Ellis (2006): Selective Attention and Transfer Phenomena in L2 Acquisition

Ellis argues that the grammatical forms adult second-language learners fail to acquire are the ones associative learning predicts they will not attend to. His central case is past time reference. A tense inflection is low in salience and is usually accompanied by a temporal adverb that marks the same meaning, as in Terrell's Hier nous sommes allés au cinéma, where hier makes the auxiliary and the participle redundant. Under the Rescorla–Wagner rule the cues of an utterance share one prediction error, and two of the mechanisms Ellis names follow. A learner to whom both cues are new divides the association between them by salience, so the adverb overshadows the inflection. A learner whose cues already predict the outcome, as a second-language learner's adverb does, has no error left to divide, so the inflection is blocked however much input follows.

Main statements

• overshadowing, tendsto_strength: learned together from nothing, the more salient cue is the stronger after every utterance, and in the limit each cue holds its salience's share of the outcome.
• blocking: for a learner whose cues already predict the outcome, utterances pairing adverb and inflection leave every association unchanged.

Implementation notes

• The rule is ErrorDrivenLearning.update on the utterance in which both cues are present. Ellis describes it in a companion article.
• Only the Rescorla–Wagner account is formalised. Ellis also explains blocking by the probabilistic contrast model of Cheng and Holyoak and by learned inattention, under which a blocked cue is harder to learn afterwards.

References

• N. C. Ellis, Selective attention and transfer phenomena in L2 acquisition (2006)
• P. W. Cheng and K. J. Holyoak, Adaptive systems as intuitive statisticians (1995)
• R. A. Rescorla and A. R. Wagner, A theory of Pavlovian conditioning (1972)

inductive Ellis2006.Cue : Type

The temporal adverb and the tense inflection are the two cues to past time reference.

• adverb : Cue
• inflection : Cue

@[instance_reducible]

instance Ellis2006.instDecidableEqCue : DecidableEq Cue

Equations

• Ellis2006.instDecidableEqCue x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[instance_reducible]

instance Ellis2006.instFintypeCue : Fintype Cue

Equations

• Ellis2006.instFintypeCue = { elems := { val := ↑Ellis2006.Cue.enumList, nodup := Ellis2006.Cue.enumList_nodup }, complete := Ellis2006.instFintypeCue._proof_1 }

Overshadowing

theorem Ellis2006.iterate_update_zero_apply {κ : Type u_1} (α : Cue → ℝ) (β lam : κ → ℝ) (j : κ) (n : ℕ) (c : Cue) : (ErrorDrivenLearning.update α β 1 lam)^[n] 0 c j = α c * (lam j * β j * ∑ k ∈ Finset.range n, (1 - (α Cue.adverb + α Cue.inflection) * β j) ^ k)

After n utterances a learner starting from nothing holds, for each cue, its salience times a strength common to both cues.

theorem Ellis2006.overshadowing {κ : Type u_1} {α : Cue → ℝ} {β lam : κ → ℝ} {j : κ} (hα : α Cue.inflection < α Cue.adverb) (hβ : 0 < β j) (hlam : 0 < lam j) (h : (α Cue.adverb + α Cue.inflection) * β j < 2) {n : ℕ} (hn : n ≠ 0) : (ErrorDrivenLearning.update α β 1 lam)^[n] 0 Cue.inflection j < (ErrorDrivenLearning.update α β 1 lam)^[n] 0 Cue.adverb j

When two cues jointly predict an outcome the less salient one is overshadowed, so after every utterance the inflection is the weaker cue.

theorem Ellis2006.tendsto_strength {κ : Type u_1} {α : Cue → ℝ} {β lam : κ → ℝ} {j : κ} (c : Cue) (h : |1 - (α Cue.adverb + α Cue.inflection) * β j| < 1) : Filter.Tendsto (fun (n : ℕ) => (ErrorDrivenLearning.update α β 1 lam)^[n] 0 c j) Filter.atTop (nhds (α c / (α Cue.adverb + α Cue.inflection) * lam j))

In the limit each cue holds its salience's share of the outcome's maximal activation.

Blocking

theorem Ellis2006.blocking {κ : Type u_1} {α : Cue → ℝ} {β lam : κ → ℝ} {V : Matrix Cue κ ℝ} (hV : V Cue.adverb + V Cue.inflection = lam) (n : ℕ) : (ErrorDrivenLearning.update α β 1 lam)^[n] V = V

A learner whose cues already predict the outcome learns nothing from utterances that pair adverb and inflection, so "all the extra input in the world might sum to naught". The second-language learner, who knows the adverb and not the inflection, is the case V .adverb = lam and V .inflection = 0.

theorem Ellis2006.inflection_pos {κ : Type u_1} {α : Cue → ℝ} {β lam : κ → ℝ} {j : κ} (hα : 0 < α Cue.inflection) (hβ : 0 < β j) (hlam : 0 < lam j) (h : (α Cue.adverb + α Cue.inflection) * β j < 2) {n : ℕ} (hn : n ≠ 0) : 0 < (ErrorDrivenLearning.update α β 1 lam)^[n] 0 Cue.inflection j

The same utterances teach the inflection to a learner for whom both cues are new, as they are for a child.