2.1 Positive feedback in language

A growing body of work ranging over the evolution of semantics (e.g. Kirby & Hurford Reference Kirby, James, Cangelosi and Parisi2002, Oliphant Reference Oliphant2002, Steels & Kaplan Reference Steels and Frederic2002, Smith Reference Smith2005), semantics–syntax mappings (e.g. Steels Reference Steels1998, Batali Reference Batali2002, Brighton et al. Reference Brighton, Kirby, Smith and Tallerman2005), morphology (e.g. Hare & Elman 1995, Batali Reference Batali2002, Bybee Reference Bybee2002a), syllable structure (Redford et al. Reference Redford, Chen and Miikkulainen2001) and vowel systems (e.g. Lindblom et al. Reference Lindblom, MacNeilage, Studdert-Kennedy, Butterworth, Comrie and Dahl1984, Joanisse & Seidenberg Reference Joanisse, Mark, G. Shafto and Langley1997, de Boer 2001) suggests that feedback cycles within languages communities can contribute to accounts of a broad range of linguistic patterns. At the same time, work within a generative framework has also provided evidence that a similarity-attraction effect plays a role in generating patterns in the lexicon (e.g. Burzio Reference Burzio, Downing, Hall and Raffelsiefen2005). In this section I provide a brief overview of several mechanistic routes for positive feedback in phonological categories that increase local similarity, using three linguistic phenomena as examples: (i) the observed coherence of sound categories, (ii) sharing of features between sound categories and (iii) segment substitution within words.

At the level of non-contrastive phonetic detail, at least two general mechanisms have been proposed that independently serve to create attractive biases toward frequently encountered variants: motor entrenchment in production, and the magnet effect in perception. Motor entrenchment is a general property of motor systems in which practised routines form attractors which bias future motor execution in relation to similarity (Zanone & Kelso Reference Zanone and Scott Kelso1997; see also Saltzman & Munhall Reference Saltzman and Munhall1989, Kelso Reference Kelso1995). As a consequence, for a given segment or segment sequence, the most frequent motor routine variants should be steadily reinforced, relative to less common variants (Bybee Reference Bybee2002a). This is relevant for the current discussion because, as pointed out earlier, a conceptual issue with a rich-memory model of the lexicon is that it, in principle, allows nearly unlimited variation. If all variant pronunciations of a given sound are stored and then potentially reproduced, a sound category should inexorably broaden over time as it collects variants, and then variants of variants and so on (for further discussion, see Pierrehumbert Reference Pierrehumbert, Gussenhoven and Warner2002). However, sound categories do not broaden over an individual's lifetime; rather, they become steadily more coherent through childhood (e.g. Barton Reference Barton, Yeni-Komshian, Kavanagh and Ferguson1980, Lee et al. Reference Lee, Potamianos and Narayanan1999). Motor entrenchment provides a mechanism to counterbalance the steady accumulation of variation, promoting a steady state level of category coherence.

On the perceptual side, the perceptual magnet effect (Kuhl Reference Kuhl1991, Reference Kuhl, Elenius and Branderud1995) provides another source of positive feedback bias which can act to promote category coherence. The perceptual magnet effect refers to the finding that the perceptual space is warped toward category centres relative to objective stimuli dimensions, such that percepts near category centres are perceived as closer together than they actually are, while percepts near category boundaries are perceived as further apart. The result is that percepts tend to be biased systematically toward the centres of categories relative to the stimuli that gave rise to them. Within models of the perceptual magnet effect in which warping precedes categorisation (Guenther & Gjaja Reference Guenther and Gjaja1996; cf. Kuhl Reference Kuhl, Elenius and Branderud1995, Lacerda Reference Lacerda, Elenius and Branderud1995), this systematic warping pulls similar pronunciations closer together over time through feedback between perception and production (Wedel Reference Wedel2006).

