorem Phenomena.PhonologicalAlternation.Studies.Zuraw2010.factorial_percentages : subCount Fragments.Tagalog.Phonology.StemC.p = 360 ∧ subCount Fragments.Tagalog.Phonology.StemC.t = 288 ∧ subCount Fragments.Tagalog.Phonology.StemC.k = 240 ∧ subCount Fragments.Tagalog.Phonology.StemC.b = 240 ∧ subCount Fragments.Tagalog.Phonology.StemC.d = 180 ∧ subCount Fragments.Tagalog.Phonology.StemC.g = 144

Factorial percentage of rankings favoring substitution for each consonant. The percentages strictly decrease from labials to velars within each voicing class, and voiceless > voiced at each place. Values match @cite{zuraw-2010} footnote 17.

theorem Phenomena.PhonologicalAlternation.Studies.Zuraw2010.factorial_rates : ↑(subCount Fragments.Tagalog.Phonology.StemC.p) / 720 = 1 / 2 ∧ ↑(subCount Fragments.Tagalog.Phonology.StemC.t) / 720 = 2 / 5 ∧ ↑(subCount Fragments.Tagalog.Phonology.StemC.k) / 720 = 1 / 3 ∧ ↑(subCount Fragments.Tagalog.Phonology.StemC.b) / 720 = 1 / 3 ∧ ↑(subCount Fragments.Tagalog.Phonology.StemC.d) / 720 = 1 / 4 ∧ ↑(subCount Fragments.Tagalog.Phonology.StemC.g) / 720 = 1 / 5

The factorial percentages as fractions of 720 total rankings: p=50%, t=40%, k=33⅓%, b=33⅓%, d=25%, g=20%.

theorem Phenomena.PhonologicalAlternation.Studies.Zuraw2010.place_monotonicity : subCount Fragments.Tagalog.Phonology.StemC.p > subCount Fragments.Tagalog.Phonology.StemC.t ∧ subCount Fragments.Tagalog.Phonology.StemC.t > subCount Fragments.Tagalog.Phonology.StemC.k ∧ subCount Fragments.Tagalog.Phonology.StemC.b > subCount Fragments.Tagalog.Phonology.StemC.d ∧ subCount Fragments.Tagalog.Phonology.StemC.d > subCount Fragments.Tagalog.Phonology.StemC.g

Place monotonicity: the factorial percentage strictly decreases from labial to velar within each voicing class.

theorem Phenomena.PhonologicalAlternation.Studies.Zuraw2010.voicing_monotonicity : subCount Fragments.Tagalog.Phonology.StemC.p ≥ subCount Fragments.Tagalog.Phonology.StemC.b ∧ subCount Fragments.Tagalog.Phonology.StemC.t ≥ subCount Fragments.Tagalog.Phonology.StemC.d ∧ subCount Fragments.Tagalog.Phonology.StemC.k ≥ subCount Fragments.Tagalog.Phonology.StemC.g

Voicing monotonicity: voiceless substitution percentage is at least as high as voiced at every place.

theorem Phenomena.PhonologicalAlternation.Studies.Zuraw2010.tagalog_is_pattern_j : factorialPatterns.contains pat_j = true

Tagalog exhibits nasal substitution for all six consonants, corresponding to pattern j — the maximal substitution pattern. This is consistent with @cite{zuraw-2010}'s analysis where Tagalog's grammar ranks both *NC and *NC̥ above all anti-substitution constraints. The Tagalog fragment in Fragments.Tagalog.Phonology models the probabilistic (gradient) version of this pattern using @cite{magri-2025}'s MaxEnt analysis.