Bakay, Akkuş & Dillon (2026): hierarchical relations guide memory retrieval

[BAD26] (JML 148, 104747): three visual-world experiments showing that c-command between NPs within a single clause guides antecedent retrieval for the Turkish reciprocal birbirleri, deconfounded from clause-mateness, case marking, subjecthood, and linear order/recency — cues the paper argues are "plausibly construed as item-level features". This file derives the retrieval predictions from the Turkish.Anaphors fragment (its plurality requirement yields the number cue), Principle A (the c-command and clause-mate cues), and BarkerPullum1990.cCommand on tree addresses; each experiment's target advantage is an instance of dominance. The retrieval model is the ordinal core of [LV05] spreading activation — a weighted count of cue matches; base-level decay, fan, mismatch penalty, and noise are not modeled.

Main definitions

• CueSource, Cue — cues tagged relational / item-level / positional, the paper's deconfound (clause-mateness is item-level, e.g. a clause index).
• birbirleriCues — the cue bundle birbirleri generates, from Principle A plus the fragment's plurality requirement.
• weightedActivation — activation as a weighted cue-match count.
• privileged — the rival representational account: direct access by structural position ([McE06]; [Obe02]; gated retrieval, [DMSP13]), defined on the tree configuration, not on cue matches.

Main results

• dominance — pointwise dominance of cue-match vectors gives strictly higher activation under every positive weighting.
• exp1_target_retrieved, exp1_target_retrieved_mismatch, exp2_io_retrieved — the target advantages (Exp 1: subject vs. possessor, both GEN; Exp 2: indirect object vs. adjunct NP, both DAT), for any positive relational weight.
• exp1_target_privileged, exp1_distractor_not_privileged — the same contrast under privileged access.

Implementation notes

The paper's General Discussion opposes differentially weighted cues to a privileged representation; both predict the target advantage and diverge only on early interference from feature-matching distractors, which needs the unmodeled fan/mismatch machinery — the paper finds limited, inconsistent number interference and leaves the distinction open. The relational cue is realized as a dynamically assigned item feature approximating c-command ([Kus13], in the paper's summary). Not yet formalized: the c-command vs. coargumenthood alternative the paper leaves open ([PS94]) — Binding.SimpleClause cannot represent the possessor/IO configurations used here — and the monotone activation-to-looks linking to Processing.VisualWorld observables.

Cue-based retrieval: ordinal core

inductive BakayEtAl2026.CueSource : Type

Source of a retrieval cue, following the paper's deconfound: relational information holds between the retrieval site and the candidate (c-command); itemLevel features are stored with the candidate (number, case, clause index); positional cues track linear order/recency.

• relational : CueSource
• itemLevel : CueSource
• positional : CueSource

@[implicit_reducible]

instance BakayEtAl2026.instDecidableEqCueSource : DecidableEq CueSource

Equations

• BakayEtAl2026.instDecidableEqCueSource x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[implicit_reducible]

instance BakayEtAl2026.instFintypeCueSource : Fintype CueSource

Equations

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

@[implicit_reducible]

instance BakayEtAl2026.instReprCueSource : Repr CueSource

Equations

• BakayEtAl2026.instReprCueSource = { reprPrec := BakayEtAl2026.instReprCueSource.repr }

def BakayEtAl2026.instReprCueSource.repr : CueSource → ℕ → Std.Format

Equations

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

structure BakayEtAl2026.Cue (F : Type u_1) : Type u_1

A retrieval cue: a required feature tagged with its source.

• source : CueSource
• feature : F

@[implicit_reducible]

instance BakayEtAl2026.instReprCue {F✝ : Type u_1} [Repr F✝] : Repr (Cue F✝)

Equations

• BakayEtAl2026.instReprCue = { reprPrec := BakayEtAl2026.instReprCue.repr }

def BakayEtAl2026.instReprCue.repr {F✝ : Type u_1} [Repr F✝] : Cue F✝ → ℕ → Std.Format

Equations

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

def BakayEtAl2026.matchCount {F : Type u_1} [DecidableEq F] (feats : List F) (cues : List (Cue F)) (s : CueSource) : ℕ

Number of cues from source s that a memory item's feature bundle matches.

Equations

• BakayEtAl2026.matchCount feats cues s = List.countP (fun (c : BakayEtAl2026.Cue F) => decide (c.source = s ∧ c.feature ∈ feats)) cues

def BakayEtAl2026.weightedActivation {F : Type u_1} [DecidableEq F] (w : CueSource → ℕ) (feats : List F) (cues : List (Cue F)) : ℕ

Activation of an item as a weighted count of cue matches.