Just as distinct motor routines within a sound category influence one another, motor theory suggests that motor routines belonging to distinct, but similar, segment and segment-sequence categories should influence one another to become more similar over time. Consistent with this, recent evidence suggests that gradient, subsegmental gestural errors creating variously ‘hybrid’ segments do frequently occur (Pouplier et al. Reference Pouplier, Chen, Goldstein and Byrd1999, Wright & Frisch Reference Wright and Frisch2002). If such a hybrid variant is stored as a motor exemplar for a given sound category, it brings the average of that category closer to the other category contributing to the hybrid. Recall that within exemplar models with a production–perception feedback loop, an exemplar stored in a category contributes both to future production events from that category and to identification of percepts with that category. A resulting prediction of these models is that hybrid variants should serve as catalysts providing a pathway for increased sharing of gestures between similar sounds, up to and including full category merger (see Hansson Reference Hansson2007).

Finally, at the level of the segment as an abstract category, full segmental substitution errors are more likely to occur between similar than dissimilar segments (Shattuck-Hufnagel & Klatt Reference Sproat and Osamu1979, Frisch Reference Frisch1996), and appear to be more likely the more frequent the resulting segment sequence (Dell et al. Reference Dell, Reed, Adams and Meyer2000). Long-term feedback over this bias would be expected to produce a lexicon in which licit properties and sequences are not randomly distributed, but are concentrated in islands of local regularity (see e.g. Ernestus & Baayen Reference Ernestus and Harald Baayen2003).Footnote ¹

Recording a single variant is unlikely to substantially change the character of a category. However, because every stored exemplar makes production and processing of a similar form more efficient in the future, any variant can occasionally serve as a catalyst for more substantive category change, particularly during language acquisition.

2.2 Similarity-based error biases and the emergence of regularity

In many models of language change, error provides a significant proportion of the variation that makes up the raw material for reanalysis (e.g. Ohala Reference Ohala, Masek, Hendrick and Miller1981, 1989, Labov Reference Labov1994, Blevins Reference Blevins2004).Footnote ² Here I will argue that error biases that favour previously encountered forms such as those discussed above provide a mechanism for the development and entrenchment of language-internal regular patterns. Regularity arises in this situation because, under positive feedback, gradient behaviour is unstable over time. To see why this is true, consider a simple simulation starting with a field of randomly distributed squares of two different shades (Fig. 1a). In each round of the simulation, each square checks the shades of the squares immediately surrounding it, and if the majority shade is different, it changes shade to match; if the distribution is equal it has even chances of changing or staying the same. Figure 1b illustrates the distribution of the two shades after several rounds. Note that the distribution has become strikingly more regular, as squares progressively change to match their local neighbourhood. As each round passes, curves and bends in the distribution are straightened out in favour of the locally more densely populated shade, producing progressively larger uniform areas and correspondingly smaller areas of contact (Fig. 1c). The steady reduction of area of contact is driven by the fact that squares at a shade boundary are likely to switch back and forth, while those surrounded by uniform shade do not change. As a boundary shifts back and forth randomly, any change that places a given square in a majority one-shade neighbourhood will be relatively more stable, minimising boundaries over time.Footnote ³ This is an example of how a bias toward local similarity can result in global regularity over time.

Figure 1

Positive feedback drives the development of regularity.

Returning to language, in a rich-memory system in which experience leaves a lasting trace that can influence subsequent production and perception, error biases toward similarity at some level of categorisation should promote the development of crisp boundaries in behaviour at that level of categorisation (Wedel Reference Wedel2004). In the next section, I introduce a simple heuristic model of a production–perception loop that demonstrates the development of regularity under positive feedback, and then turn to more complex regular patterns that arise when multiple contextual biases compete.

3.1 Analogical error promotes uniformity

To illustrate the contribution of analogical error to evolution of the lexicon in this model, let us start by comparing two simulations: a control simulation lacking both analogical and external error, and a simulation that includes analogical error alone. Both simulations include a further random error set at a 1% chance that an /x/ value will flip in any production event. These simulations follow the evolution of a lexicon with 100 lexical entries each containing the category /x/ and a randomly assigned additional category taken from the set {d, e, f, … , w}, which functions to supply additional random similarities and differences between words. At the start, the lexicon is balanced with 50 entries containing [x: +1] and 50 containing [x: −1]. Because the model presented here abstracts away from contrast maintenance to focus on feedback mechanisms promoting development of analogical patterns, no cost is imposed if lexical entries happen to be identical (cf. Wedel Reference Wedel2004, Reference Wedel2006 for simulations of contrast maintenance in a feedback-driven model).

