@cite{mueller-2013}: Unifying Everything #

@cite{mueller-2013}

Cross-theory comparison formalizing Müller's central thesis: Minimalism, HPSG, CCG, Construction Grammar, and Dependency Grammar converge on three universal combination schemata (Head-Complement, Head-Specifier, Head-Filler).

Structure #

§1. Classification functions: map each theory's operations to CombinationKind
§2. Labeling convergence: head determines category of result
§3. External Merge ↔ Head-Complement ↔ Application
§4. Internal Merge ↔ Head-Filler ↔ Composition
§5. Coordination diagnostic: same category required
§6. Directional MG ≈ CCG (placeholder)
§7. Both directions right: need Merge AND phrasal constructions
§8. Concrete cross-theory examples
§9. Labeling failures: free relatives + coordination
§10. Monovalent verb serialization problem
§11. Iterable valence operations

§1. Classification Functions #

Lightweight mappers from each theory's combination operations to the theory-neutral CombinationKind.

CCG classification #

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def Comparisons.Mueller2013.classifyCCGDerivStep :

CCG.DerivStep → Option Core.CombinationKind

Classify a CCG derivation step as one of the three schemata.

Forward/backward application → Head-Complement (functor selects argument)
Forward/backward composition → Head-Filler (enables extraction; this is an approximation — composition also serves non-extraction functions like heavy NP shift and right-node raising)
Type-raising → none (unary operation, not a binary combination)
Coordination → none (symmetric, not one of the three headed schemata)

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Instances For

HPSG classification #

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def Comparisons.Mueller2013.classifyHPSGSchema :

HPSG.HPSGSchema → Option Core.CombinationKind

Classify an HPSG schema application as one of the three schemata.

@cite{mueller-2013}'s three universal schemata are Head-Complement, Head-Subject, and Head-Filler. HPSG's fourth schema, Head-Modifier (adjunction), falls outside this classification — Müller does not include adjunction in the convergence claim.

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Dependency Grammar classification #

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def Comparisons.Mueller2013.classifyDepType :

UD.DepRel → Core.CombinationKind

Classify a UD dependency relation as one of the three schemata.

Subject dependencies are Head-Specifier; all other core dependencies are Head-Complement. Non-projective dependencies (handled separately) correspond to Head-Filler.

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CxG classification #

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def Comparisons.Mueller2013.classifyCxGFullyCompositional (c : ConstructionGrammar.Construction) :

Bool

Classify whether a CxG construction is fully compositional.

Fully abstract constructions without pragmatic function decompose into sequences of Head-Complement and Head-Specifier steps. Other constructions are irreducible phrasal patterns.

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Comparisons.Mueller2013.classifyCxGFullyCompositional c = ConstructionGrammar.isFullyCompositional c

Instances For

§2. Labeling Convergence (Müller §2.1) #

The head determines the category of the result. This is called:

Minimalism: the labeling algorithm (selector projects)
HPSG: Head Feature Principle (head features percolate)
CCG: the functor's result category is the output

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theorem Comparisons.Mueller2013.ccg_fapp_result_category (x y : CCG.Cat) :

CCG.forwardApp (x.rslash y) y = some x

CCG forward application preserves the functor's result category.

When X/Y combines with Y via forward application, the result is X — the left part of the functor's slash category.

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theorem Comparisons.Mueller2013.ccg_bapp_result_category (x y : CCG.Cat) :

CCG.backwardApp y (x.lslash y) = some x

CCG backward application preserves the functor's result category.

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theorem Comparisons.Mueller2013.min_selector_projects (a b : Minimalism.SyntacticObject) (h : Minimalism.selects a b) :

Minimalism.label (Minimalism.merge a b) = Minimalism.label a

Minimalist labeling: when α selects β, the label of {α, β} = label of α.

The selector projects, giving the result the same category as the head.

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theorem Comparisons.Mueller2013.labeling_convergence :

(∀ (a b : Minimalism.SyntacticObject), Minimalism.selectsB a b = true → Minimalism.labelCat (a.node b) = Minimalism.labelCat a) ∧ (∀ (x y : CCG.Cat), CCG.forwardApp (x.rslash y) y = some x) ∧ ∀ (x y : CCG.Cat), CCG.backwardApp y (x.lslash y) = some x

Labeling convergence across theories.

Three independent formulations of "the head determines the result's category" all hold simultaneously.

§3. External Merge ↔ Head-Complement ↔ Application (§2.1–2.2) #

All theories implement the head-complement combination:

Minimalism: External Merge where one SO selects the other
HPSG: Head-Complement Schema (head word combines with complements)
CCG: Forward/backward application (functor consumes argument)
DG: Core dependency relations (obj, det, comp,...)

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External Merge with selection is Head-Complement across all theories.

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External Merge without selection is Head-Specifier across theories.