Equations

• BakayEtAl2026.weightedActivation w feats cues = ∑ s : BakayEtAl2026.CueSource, w s * BakayEtAl2026.matchCount feats cues s

theorem BakayEtAl2026.dominance {F : Type u_1} [DecidableEq F] {w : CueSource → ℕ} {a b : List F} {cues : List (Cue F)} (hle : ∀ (s : CueSource), matchCount b cues s ≤ matchCount a cues s) (hlt : ∃ (s : CueSource), 0 < w s ∧ matchCount b cues s < matchCount a cues s) : weightedActivation w b cues < weightedActivation w a cues

If a's cue-match vector pointwise dominates b's, strictly at some source carrying positive weight, then a out-activates b under every such weighting — Core.Optimization.sum_mul_lt_sum_mul on cue-match vectors.

Retrieval cues for birbirleri

inductive BakayEtAl2026.Feature : Type

Features relevant to antecedent retrieval for birbirleri. cCommanding is the dynamically assigned feature realizing the relational cue.

• cCommanding : Feature
• clauseMate : Feature
• plural : Feature
• singular : Feature
• genCase : Feature
• datCase : Feature

@[implicit_reducible]

instance BakayEtAl2026.instDecidableEqFeature : DecidableEq Feature

Equations

• BakayEtAl2026.instDecidableEqFeature x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[implicit_reducible]

instance BakayEtAl2026.instReprFeature : Repr Feature

Equations

• BakayEtAl2026.instReprFeature = { reprPrec := BakayEtAl2026.instReprFeature.repr }

def BakayEtAl2026.instReprFeature.repr : Feature → ℕ → Std.Format

Equations

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

def BakayEtAl2026.numberCues : List (Cue Feature)

Item-level number cue, generated exactly when the fragment's anaphor type imposes a plurality requirement on its antecedent.

Equations

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

def BakayEtAl2026.birbirleriCues : List (Cue Feature)

Retrieval cues generated on encountering birbirleri: Principle A supplies the relational c-command cue and the clause-mate cue; the fragment's plurality requirement supplies the number cue.

Equations

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

Experiment 1: subject targets vs. possessor distractors

Target = embedded subject (c-commanding clause-mate, GEN, plural). Distractor = possessor inside the subject NP (clause-mate, GEN, plural or singular, not c-commanding). Same clause, same case, and — in the Match condition — same number: only c-command distinguishes them.

        CP_emb
       /      \
   NP_subj     VP_emb
   /    \       /   \
NP_poss  N'  anaph   V

def BakayEtAl2026.exp1TargetAddr : BarkerPullum1990.Address

Equations

• BakayEtAl2026.exp1TargetAddr = [BarkerPullum1990.Dir.L]

def BakayEtAl2026.exp1DistractorAddr : BarkerPullum1990.Address

Equations

• BakayEtAl2026.exp1DistractorAddr = [BarkerPullum1990.Dir.L, BarkerPullum1990.Dir.L]

def BakayEtAl2026.exp1AnaphorAddr : BarkerPullum1990.Address

Equations

• BakayEtAl2026.exp1AnaphorAddr = [BarkerPullum1990.Dir.R, BarkerPullum1990.Dir.L]

theorem BakayEtAl2026.exp1_target_ccommands : BarkerPullum1990.cCommand exp1TargetAddr exp1AnaphorAddr = true

The embedded subject c-commands the anaphor.

theorem BakayEtAl2026.exp1_distractor_no_ccommand : BarkerPullum1990.cCommand exp1DistractorAddr exp1AnaphorAddr = false

The possessor does not c-command the anaphor.

def BakayEtAl2026.exp1Target : List Feature

Target subject (kameramanlar 'cameramen').

Equations

• BakayEtAl2026.exp1Target = [BakayEtAl2026.Feature.cCommanding, BakayEtAl2026.Feature.clauseMate, BakayEtAl2026.Feature.plural, BakayEtAl2026.Feature.genCase]

def BakayEtAl2026.exp1DistractorMatch : List Feature

Possessor distractor, Match condition (plural yönetmenler 'directors').

Equations

• BakayEtAl2026.exp1DistractorMatch = [BakayEtAl2026.Feature.clauseMate, BakayEtAl2026.Feature.plural, BakayEtAl2026.Feature.genCase]

def BakayEtAl2026.exp1DistractorMismatch : List Feature

Possessor distractor, Mismatch condition (singular yönetmen 'director').