Figure 3 shows the evolution of the average value of /x/ over the entire lexicon for 10,000 cycles in the presence of random error alone. Since the two values for /x/ are coded as [+1] and [−1], a lexicon with equal numbers of each value will show an average /x/ value of zero, while a lexicon with a tilt toward one or the other value will show an average closer to [+1] or [−1] respectively. Note that the lexicon could only rarely ever show an average /x/ value of exactly [+1] or [−1], because some lexical items in any round are likely to be produced with the minority /x/ value.

Figure 3

In the presence of random error alone, the average value of /x/ over the entire lexicon stays close to zero over many cycles. This is because any change in /x/ value is independent of all other /x/ values, leading to approximately equal numbers of [+1] and [−1] /x/ values over time.

In the control simulation lacking analogical error shown in Fig. 3, the mean /x/ value of the lexicon as a whole at any time remains close to 0, as we expect, since every lexical entry is fully independent of every other. When many variables (in this case the values of /x/ in the lexicon) are randomly distributed, the central limit theorem states that the sum will be approximately normally distributed. As a consequence, the probability that the mean value of /x/ will approach the limits of [+1] or [−1] is very low.Footnote ⁷ This means that within this model, without the action of analogical error the distribution of feature values in the lexicon would tend to maximal non-regularity.

The simulation illustrated in Fig. 4 is the same, except for the inclusion of analogical error. Here we see a strikingly different pattern, as the value of /x/ veers immediately to an extreme and stays there, switching only occasionally to the other extreme over the course of the simulation. This pull to extremes occurs because when a particular /x/ value attains a sufficient majority in the lexicon through random chance, analogical error favouring that value becomes more likely, pulling the entire lexicon further in the same direction. This is conceptually the same as the process we saw above in Fig. 1, where gradience (there in the form of a uniform distribution of shades across the field of squares) is rendered unstable by positive feedback. Within the simulation in Fig. 4, however, continual external error at a rate of 1% per lexical entry per cycle occasionally injects enough of the alternate /x/ value into the lexicon to weaken the direction of analogical error and allow a switch to the alternate value. Note that the push toward categorical behaviour comes entirely from positive feedback in this system – there is no extrinsic bias in favour of either of the two stable states in this system.

Figure 4

With the addition of analogical error, changes in /x/ value are no longer independent of all other /x/ values, but are biased toward those values in proportion to frequency. As a result, lexicon-wide /x/ values tend toward uniformity, falling quickly to one extreme value. Random error allows occasional transitions to the other value.

3.2 Bias conflict and pattern-levelling

Consider the cross-linguistic distribution of velarised and plain /l/ mentioned earlier. There are some languages with only velarised or plain lateral approximants in all contexts. And there are other languages where velarised and plain laterals are variants or ‘allophones’ of the same sound category or ‘phoneme’. In some languages, one variant is associated with a particular phonological context: plain laterals may be restricted to occurring before front vowels, as in Georgian, or only in syllable-onset positions, as in some dialects of English. If one assumes that the context-influenced pathways for development of plain vs. velarised laterals are present to some extent in most languages which have a voiced lateral approximant phone, then two questions immediately present themselves. First, why aren't distinctly velarised and plain [l] present in all languages that have /l/? And second, how does a pattern arise associating one of these variants with one possible conditioning context to the exclusion of other conditioning contexts? For example, how can it be that the distinction in velarisation of /l/ in English is primarily associated with syllable position, while in Georgian it is dependent on the backness of a following vowel? In this section we will see how the pattern-smoothing effect of analogical error competes with the pattern-diversifying effect of external, contextual biases, resulting in a limitation on the number of factors that can condition variation. In the context of positive feedback, this competition can produce a system in which both uniform category exponence and contextually conditioned allophonic variation are stable states. As a result, lexicons under the very same starting conditions can settle unpredictably into one or the other kind of state, depending on small random variations in the initial path taken.

