Dispersion Theory

Dispersion theory (DT) is a version of functionalism: it takes as its starting point the premise that languages (in this case, inventories) are shaped by the conflicting interests of speakers and hearers: whereas the former aim to min-imise the effort of producing speech sounds, the latter demand a maximal degree of perceptibility. That the two conflict, can be demonstrated with respect to the phonetic vowel space: the speaker prefers to keep the energy investment per vowel down, and thus deviate only minimal from the resting position, roughly schwa. However, such an attitude would render the vowels in the system quite similar, making it harder for the listener to distinguish between the vowels, and thus between words. As every speaker is also a listener, s/he is familiar with these conflicting forces, and allows them to actively influence her grammar – such is the view of functionalism. In this section, we will briefly discuss the dynamic model of dispersion introduced by Liljencrants and Lindblom (1972), before turning to a more elaborate evaluation of modern incarnations of DT, mainly the one proposed in Flemming (2004) and related work. We will fol-low the criticisms expressed before in Boersma and Hamann (2008); van ‘t Veer (2008); Dresher (2009); Hall (2011). In particular, we will conclude that Dispersion Theory encodes more in the grammar then is needed (teleology), and that it makes unclear and strange predictions about the nature of human linguistic competence.

Early versions of Dispersion Theory

Liljencrants and Lindblom (1972) propose a dynamic computational model of the vowel inventory, in which vowels are endowed with mutually repelling forces, much as electrical particles with equal charge are. Starting from a circle around the centre of a two-dimensional phonetic space, the vowels move, generally, to more extreme positions in response to the repellent influence of the other vow-els. At each step, the ‘energy’ of the entire system is measured (the energy decreases as the distance between the vowels becomes larger, as the strength of the repelling force decreases with distance – analogous to the familiar In-verse Square Law in physics), and the model reaches its final state at the point at which no further reduction in total energy can be obtained. Although the model is relatively successful in some respects, there are some empirical and conceptual problems. Liljencrants and Lindblom (1972) show results obtained for vowel inventories of varying sizes, from thee to twelve members. These results are compared to descriptions of human languages, and the authors con-clude that their model performs rather well: “The model produces about nine

clear errors in a comparison involving 75 vowel qualities.” As Hall (2011) points out, however, the model as it is presented is incapable of generating schwa in inventories smaller then ten vowels, nor does it generate any non-high front rounded vowels. Also, it predicts an unattested five-way back-front contrast among the high vowels in inventories of more than nine vowels. Furthermore, although the authors state that the initial state of the model is a “random”

organisation, the vowels are initially highly ordered (at equal distances on a circle with a radius of 100 mel). As demonstrated by Hall (1999, 2007), how-ever, the outcome of the model is highly dependent on the arrangement at the initial state, while there is no principled reason to assume the shape of the initial state used by Liljencrants and Lindblom (1972).

Dispersion Theory in OT

A contemporary incarnation of the spirit of Liljencrants and Lindblom (1972) can be found in Dispersion Theory. Flemming (2004) takes as his point of de-parture that phonology functions to ‘minimally distinguish words’.⁸From this it follows, he argues, that phonology should favor larger contrasts over smaller ones. Adopting Optimality Theory as model for the phonological grammar, Flemming (2004) argues that in order to fulfil this task, the grammar provides constraints that do not concern individual segments, but that rather evaluate contrasts between pairs of phonemes. According to Flemming (2004), phonol-ogy shapes its inventories under the influence of these three ‘functional goals’

(p.7):

1. Maximise the distinctiveness of contrasts 2. Minimise articulatory effort

3. Maximise the number of contrasts

The idea that these principles play a role in phonology and/or phonetics is, of course, not new (Liljencrants & Lindblom, 1972, among others). What is specific to Dispersion Theory, however, is that they are directly encoded in the grammatical formalism. In Flemming (2004), for example, they are recorded in the following Optimality Theoretical constraints (or constraint families):

Mindist = D:n

assign a violation mark for any contrast that has a distance less then n on dimension D

*Effort

minimise articulatory effort

8Although similar to our conception of phonology as an addressing system, there is a crucial difference between the two ideas. An addressing system must involve the lexicon and relate (or translate) phonetic input to addresses in that lexicon; to ‘minimally distinguish words’ is an important step, but much more than that. We shall see below that this different conception of phonology has deep implications.

