Arnold, Wasow, Losongco & Ginstrom 2000: heaviness vs. newness

Postverbal constituent order in English — heavy NP shift and the dative alternation — has been attributed to grammatical weight (Behaghel's law of growing constituents) and to discourse status (given before new), but the two are confounded: given referents get short expressions and new ones long. A corpus study of the Aligned-Hansard debates and an elicitation experiment in which directors instruct actors to give objects to toy animals measure heaviness as the difference in word count between the two constituents and newness by prior mention, and find that both factors independently shift the order, the newer and heavier constituent coming later, with neither reducible to the other. Their relative strength varies with the data: heaviness dominates where length differences are large, newness where the given referent has just been mentioned — a system of interacting constraints whose strength grows as the competing constraint weakens. Speakers' early disfluencies favour goal-first orders, so the postponement of new and heavy material serves planning as well as comprehension.

The two factors are read as weighted constraints over the two orders of a constituent pair: a graded penalty for putting the heavier constituent first, sized by the length difference the paper measures, and a penalty for putting a new constituent before a given one. The harmony difference between the orders is the weighted sum of the two signed preferences, which is the additive independence the paper reports; the paper's interaction is a matter of effect size in the data and is not represented.

Main definitions

• Phrase, Pair, Order: a constituent by word count and givenness, a postverbal pair, and which constituent comes last.
• heavyFirst, newFirst: the two constraints; con their weighted family.
• heavyDiff, newDiff: each constraint's signed preference for the theme-last order.

References

• [ALWG00]
• [Beh09] — the law of growing constituents
• [Pri92] — discourse-given, inferable, and new

Phrases, orderings, and candidates

structure ArnoldEtAl2000.Phrase : Type

A constituent by the two measures the paper codes: word count and discourse status.

• words : ℕ
• discourse : Discourse.BinaryGivenness

def ArnoldEtAl2000.instDecidableEqPhrase.decEq (x✝ x✝¹ : Phrase) : Decidable (x✝ = x✝¹)

Equations

• ArnoldEtAl2000.instDecidableEqPhrase.decEq { words := a, discourse := a_1 } { words := b, discourse := b_1 } = if h : a = b then h ▸ if h : a_1 = b_1 then h ▸ isTrue ⋯ else isFalse ⋯ else isFalse ⋯

@[instance_reducible]

instance ArnoldEtAl2000.instDecidableEqPhrase : DecidableEq Phrase

Equations

• ArnoldEtAl2000.instDecidableEqPhrase = ArnoldEtAl2000.instDecidableEqPhrase.decEq

@[reducible, inline]

abbrev ArnoldEtAl2000.Pair : Type

A postverbal pair: theme and goal for the dative alternation, direct object and PP for heavy NP shift.

Equations

• ArnoldEtAl2000.Pair = (ArnoldEtAl2000.Phrase × ArnoldEtAl2000.Phrase)

inductive ArnoldEtAl2000.Order : Type

Which constituent comes last: themeLast is the double object or the shifted order, goalLast the prepositional dative or the canonical order.

• themeLast : Order
• goalLast : Order

@[instance_reducible]

instance ArnoldEtAl2000.instDecidableEqOrder : DecidableEq Order

Equations

• ArnoldEtAl2000.instDecidableEqOrder x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[reducible, inline]

abbrev ArnoldEtAl2000.Candidate : Type

Equations

• ArnoldEtAl2000.Candidate = (ArnoldEtAl2000.Pair × ArnoldEtAl2000.Order)

The two constraints

def ArnoldEtAl2000.heavyFirst : OptimalityTheory.Constraint Candidate

The weight penalty: the number of words by which the first constituent exceeds the second — the relative length the paper measures.

Equations

• ArnoldEtAl2000.heavyFirst ((th, gl), ArnoldEtAl2000.Order.themeLast) = gl.words - th.words
• ArnoldEtAl2000.heavyFirst ((th, gl), ArnoldEtAl2000.Order.goalLast) = th.words - gl.words

def ArnoldEtAl2000.newFirst : OptimalityTheory.Constraint Candidate

The newness penalty: a new constituent before a given one.

Equations

• ArnoldEtAl2000.newFirst ((th, gl), ArnoldEtAl2000.Order.themeLast) = if gl.discourse = Discourse.BinaryGivenness.new ∧ th.discourse = Discourse.BinaryGivenness.given then 1 else 0
• ArnoldEtAl2000.newFirst ((th, gl), ArnoldEtAl2000.Order.goalLast) = if th.discourse = Discourse.BinaryGivenness.new ∧ gl.discourse = Discourse.BinaryGivenness.given then 1 else 0

def ArnoldEtAl2000.con : OptimalityTheory.ConstraintSet Candidate (Fin 2)

The two constraints.

Equations

• ArnoldEtAl2000.con = ![ArnoldEtAl2000.heavyFirst, ArnoldEtAl2000.newFirst]

def ArnoldEtAl2000.weights (wH wN : ℝ) : Fin 2 → ℝ

The weights: wH for heaviness, wN for newness.