In the simulation shown in Fig. 4, two categorical patterns developed solely through positive feedback, without any extrinsic bias toward one /x/ value or the other. To model the interaction of distinct contextual biases, we set all lexical entries in the simulation to include either the categories /a/ or /b/, and introduce external contextual biases in error for the value of /x/ based on the presence of [a] or [b].Footnote ⁸ Specifically, the presence of [a] increases the relative probability of error toward [x: +1] to 3%, and the presence of [b] increases the relative probability of error toward [x: −1] to 3%. The context-free random probability of an /x/ value changing remains at 1%. The lexicon consists of 100 entries, 50 of which contain /a, x/ and 50 /b, x/, where the starting value of /x/ is balanced across both sets of entries. Again, an additional random category is appended to each lexical entry, here taken from the set {c, d, e, …, w}. Recall that analogical error can be created by a significant association of /x/ values with any of the category labels included in the simulation, including /a/ and /b/ as well as the added random category. There are four possible major patterns of association that are available as attractors for the system solely through positive feedback: association of the value of /x/ with the category label /x/ itself, resulting in consistent /x/ values of either [+1] or [−1] across the lexicon; or association of a particular value of /x/ with /a/, and the other with /b/ or vice versa. Minor patterns are in principle possible in the association of a particular value of /x/ with the randomly assigned category label in each lexical entry.

Consistent, divergent error biases for /x/ values conditioned by [a] and [b] encourage the development of a lexicon in which [x: +1] is associated with /a/ and [x: −1] associated with /b/. However, one might also expect global positive feedback between the value of /x/ and its own label /x/ to be able to override these more local patterns and enforce one of the two possible lexicon-wide values of /x/. Looking closely at Fig. 5, we can see that these three patterns in fact emerge. In the initial portion of the simulation through approximately cycle 1000, the average value of /x/ for the entire lexicon ins near [+1], despite the fact that low-level error in production of outputs containing /b/ promotes an /x/ value of [−1] in those outputs. Shortly after 1000 cycles, the value of /x/ flips for the entire lexicon toward [−1], and remains there for another 3000 cycles. Periods such as these, in which /x/ values are consistent across the lexicon, are stabilised by analogical associations between all category labels with one value of /x/ that overrides persistent biases toward association of a particular /x/ value with /a/ vs. /b/.

Figure 5

The interaction of analogical and external errors can result in multiple stable states. In this simulation, change in /x/ values in /ax/ sequences is biased slightly toward [+1], and in /bx/ sequences it is biased slightly toward [−1], while analogical error promotes uniformity in /x/ value with respect to any associated category. The two stable states between cycles 0 and 4200 have uniform /x/-value associations with the /x/ category label itself, while the state between cycles 4200 and 7200 has two opposing uniform /x/-value associations with the /a/ and /b/ category labels respectively.

Near cycle 4500, however, persistent error resulting in /x/ values of [+1] in outputs containing [a] create a sufficient nucleus of /a, x: +1/ entries in the lexicon to support a separate stabilising association between /a/ and /x: +1/, with the result that /x/ values for lexical entries containing /a/ jump up to [+1], leaving those containing /b/ behind. Here, the associations of the category label /x/ with its possible values [+1] and [−1] have become equivalently strong, resulting in no net analogical error bias from this source. The development of the two ‘allophones’ of /x/ here is initially prompted by external contextual biases, but, once in place, the stable pattern is maintained through their consistent associations with /a/ and /b/ within the lexicon. The three stable states that appear in Fig. 5 are parallel to the distribution of allophones of /l/ in language, where we find languages with one [l] variant that remains relatively constant across contexts, and others that have developed two conditioned variants in association with some context.

OT could account for the various output patterns in Fig. 5 through the relative ranking of context-free constraints violated by either value of /x/, and context-sensitive constraints violated by particular values of /x/ in conjunction with [a] or [b]. For example, the uniform output value of [x: +1] in the 1–1000 cycle range could be accounted for by ranking a constraint against [x: −1] above a constraint against [x: +1], while the opposite pattern found in the 2000–4000 cycle range could be accounted for by the opposite ranking. In turn, the pattern of context-dependent variation in /x/ values that emerges in the 5000–7000 cycle range could be accounted for by ranking the two context-sensitive constraints above the context-free ones. The feature of OT that allows these conflicting categorical patterns to emerge from the same general system is the stipulated property of constraint dominance: all of these constraints may be present in the grammar, but in the event of constraint conflict, the higher-ranked constraint wholly determines the outcome. In contrast, in the model simulated here pattern ‘dominance’ arises naturally under the well-supported assumption that production and perception are subject to similarity-based positive feedback. In the next section, we will discuss this feature of the model further in the context of the claim that phonological patterns are often more coarsely grained than the phonetic trends that give rise to them.