Equations

• BakayEtAl2026.exp1DistractorMismatch = [BakayEtAl2026.Feature.clauseMate, BakayEtAl2026.Feature.singular, BakayEtAl2026.Feature.genCase]

theorem BakayEtAl2026.exp1_relational_distinguishes : matchCount exp1Target birbirleriCues CueSource.relational = 1 ∧ matchCount exp1DistractorMatch birbirleriCues CueSource.relational = 0 ∧ ∀ (s : CueSource), s ≠ CueSource.relational → matchCount exp1Target birbirleriCues s = matchCount exp1DistractorMatch birbirleriCues s

In the Match condition only the relational cue distinguishes target from distractor: item-level (and positional) match counts tie.

theorem BakayEtAl2026.exp1_target_retrieved (w : CueSource → ℕ) (hw : 0 < w CueSource.relational) : weightedActivation w exp1DistractorMatch birbirleriCues < weightedActivation w exp1Target birbirleriCues

The target out-activates the Match distractor — the hardest case, where item-level cues do not distinguish them — for any weighting with positive relational weight.

theorem BakayEtAl2026.exp1_target_retrieved_mismatch (w : CueSource → ℕ) (hw : 0 < w CueSource.relational) : weightedActivation w exp1DistractorMismatch birbirleriCues < weightedActivation w exp1Target birbirleriCues

In the Mismatch condition the distractor also loses the number cue, so the target advantage holds a fortiori.

Experiment 2: indirect-object targets vs. adjunct distractors

Target = c-commanding indirect object (DAT). Distractor = NP inside a postpositional adjunct (DAT, e.g. göre 'according to'), not c-commanding. Extends the advantage to non-subject c-commanders, ruling out a composite subject-of-the-current-clause item-level cue.

    IO condition:              Distractor condition:
        CP_emb                     CP_emb
       /      \                   /      \
   NP_subj     VP             NP_subj     VP
               /  \                       /  \
           NP_IO   V'                 PP_adj   V'
                  /  \                /    \  /  \
              anaph   V          NP_dist  P anaph V

def BakayEtAl2026.exp2IOAddr : BarkerPullum1990.Address

Equations

• BakayEtAl2026.exp2IOAddr = [BarkerPullum1990.Dir.R, BarkerPullum1990.Dir.L]

def BakayEtAl2026.exp2DistractorAddr : BarkerPullum1990.Address

Equations

• BakayEtAl2026.exp2DistractorAddr = [BarkerPullum1990.Dir.R, BarkerPullum1990.Dir.L, BarkerPullum1990.Dir.L]

def BakayEtAl2026.exp2AnaphorAddr : BarkerPullum1990.Address

Equations

• BakayEtAl2026.exp2AnaphorAddr = [BarkerPullum1990.Dir.R, BarkerPullum1990.Dir.R, BarkerPullum1990.Dir.L]

theorem BakayEtAl2026.exp2_io_ccommands : BarkerPullum1990.cCommand exp2IOAddr exp2AnaphorAddr = true

The indirect object c-commands the anaphor.

theorem BakayEtAl2026.exp2_distractor_no_ccommand : BarkerPullum1990.cCommand exp2DistractorAddr exp2AnaphorAddr = false

The adjunct-internal distractor does not c-command the anaphor.

def BakayEtAl2026.exp2IO : List Feature

Indirect-object target: c-commanding clause-mate, plural, DAT.

Equations

• BakayEtAl2026.exp2IO = [BakayEtAl2026.Feature.cCommanding, BakayEtAl2026.Feature.clauseMate, BakayEtAl2026.Feature.plural, BakayEtAl2026.Feature.datCase]

def BakayEtAl2026.exp2Distractor : List Feature

Adjunct-internal distractor: clause-mate, plural, DAT, not c-commanding.

Equations

• BakayEtAl2026.exp2Distractor = [BakayEtAl2026.Feature.clauseMate, BakayEtAl2026.Feature.plural, BakayEtAl2026.Feature.datCase]

theorem BakayEtAl2026.exp2_io_retrieved (w : CueSource → ℕ) (hw : 0 < w CueSource.relational) : weightedActivation w exp2Distractor birbirleriCues < weightedActivation w exp2IO birbirleriCues

The indirect object out-activates the adjunct distractor for any weighting with positive relational weight. Experiment 3 is the paper's pre-registered, high-powered replication of the Experiment 1–2 contrasts; it introduces no new configuration.

Privileged representation

The representational account grants c-commanding items a temporary association with a privileged store — access by structural position, not cue matching — so privilege is defined on the tree configuration, not on matchCount.

def BakayEtAl2026.privileged (np anaphor : BarkerPullum1990.Address) : Prop

An NP position is privileged at a retrieval site iff it c-commands it: the region of direct access holds the current c-commanders.

Equations

• BakayEtAl2026.privileged np anaphor = (BarkerPullum1990.cCommand np anaphor = true)

theorem BakayEtAl2026.exp1_target_privileged : privileged exp1TargetAddr exp1AnaphorAddr

The Experiment 1 target is in the region of direct access.

theorem BakayEtAl2026.exp1_distractor_not_privileged : ¬privileged exp1DistractorAddr exp1AnaphorAddr

The Experiment 1 distractor is not, whatever its feature match.