Equations

• ArnoldEtAl2000.weights wH wN = ![wH, wN]

Signed preferences

def ArnoldEtAl2000.heavyDiff (p : Pair) : ℝ

Heaviness's preference for themeLast: the theme's length minus the goal's.

Equations

• ArnoldEtAl2000.heavyDiff p = ↑p.1.words - ↑p.2.words

def ArnoldEtAl2000.newDiff (p : Pair) : ℝ

Newness's preference for themeLast: 1 when the theme is new and the goal given, -1 in the mirror case, 0 otherwise.

Equations

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

theorem ArnoldEtAl2000.cast_sub_sub_cast_sub (a b : ℕ) : ↑(a - b) - ↑(b - a) = ↑a - ↑b

theorem ArnoldEtAl2000.score_diff_eq_components (wH wN : ℝ) (p : Pair) : HarmonicGrammar.harmonyScore con (weights wH wN) (p, Order.themeLast) - HarmonicGrammar.harmonyScore con (weights wH wN) (p, Order.goalLast) = wH * heavyDiff p + wN * newDiff p

The harmony difference between the two orders is the weighted sum of the two signed preferences.

theorem ArnoldEtAl2000.themeLast_iff (wH wN : ℝ) (p : Pair) : HarmonicGrammar.harmonyDominates con (weights wH wN) (p, Order.themeLast) (p, Order.goalLast) ↔ 0 < wH * heavyDiff p + wN * newDiff p

The theme-last order wins exactly when the weighted preferences sum in its favour.

Independence

theorem ArnoldEtAl2000.heaviness_alone {wH : ℝ} (hH : 0 < wH) (p : Pair) : HarmonicGrammar.harmonyDominates con (weights wH 0) (p, Order.themeLast) (p, Order.goalLast) ↔ p.2.words < p.1.words

Heaviness alone places the heavier constituent last.

theorem ArnoldEtAl2000.newness_alone {wN : ℝ} (hN : 0 < wN) (p : Pair) : HarmonicGrammar.harmonyDominates con (weights 0 wN) (p, Order.themeLast) (p, Order.goalLast) ↔ p.1.discourse = Discourse.BinaryGivenness.new ∧ p.2.discourse = Discourse.BinaryGivenness.given

Newness alone places the new constituent after the given one.

theorem ArnoldEtAl2000.score_diff_add_word (wH wN : ℝ) (th gl : Phrase) : HarmonicGrammar.harmonyScore con (weights wH wN) (({ words := th.words + 1, discourse := th.discourse }, gl), Order.themeLast) - HarmonicGrammar.harmonyScore con (weights wH wN) (({ words := th.words + 1, discourse := th.discourse }, gl), Order.goalLast) = HarmonicGrammar.harmonyScore con (weights wH wN) ((th, gl), Order.themeLast) - HarmonicGrammar.harmonyScore con (weights wH wN) ((th, gl), Order.goalLast) + wH

Weight effects are graded: lengthening the theme raises the harmony of the theme-last order by the heaviness weight per word.

theorem ArnoldEtAl2000.one_factor_decides (wH wN : ℝ) (p : Pair) : (newDiff p = 0 → (HarmonicGrammar.harmonyDominates con (weights wH wN) (p, Order.themeLast) (p, Order.goalLast) ↔ 0 < wH * heavyDiff p)) ∧ (heavyDiff p = 0 → (HarmonicGrammar.harmonyDominates con (weights wH wN) (p, Order.themeLast) (p, Order.goalLast) ↔ 0 < wN * newDiff p))

When newness is neutral, heaviness decides; when weight is neutral, newness decides.

The paper's examples

def ArnoldEtAl2000.words (s : String) : ℕ

The number of words in a constituent as printed.

Equations

• ArnoldEtAl2000.words s = (List.filter (fun (x : Char) => decide (x = ' ')) s.toList).length + 1

def ArnoldEtAl2000.constituents (r : Datum) : Option (ℕ × ℕ)

The word counts of a row's two constituents in surface order.

Equations

• ArnoldEtAl2000.constituents r = do let a ← r.feature? "first" let b ← r.feature? "second" pure (ArnoldEtAl2000.words a, ArnoldEtAl2000.words b)

theorem ArnoldEtAl2000.rows_heavier_last (r : Datum) : r ∈ Examples.all → r.feature? "exception" = none → ∀ c ∈ (constituents r).toList, c.1 ≤ c.2

In every corpus and text example the paper offers as ordered by weight, the heavier constituent comes last.

theorem ArnoldEtAl2000.rows_exceptions_lighter_last (r : Datum) : r ∈ Examples.all → (r.feature? "exception").isSome = true → ∀ c ∈ (constituents r).toList, c.2 < c.1

The examples the paper sets aside — the announcer's That brings to the plate Barry Bonds and (17) — put the lighter constituent last: other factors, planning among them.