3.3 The coarseness of the grammar

It has been noted that statistical regularities in the speech stream seem to be more complex and fine-grained than the phonological generalisations that develop from them (Hyman Reference Hyman and Juilland1976, Pierrehumbert Reference Pierrehumbert2001b, Reference Pierrehumbert2003, Gordon Reference Gordon2002). In a rich-memory model, analogical error minimises diversity of behaviour along some dimension over time, while any sources of variation promote greater diversity in behaviour. Within this model, the competition between these two general tendencies results in the stabilisation of phonological patterns that are on average more coarsely grained than would arise from a straightforward mapping of all external biases onto the lexicon. For example, in the simulation shown in Fig. 5 above, the evolution of a category under the influence of two opposing context-specific biases often still results in relatively uniform category exponence, because positive feedback favours uniformity. In this section, we will see that analogical bias can transform a more complex set of interacting external biases into simpler patterns that would be accounted for in OT through the mechanism of strict constraint domination.

As discussed in the introduction, generative models of phonological competence favour coarse-grained phonological patterns through limitations on the alphabet of underlying symbols and their combinatorial possibilities, and/or on the rules or constraints that govern output forms. Much of OT's ability to account for the coarse-grained nature of grammatical patterns lies in its claim that there is a limited set of universal constraints, and that there is a limited mechanism for their interaction, in particular that the choice of optimal outputs proceeds through satisfaction of constraints in ranked order.

Standard Optimality Theory (Prince & Smolensky Reference Prince and Paul1993) stipulates that the choice of optimal outputs proceeds through satisfaction of constraints in ranked order. The principle of strict domination further specifies that ranking is absolute: no degree of potential violation of lower-ranked constraints can ever compel violation of a higher-ranked constraint. This principle functions to limit the complexity of the patterns that can be generated with an optimality-theoretic grammar. Strict domination can often be found paraphrased informally as ‘lower-ranked constraints can't gang up against a higher-ranked constraint’, but another informal restatement that will be especially useful here can be given as ‘the outcome of multiple constraint conflicts reproduces the outcomes of the component pairwise constraint conflicts’. This way of stating the principle makes it clear that its operation keeps phonological patterns simpler than they would be otherwise. Further, because the set of forms in which two particular constraints conflict must be equal to or larger than the set of forms in which those two plus another constraint all conflict, the principle of strict domination can be seen as a prediction that winning patterns are associated with typologically more frequent constraint conflicts. This limitation allows OT equipped with a particular finite set of constraints to predict that certain complex patterns cannot exist.

However, while strict domination allows OT to accurately describe many phonological systems, it sits uneasily with the notion, increasingly well represented within the field, that constraints on grammatical patterns are directly or indirectly related to phonetic biases based in articulatory and perceptual difficulty (e.g. Donegan & Stampe Reference Donegan, David and Dinnsen1979, Archangeli & Pulleyblank Reference Archangeli and Pulleyblank1994, Blevins & Garrett Reference Blevins and Andrew1998, Hayes Reference Hayes, Darnell, Moravcsik, Newmeyer, Noonan and Wheatley1999, Steriade Reference Steriade1999, Blevins Reference Blevins2004, Reference Blevins2006b, Hayes & Steriade Reference Hayes, Steriade, Hayes, Kirchner and Steriade2004). This unease arises because it is difficult to see how biases with sources outside the grammar, such as physiological constraints on articulation or perception, would not interact additively in some overall performance cost. Within OT, however, the inability of low-ranked constraints to gang up on higher-ranked constraints amounts to a statement that though costs may sum at the level of articulation and perception, they cannot at the level of phonology, unless a specific constraint exists that by itself penalises that combination of costs (as in, for example, a constraint that conjoins a penalty for voicing in obstruents with a penalty for codas to account for syllable-final obstruent devoicing; Ito & Mester Reference Ito and Mester2003). The fact that constraint conjunction has indeed been frequently invoked in optimality-theoretic work suggests that strict domination does not serve as an accurate mechanism to limit the power of OT. It is too strong because independent constraints can show additive effects when operative within particular domains (e.g. Ito & Mester Reference Ito and Mester2003). And when it is weakened by the allowance of constraint conjunction to allow these kinds of additive effects, there is significant overgeneration of unattested systems (discussed in Padgett Reference Padgett2002).

