Futrell, Levy and Gibson (2020): Dependency Locality as an Explanatory Principle for Word Order

This file formalizes the worked examples of [FLG20]'s case for dependency length minimization as a source of word-order universals: the dependency trees of its sections 2.3 and 2.4, with every total dependency length checked against the printed figure. Displacement makes trees nonprojective, but only mildly, at gap degree one, in extraposition and wh-movement ((3) and (4)). Short-before-long order minimizes dependency length after a head-initial head, and its mirror image, long-before-short, before a head-final one ((7) and (8)); a chain of single dependents is shortest with consistent head direction (9), while a head with several short dependents does better splitting them across itself (10), the exceptions to harmony that [GT10] derived. Heavy NP shift is the flagship case (11): keeping the object before the particle costs one unit for a light object and five for a heavy one. The random-order baseline of the corpus studies is illustrated by (13), the attested order against a reordering of the same tree. The corpus results of sections 4 and 5, the Monte Carlo comparisons with random baselines, are not formalized; the per-language head-final proportions and mean dependency lengths of Table 2 are the rows of Data.UD.DependencyLength.FutrellEtAl2020.

Implementation notes

• English words come from the Fragment lexicon, and the trees follow the paper's drawing convention, on which a preposition heads its noun, so arc lengths match the printed diagrams.
• Mirror-image and reordering claims go through Graph.mirror and Graph.map, so they hold by the general invariance theorems rather than by inspection of hand-typed twins.

References

• [FLG20]
• [GT10]
• [Beh09]

@[reducible, inline]

abbrev FutrellEtAl2020.this_ : Morphology.Word

Equations

• FutrellEtAl2020.this_ = English.Determiners.this.toWord

@[reducible, inline]

abbrev FutrellEtAl2020.ap : Morphology.Word

Equations

• FutrellEtAl2020.ap = Morphology.Word.mk' "AP" UD.UPOS.PROPN

def FutrellEtAl2020.tok (s : String) : Morphology.Word

Schematic token for the paper's abstract tree diagrams (A, B, C, …).

Equations

• FutrellEtAl2020.tok s = Morphology.Word.mk' s UD.UPOS.X

Examples (3)–(4): displacement and nonprojectivity

Displacement produces nonprojective trees: right extraposition in (3), where who you know modifies woman across the intervening verb, and wh-movement in (4), where what is the object of did. Both sit at gap degree 1 — the paper's point that natural languages deviate from context-freeness only mildly.

def FutrellEtAl2020.extraposition : DependencyGrammar.Graph 8

Example (3): "I think a woman arrived who you know".

Equations

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

def FutrellEtAl2020.whMovement : DependencyGrammar.Graph 8

Example (4): "I know what he thinks you did yesterday".

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

theorem FutrellEtAl2020.displacement_gap_degree_one : extraposition.gapDegree = 1 ∧ whMovement.gapDegree = 1

Displacement stays mildly non-context-free: both trees have gap degree 1.

Examples (7)–(8): short-before-long and its head-final mirror

A head with three dependent phrases of sizes 1, 2, 3. In head-initial contexts (7), placing them short-to-long after the head minimizes total dependency length; the head-final long-before-short preference (8) is its exact mirror — here literally Graph.mirror, so the equal-cost claim is Graph.totalLength_mirror, not an inspection of hand-typed twins.

def FutrellEtAl2020.shortBeforeLong : DependencyGrammar.Graph 7

(7a) A [B] [C D] [E F G]: dependents short-to-long after the head.

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

def FutrellEtAl2020.longBeforeShort : DependencyGrammar.Graph 7

(7b), the same head with its phrases reordered long to short: A [E F G] [C D] [B].

Equations

• FutrellEtAl2020.longBeforeShort = FutrellEtAl2020.shortBeforeLong.reorderDependents 0 (Equiv.swap ⟨1, FutrellEtAl2020.longBeforeShort._proof_1⟩ ⟨4, FutrellEtAl2020.longBeforeShort._proof_2⟩)

theorem FutrellEtAl2020.short_before_long_head_initial : shortBeforeLong.totalLength < longBeforeShort.totalLength

(7): short-before-long wins in head-initial contexts, by Behaghel's law at the head A: the phrases of one, two and three words grow outward in (7a) and not in (7b).

theorem FutrellEtAl2020.long_before_short_head_final : shortBeforeLong.mirror.totalLength < longBeforeShort.mirror.totalLength

(8): the head-final regime is the mirror image, so long-before-short wins there at exactly the head-initial costs — by the general mirror invariance, no separate fixtures needed.