Maximise Contrast

Assign a satisfaction mark for every member in the candidate inventory

The first constraint, Mindist, evaluates for each pair in the inventory the distance on a scale. For vowels, Flemming (2004) proposes a somewhat abstract distance scale on three dimensions, corresponding to the first three formants.

The second constraint against articulatory effort is not discussed in much detail;

it remains entirely unclear how ‘effort’ is defined and/or quantified. Flemming (2004) goes no further then to assert that to minimise effort ‘. . . appears to be a general principle of human motor behaviour not specific to language.’ The third constraint is a positive constraint, that is satisfied by the candidate containing the most members.

This brings us to a rather peculiar characteristic of Flemming (2004)’s Dis-persion Theory, briefly hinted to above: although it employs the formal ap-paratus provided by Optimality Theory, it does not perform the task usually ascribed to grammar, namely to map input forms to output forms. Dispersion theory is not a theory of derivation, which leaves its ontological status, or place in the grammar rather unclear. This is not necessarily a problem, as long as it is acknowledged. It does, however, beg the question as to what exactly is modeled.

The difficulty that DT encounters with derivations is extended to its incom-patibility with faithfulness constraints. As Flemming (2004) notes, faithfulness

‘subverts the intended effect of the Mindist and Maximise Contrast con-straints’, by enforcing a relation between input and output. Take, for example, the tableaux in (19) (examples 9 and 23 in Flemming (2004)), representing a part of the vowel inventory of Italian. Dispersion constraints correctly pre-dict the optimal inventory to be [i-e-a], as we can see in 19a: the two-member inventory [i-a] is excluded because other candidates fare better on the Max-imise Contrast-constraint, whereas the four-member candidate [i-e-E-a] fails on a MinDist-constraint. By placing four segments on the F1 continuum, the distance between the individual segments is smaller than enforced by the high-ranked MinDist=F1:3.

If a faithfulness constraint enters the evaluation, however, the system derails easily (19b). Remember that Richness of the Base holds that the correct output must be selected regardless of the input, and that no constraints hold at the level of the input. Hence, /I/ is a valid input. In 19b, candidates a. and d.

are discarded based on their unfaithfulness to the input on the F1 dimension.

Candidate b. retains faithfulness and does better on Maximise Contrast than candidate c. Candidate b. is minimally different from the actual sub-inventory, but it crucially fails on MinDist=F1:3.

(19) a. Dispersion constraints without faithfulness

Input: //I// Mindist

= F1:2

Mindist

= F1:3

Maximise con-trasts

Mindist

= F1:4

Mindist

= F1:5

a. i-a ✓✓!

b. ☞ i-e-a ✓✓✓ ∗!∗ ∗∗

c. i-e-E-a ∗!∗∗∗ ✓✓✓✓ ∗∗∗ ∗∗∗∗∗

b. Dispersion constraints with faithfulness Input: //I// Ident

[F1]

Mindist=

F1:3

Maximise contrasts

Mindist=

F1:4

a. i-e-a ∗! ✓✓✓ ∗∗

b. I-e-a ∗! ✓✓✓ ∗∗

c. ☞ I-a ✓✓

d. i-a ∗! ✓✓

Flemming (2004) solves this problem by relegating the burden of faithfulness effects to constraints on output resemblance, such as output-output correspon-dence constraints (Benua, 1997) or constraints enforcing paradigm uniformity.

It is not shown how this is done exactly, however, and the problem bears di-rectly on the issue of the Italian vowel inventory outlined in example (19).

Whereas the tableaux presented by Flemming derive the intended invento-ries for stressed and unstressed positions, the theory does not predict which vowels will neutralise to what position. The inability to deal with faithfulness restrictions leads to another complication: whereas Flemming stated the goal of phonology to be to ‘minimally distinguish words,’ this apparently must be read as ‘minimally distinguish surface forms.’