Here we will see that when the interaction of multiple external error biases is simulated within our simple model, regular patterns can evolve that are consistent with the strict domination principle of OT, despite the fact that these interactions do compound in production, as we expect they should. In particular, when multiple patterns potentially conflict in a single output form, we will see that pattern smoothing from analogical error promotes outcomes that follow the individual pairwise outcomes of pattern conflict, just as predicted by the principle of strict domination. Interestingly, we will see that patterns can also arise that violate strict domination (i.e. would require the addition of a new constraint), but only when violations of multiple external biases are associated in a sequence that is typologically frequent in the lexicon.

Here, we will consider the interaction of three external biases: (a) a context-free 5% error bias toward [x: +1] in production, (b) a 3% bias toward [x: −1] conditioned by the presence of [a], and (c) a 3% bias toward [x: −1] conditioned by the presence of [b]. These error biases are summarised in (1).

1. (1)

   

Within the simulation, these biases are modelled as independent of one another, as we expect for biases with distinct sources, e.g. those that arise from independent articulators, or those grounded in articulation in contrast to perception. Bias independence is modelled within the simulation by applying multiple biases that apply to one output in random order. To probe the interaction of these biases with each other and with analogical error, we will compare the stable states of three different lexicons containing distinct sets of lexical entries, where each lexicon has 50 categories corresponding to each given combination of {a, b, x}. The three lexicons are given in (2), where, as before, an additional random category is included in each entry.

1. (2)

   

Figure 6 shows a 10,000 round simulation using Lexicon 1, showing that entries for all words containing {a, x} and {b, x} maintain an average /x/ value near [+1].Footnote ⁹ If we examine the relative strengths of the external biases used, we can see why this is: the tendency to shift towards [+1] provided by /x/ is stronger than the bias towards [−1] provided by [a] or [b], so each lexical entry has a small net bias towards [x: +1] in production.

Figure 6

The lexicon in this simulation consists of /ax/ and /bx/ lexical entries, where a context-free external bias on /x/ values toward [+1] slightly exceeds independent context-sensitive external error biases on /ax/ and /bx/ values toward [−1]. Because the cumulative external error bias on /x/ value is therefore slightly toward [+1] in every lexical entry, the average value of /x/ in the lexicon remains near [+1].

However, when [a] and [b] are present together in the same output, as in Lexicon 2, their independent biases should be able to interact additively, and overwhelm the bias from [x]. This is shown in Fig. 7, where we see that the joint influence of [a] and [b] together on output form is greater than that of [x], leading to a stable lexicon-wide /x/ value near [−1].

Figure 7

The lexicon in this simulation consists solely of /abx/ lexical items. As in the simulation shown in Fig. 6, a context-free external bias on /x/ values toward [+1] slightly exceeds context-sensitive external error biases on /ax/ and /bx/ values toward [−1]. However, because the context-sensitive /ax/ and /bx/ biases influence error independently, they interact additively in /abx/ entries to promote the development of lexicon-wide /x/ value/s of [−1].

But what happens when the two patterns are put in conflict with one another in Lexicon 3, where all three types of entries are present? All lexical entries contain an /x/ category, and two pairs out of the three sets share an /a/ or /b/ category, so analogical error has a number of similarities over which to pattern-smooth. And as a consequence of this pattern smoothing, in Fig. 8, we see that in the context of /a, x/ and /b, x/ entries, the biases acting on /a, b, x/ entries no longer appear additive at the level of the lexicon, even though at the level of actual error they are in fact additive.Footnote ¹⁰ Because the stronger bias on /x/ values in the /a, x/ and /b, x/ entries tends to push these outputs toward [x: +1], the /a, b, x/ outputs are pulled toward [x: +1] as well, through persistent analogical error on the basis of shared internal structure. As a result, the entries of the form /a, b, x/ come to conform to the larger pattern set up by the forms /a, x/ and /b, x/ despite the fact that in production the net error bias for /a, b, x/ consistently pushes in the opposite direction. This pattern conforms to the OT principle of strict domination, without stipulating that the actual biases involved cannot interact additively. In this model, then, a strict domination pattern is not a fundamental property of the grammar, but is an emergent result of positive feedback that tends to arise when lexical forms in which multiple constraints conflict are less frequent than forms in which subsets of those constraints conflict. This property of the model predicts that independent biases targeting distinct features of an output will often fail to pattern as additive unless there is a strong association in the lexicon between those features.

