1. The “warping” view

In the paper that presented the Gradual Learning Algorithm, I modelled perceptual categorization as a mapping from an infinite number of values along an auditory continuum (e.g. F1) to a finite number of values along that same continuum, with constraints like PERCEIVE, *CATEG and *WARP:

2. Arbitrary categories

As pointed out by Robert Kirchner, the drawback of the “warping” approach was that it could not handle mappings from more than one continuum to a single phonological category. In work with Paola Escudero, who had investigated cue weighting in L2 acquisition, we therefore modelled perceptual categorization as a mapping from an auditory continuum to arbitrary abstract categories, with cue constraints like “an F1 of 360 Hz is not /i/”. The following papers model (in OT, with the GLA) the acquisition of the perception of the English /ɪ/–/i/ contrast by L1 and L2 learners on the basis of two auditory cues, namely F1 and duration:

3. Why are cue constraints formulated negatively?

Wouldn't it be easier to have positively formulated constraints like “an F1 of 360 Hz is /i/” rather than negatively formulated constraints like “an F1 of 360 Hz is not /i/”?

There are two reasons for the negative formulation. The first reason is that if the number of phonological categories is greater than 2, modelling of cue weighting will fail spectacularly. This is shown in the following paper, which models the acquisition of the perception of 9 Dutch vowels by L1 and L2 learners on the basis of two auditory cues, namely F1 and F2:

The second reason is that their negative formulation allows the cue constraints to be used for modelling comprehension as well as production. For instance, the constraint “an F1 of 360 Hz is not /i/” logically means the same as “/i/ does not have an F1 of 360 Hz”, so that it can be used both for modelling perception and for modelling phonetic implementation. All my papers on Parallel Bidirectional Phonology and Phonetics (BiPhon) take advantage of this bidirectionality.

4. Distributional learning

The acquisition algorithm in the above examples was lexicon-driven learning of perception: if you perceive auditory event [x] as the category /y/, but the lexicon subsequently (i.e. after recognition) tells you that the speaker's intended category was /z/, then the constraint rankings are modified in such a way that the next time you hear [x] you will be more likely to perceive it as /z/ than before.

This acquisition algorithm relies on correct lexical representations. Infants do not have those; they can only learn from the distributions of auditory events alone. Rachel Hayes noted that the old *WARP family of constraints might be able to handle this:

5. Category emergence in neural networks

The Optimality-Theoretic simulations by Boersma, Escudero & Hayes (2003) suffered from the need for a “Category Creation Day”. In neural networks, by contrast, categories can emerge gradually.

Here this works along a single auditory continuum, using the inoutstar learning algorithm:

And here it works along one or two auditory continua, using a Deep Boltzmann Machine learning algorithm:

Both of the above had only two levels of representation. Here is an example with three levels, in which things that look like phonological features emerge:

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