§4. Internal Merge ↔ Head-Filler ↔ Composition (§2.3) #

All theories handle long-distance dependencies via the third schema:

Minimalism: Internal Merge (re-merge of a contained element)
HPSG: Head-Filler Schema (filler XP + S[SLASH {XP}])
CCG: Forward/backward composition (enables extraction)
DG: Non-projective (crossing) dependencies

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theorem Comparisons.Mueller2013.internal_merge_is_head_filler :

Minimalism.classifyInternalMerge = Core.CombinationKind.headFiller ∧ (∀ (r : HPSG.HeadFillerRule), classifyHPSGSchema (HPSG.HPSGSchema.headFiller r) = some Core.CombinationKind.headFiller) ∧ (∀ (d1 d2 : CCG.DerivStep), classifyCCGDerivStep (d1.fcomp d2) = some Core.CombinationKind.headFiller) ∧ ∀ (d1 d2 : CCG.DerivStep), classifyCCGDerivStep (d1.bcomp d2) = some Core.CombinationKind.headFiller

Internal Merge / Head-Filler / Composition across theories.

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theorem Comparisons.Mueller2013.dg_nonproj_is_filler_gap :

DepGrammar.hasFillerGap DepGrammar.nonProjectiveTree = true

Non-projective dependencies in DG correspond to Head-Filler.

A non-projective (crossing) dependency represents displacement — the DG analogue of Internal Merge and the Head-Filler Schema.

§5. Coordination Diagnostic (§2.2) #

Coordination requires matching categories across all theories. This is a diagnostic for whether two expressions have the same category.

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theorem Comparisons.Mueller2013.ccg_coordination_same_cat (c1 c2 : CCG.Cat) :

CCG.coordinate c1 c2 ≠ none → c1 = c2

CCG coordination requires matching categories.

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theorem Comparisons.Mueller2013.ccg_coordination_mismatch (c1 c2 : CCG.Cat) (h : c1 ≠ c2) :

CCG.coordinate c1 c2 = none

CCG coordination of mismatched categories fails.

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theorem Comparisons.Mueller2013.hpsg_lexrule_enables_coordination (rule : HPSG.LexicalRule) (s1 s2 : HPSG.Sign) (h : s1.synsem.cat = s2.synsem.cat) :

(HPSG.applyLexRule rule s1).synsem.cat = (HPSG.applyLexRule rule s2).synsem.cat

HPSG lexical rules preserve head features, enabling coordination.

When two signs undergo the same lexical rule, they retain the same category — which is why passivized verbs can coordinate with each other.

§6. Directional MG ≈ CCG (§2.3) #

Stabler's directional Minimalist Grammar uses features =x (looking right) and x= (looking left), which correspond directly to CCG's X/Y and X\Y.

This formal correspondence is not yet formalized because directional MG is not in the codebase.

§7. Both Directions Right (§3) #

Müller's conclusion: the three universal schemata handle fully abstract constructions, but Construction Grammar's phrasal constructions are irreducible — they cannot be decomposed into the three schemata.

"Both directions right": we need BOTH Merge/schemata AND constructions.

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theorem Comparisons.Mueller2013.ditransitive_decomposes :

ConstructionGrammar.decompose ConstructionGrammar.ditransitive = [Core.CombinationKind.headSpecifier , Core.CombinationKind.headComplement , Core.CombinationKind.headComplement ]

Concrete examples: fully abstract constructions decompose.

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theorem Comparisons.Mueller2013.causedMotion_decomposes :

ConstructionGrammar.decompose ConstructionGrammar.causedMotion = [Core.CombinationKind.headSpecifier , Core.CombinationKind.headComplement , Core.CombinationKind.headComplement ]

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theorem Comparisons.Mueller2013.pal_irreducible :

ConstructionGrammar.isFullyCompositional ConstructionGrammar.Studies.GoldbergShirtz2025.palConstruction = false

Concrete examples: phrasal constructions are irreducible.

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theorem Comparisons.Mueller2013.let_alone_irreducible :

ConstructionGrammar.isFullyCompositional ConstructionGrammar.Studies.FillmoreKayOConnor1988.letAloneConstruction = false

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theorem Comparisons.Mueller2013.both_directions_right :

(∀ (c : ConstructionGrammar.Construction), c.specificity = ConstructionGrammar.Specificity.fullyAbstract → c.pragmaticFunction = none → ConstructionGrammar.isFullyCompositional c = true) ∧ ∃ (c : ConstructionGrammar.Construction), ConstructionGrammar.isFullyCompositional c = false

Both directions right: the three schemata AND phrasal constructions are needed.

Fully abstract constructions without pragmatic functions are fully compositional — decomposable into Head-Complement and Head-Specifier steps.
There exist constructions that are not fully compositional — they cannot be captured by the three schemata alone, requiring CxG's phrasal patterns.

§8. Concrete Cross-Theory Examples #

Verify that specific combinations classify identically across theories.

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theorem Comparisons.Mueller2013.min_external_exhaustive (a b : Minimalism.SyntacticObject) :

Minimalism.classifyExternalMerge a b = Core.CombinationKind.headComplement ∨ Minimalism.classifyExternalMerge a b = Core.CombinationKind.headSpecifier

The three schemata are exhaustive for External Merge in Minimalism.