Examples (9)–(10): head-direction consistency and its exceptions

def FutrellEtAl2020.consistentChain : DependencyGrammar.Graph 4

(9a) chain A → B → C → D linearized consistently: A B C D.

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

def FutrellEtAl2020.mixedChain : DependencyGrammar.Graph 4

(9b) the same chain linearized with mixed head direction: A C D B.

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

theorem FutrellEtAl2020.consistent_chain_shorter : consistentChain.totalLength < mixedChain.totalLength

(9): consistent head direction minimizes chain dependency length — the DLM route to the Greenbergian harmonic correlations.

def FutrellEtAl2020.dependentsSameSide : DependencyGrammar.Graph 4

(10a) A B C D: both dependents (B and C–D) after the head A.

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

def FutrellEtAl2020.dependentsSplit : DependencyGrammar.Graph 4

(10b) B A C D: the one-word dependent moved before the head.

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

theorem FutrellEtAl2020.split_beats_consistency : dependentsSplit.totalLength < dependentsSameSide.totalLength

(10): with several short dependents, splitting them across the head beats consistency ([GT10]), predicting the documented exceptions (e.g. prenominal determiners in head-initial Spanish).

Example (11): heavy NP shift

The paper's flagship worked example: the cost of the verb–object–particle order is 1 for a light object (6 vs. 7) and 5 for a heavy one (11 vs. 16), deriving the weight-sensitivity of heavy NP shift (cf. example (6)).

def FutrellEtAl2020.lightParticleEarly : DependencyGrammar.Graph 5

(11a) "John threw out the trash", total dependency length 6.

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

def FutrellEtAl2020.lightParticleLate : DependencyGrammar.Graph 5

(11b) "John threw the trash out", total dependency length 7.

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

def FutrellEtAl2020.heavyParticleEarly : DependencyGrammar.Graph 9

(11c) "John threw out the trash sitting in the kitchen", total 11.

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

def FutrellEtAl2020.heavyParticleLate : DependencyGrammar.Graph 9

(11d) "John threw the trash sitting in the kitchen out", total 16.

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

theorem FutrellEtAl2020.dlm_penalty_grows_with_weight : lightParticleLate.totalLength - lightParticleEarly.totalLength < heavyParticleLate.totalLength - heavyParticleEarly.totalLength

The DLM penalty for late particle placement grows with object weight: 1 for the light object, 5 for the heavy one.

Example (13): the random-order baseline

The attested sentence against reorderings of the same structure — stated through Graph.map, so "same structure" is by construction, which is what the paper's random-baseline methodology asserts.

def FutrellEtAl2020.attestedOrder : DependencyGrammar.Graph 6

(13a) "this story comes from the AP", the attested order, total 6.

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

def FutrellEtAl2020.σB : Equiv.Perm (Fin 6)

The position permutation taking the attested order to the paper's first reordering "from AP the this story comes".

Equations

• FutrellEtAl2020.σB = { toFun := ![3, 4, 5, 0, 2, 1], invFun := ![3, 5, 4, 0, 1, 2], left_inv := FutrellEtAl2020.σB._proof_1, right_inv := FutrellEtAl2020.σB._proof_2 }

def FutrellEtAl2020.reorderingB : DependencyGrammar.Graph 6

(13b) the reordering, as a relabeling of the attested structure.

Equations

• FutrellEtAl2020.reorderingB = FutrellEtAl2020.attestedOrder.map FutrellEtAl2020.σB

theorem FutrellEtAl2020.attested_below_reordering : attestedOrder.totalLength < reorderingB.totalLength

The attested order beats the reordering — and since reorderingB is a Graph.map of attestedOrder, that they share a structure is not an assertion but a definition.