One of the more conceptual objections one might have raise against Dis-persion Theory is that it is overly teleological. Although Flemming (2004) is correct in arguing that phonology distinguishes between words, it is a non-sequitur to then assert that this entails that the phonological grammar must encode a preference for more extreme contrasts over lesser ones. As shown by, among others, Boersma and Hamann (2008) and Hall (2011), dispersive effects can be modeled very well as epiphenomenal to a system designed to map lexical forms to surface forms, but which is not aimed at these effects per se. A re-lated issue is that of degree, or gradience. While it is true that the phonological grammar must conserve certain contrasts (but also neutralise them elsewhere), this by no means automatically implies that this same grammar is concerned with the degree of differentness. Again, although dispersion in vowel inventories is a real observation, this maw well be an epiphenomenal effect of enhancement (Hall, 2011), or of diachrony.

Summary

Whereas the model of conflicting interests of speakers and hearers is often successfully used to model diachronic change (Blevins, 2007, for example), it is

by no means a given that it is an active ingredient of synchronic grammars. For one thing, both articulatory effort and perceptual ease are difficult to quantify.

Related to this is the question whether it is desirable for a theory of grammar to encode physical properties (other than perhaps instructions to the speech organs at the phonetics–phonology interface). Doing so effectively duplicates the explanatory burden for some effects both in the physical realm, and in the grammar. Parsimony is not served well. Note that this does not hold to the same degree for Boersma and Hamann (2008), because although physical properties in their model are treated with the same apparatus as grammatical properties, they are computed in parallel, and do not replace grammar.

In this thesis, we entertain a minimal view on phonology, where it com-putes only on features and their paradigmatic (co-occurrence) and syntagmatic (phonotactic) relations.⁹ There are other elements in the phonological alpha-bet, most notably prosodic categories, but they serve mainly to describe these relations. Systemic relations (such as contrast) are not part of phonological computation; even in a framework that promotes contrast to the center stage (such as the Modified Contrastive Hierarchy, see Dresher (2009)), contrast is not seen as part of grammatical knowledge. Instead, it is epiphenomenal (or emergent) to the proposed algorithm to arrive at the featural specification of the segment inventory. The reason why systemic relations are not usually thought to be part of linguistic knowledge is because it is unclear when such relations are relevant for on-line computation of linguistic forms – either in production or perception. Phonology takes care of mapping lexical forms of morphemes to surface forms, forms which are interpretable by phonetics. This mapping function is, as shown extensively by Hall (2011), thrown overboard by Disper-sion Theory, and in any case in the verDisper-sion presented in (Flemming, 2004). In Flemming (2004) systemic relations are part of what the grammar computes.

In evaluating inventories rather than individual forms, the theory is no longer a theory of derivation, which is what Optimality Theory is usually conceived as.

Coming back to the main matter of this dissertation, which is the acquisition of the segment inventory, there are two questions we may ask with respect to Dispersion Theory. The first is whether there is any evidence that children eval-uate entire inventories during the course of acquisition, the second is whether dispersion is an active force in it.

As with any theory of phonology framed in Optimality Theory, Dispersion Theory is in principle learnable (Tesar & Smolensky, 2000). That is to say, con-straints are rankable when the learner is faced with learning data. The issue, however, is with the learning data themselves: what is it that the child eval-uates? Does she evaluate entire phonological inventories or individual forms?

9Flemming (2006) uses the term ‘paradigmatic’ for his constraints on contrast. I will not adopt that use, but rather take ‘paradigmatic’ to mean ‘referring to distributional properties’

(what fits in what position), ‘syntagmatic’ to denote sequential relations (phonotactics), and

‘systemic’ to relations that hold between elements regardless of their context in lexical or derived forms, but rather in abstract constellations such as the inventory.

To my knowledge, there is no evidence that children perform the former kind of computation. Furthermore, although there is some evidence that larger con-trasts facilitate learning (Stager & Werker, 1997), this has been brought into question (Fikkert, 2008; White & Morgan, 2008).