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The three primary HPSG schemata map to the three universal schemata; Head-Modifier (adjunction) falls outside the classification.

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theorem Comparisons.Mueller2013.hpsg_headMod_not_classified (r : HPSG.HeadModRule) :

classifyHPSGSchema (HPSG.HPSGSchema.headMod r) = none

Head-Modifier falls outside Müller's three schemata. Adjunction is HPSG-specific and not part of the universal convergence claim.

§9. Labeling Failures (§2.1) #

Müller shows that Chomsky's labeling algorithm fails in two ways:

Free relatives: rules 14a and 14b give contradictory labels (D vs V)
Coordination of phrases: neither rule applies (neither daughter is an LI, neither selects the other)

Note: The free relative SO freeRelSO models the surface structure {what, [wrote ___]} without explicitly modeling Internal Merge — "what" and the gap have different token IDs rather than being literal copies. The labeling conflict holds regardless of how the gap is represented.

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theorem Comparisons.Mueller2013.free_relative_labeling_conflict :

Minimalism.labelRule14a Minimalism.freeRelSO = some Minimalism.Cat.D ∧ Minimalism.labelCat Minimalism.freeRelSO = some Minimalism.Cat.V

Free relatives expose a labeling conflict between Chomsky's two rules.

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theorem Comparisons.Mueller2013.coordination_labeling_failure :

Minimalism.labelRule14a Minimalism.coordDP = none ∧ Minimalism.selectsB Minimalism.theDP Minimalism.aBookDP = false ∧ Minimalism.selectsB Minimalism.aBookDP Minimalism.theDP = false

Coordination of two phrases: rule 14a fails (no LI daughter) and neither phrase selects the other.

§10. Monovalent Verb Serialization Problem (§2.3) #

Merge classifies a monovalent verb's sole argument as a complement, yielding wrong linearization ("*Sleeps Max" instead of "Max sleeps").

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theorem Comparisons.Mueller2013.monovalent_verb_serialization_problem :

Minimalism.classifyExternalMerge (Minimalism.SyntacticObject.leaf Minimalism.sleepsToken) (Minimalism.SyntacticObject.leaf Minimalism.maxToken) = Core.CombinationKind.headComplement ∧ (Minimalism.merge (Minimalism.SyntacticObject.leaf Minimalism.sleepsToken) (Minimalism.SyntacticObject.leaf Minimalism.maxToken)).phonYield = ["sleeps", "Max"]

Monovalent verbs: sole argument classified as complement → wrong order.

§11. Iterable Valence Operations (§1) #

Lexical rules compose freely, capturing double passivization (Turkish, Lithuanian) without phrasal machinery.

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theorem Comparisons.Mueller2013.valence_iteration_works (s : HPSG.Sign) :

(HPSG.applyLexRule HPSG.passiveRule (HPSG.applyLexRule HPSG.passiveRule s)).synsem.head = s.synsem.head

Any chain of lexical rules preserves head features.

Summary Table #

Claim	Min	HPSG	CCG	DG	CxG	Status
Head-Complement	Ext Merge + sel	HeadComp	fapp/bapp	obj/det/...	slot decomp	Proved
Head-Specifier	Ext Merge − sel	HeadSubj	(= bapp)	subj	slot decomp	Proved
Head-Filler	Int Merge	HeadFiller	fcomp/bcomp	nonproj	irreducible	Proved
Head-Modifier	—	HeadMod	—	—	—	Not in 3 schemata
Labeling	selector proj	HFP	functor result	head	—	Proved
Coordination	same cat	same cat	same cat	—	—	Proved
Labeling failure (FR)	14a≠14b	—	—	—	—	Proved
Labeling failure (coord)	no rule applies	—	—	—	—	Proved
Monovalent verb	*Sleeps Max	—	—	—	—	Proved
Valence iteration	—	double passive	—	—	—	Proved
Directional MG ≈ CCG	=x ≈ X/Y	—	—	—	—	Sorry
Both directions right	—	—	—	—	abstract ∨ phrasal	Proved

Note: CCG has no separate Head-Specifier mechanism. Subject combination uses backward application (the verb S\NP is the functor), which is Head-Complement in the classification. The subject/complement distinction is syntactic, not combinatory, in CCG.

Documentation

Linglib.Comparisons.Mueller2013

@cite{mueller-2013}: Unifying Everything #

Structure #

§1. Classification Functions #

CCG classification #

HPSG classification #

Dependency Grammar classification #

CxG classification #

§2. Labeling Convergence (Müller §2.1) #

§3. External Merge ↔ Head-Complement ↔ Application (§2.1–2.2) #

§4. Internal Merge ↔ Head-Filler ↔ Composition (§2.3) #

§5. Coordination Diagnostic (§2.2) #

§6. Directional MG ≈ CCG (§2.3) #

§7. Both Directions Right (§3) #

§8. Concrete Cross-Theory Examples #

§9. Labeling Failures (§2.1) #

§10. Monovalent Verb Serialization Problem (§2.3) #

§11. Iterable Valence Operations (§1) #

Summary Table #