Figure 8

The lexicon in this simulation contains /ax/, /bx/ and /abx/ lexical entries, with the same error biases as in Figs 6 and 7. As in Figs 6 and 7, external error interactions push /ax/ and /bx/ entries toward [+1], and /abx/ entries toward [−1]. However, analogical error promotes uniformity between entries with shared categories, resulting here in a single lexicon-wide /x/ value, despite additivity of independent error biases. This demonstrates that analogical error can create patterns exhibiting the non-additivity of markedness constraints stipulated by the OT principle of strict domination.

4.1 Modelling the interaction of edge-aligned stress and syllable weight

Within this model, external biases on the evolution of each lexical entry have a cumulative influence on the evolution of the entire lexicon through feedback. As a result, the frequency of category types and sequences should play a significant role in the evolution of grammatical patterns. Stated slightly differently, this model predicts that, all else being equal, the larger the number of lexical entries a bias potentially applies to, the more likely its effects will come to be reflected in the behaviour of the lexicon as a whole.

Gordon (Reference Gordon1999, Reference Gordon2002) reports precisely such a phenomenon in his work on the relationship between the proportion of sonorants in a language's coda inventory and the likelihood that coda consonants contribute to syllable weight. Out of a sample of 400 languages that display quantity-sensitive stress, Gordon found that the two most common weight distinctions are those that treat (i) CVV but not CVC syllables as heavy, or (ii) both CVV and CVC syllables as heavy. Based on the range of phenomena that are cross-linguistically correlated with weight, as well as results from his own phonetic study, Gordon concluded that total acoustic energy in the rhyme is the most significant phonetic factor associated with a phonological heavy/light distinction for stress. Gordon found that the occurrence of these two weight distinctions was not random, but remarkably predictable on the basis of the relative number of high-energy to low-energy consonants in the coda inventory. Out of a set of 24 languages in which the sonorant/obstruent and voiced/voiceless ratios in the coda inventory were lower than one, 20 languages showed the CVV=heavy pattern, while out of a set of 23 languages in which the sonorant/obstruent and voiced/voiceless ratios in the coda inventory were greater than one, 22 showed the CVV, CVC=heavy pattern. Given that sonorants characteristically have more energy than obstruents, and voiced sounds more than voiceless sounds, this finding is consistent with the distributional data suggesting that acoustic energy in the rhyme is a significant factor influencing the development of quantity-sensitive stress patterns.

But this line of reasoning leaves the following question unanswered. If, for example, as suggested by the influence of sonorant-to-obstruent coda ratio on the probability of having CVC as heavy, sonorant codas make good stressed syllables and obstruent codas make bad ones, why don't languages develop quantity-sensitive stress that targets CVV and CV[+son] syllables, rather than just CVV and CVC? There are languages that display just this sort of split (e.g. Kwakwala and Inga Quechua; Zec 1988), again suggesting that this reasoning is on the right track, but this pattern remains relatively rare. Gordon explains this rarity as a consequence of the pattern's greater phonological complexity and specificity relative to the more general CVV and CVC pattern. In the following section, we will see how these patterns can arise through the feedback-driven mechanism proposed here.

To model the influence of sonorant and obstruent coda consonants on stress patterns, we add to our inventory of features the set /S/ (sonorant), /O/ (obstruent) and /C/ (consonant), and include them in a single syllable of a subset of lexical items in a lexicon containing 20 each of two- and three-syllable words, as illustrated in Table II. In the starting lexicon, each word class contains equal numbers of words with initial- and final-aligned stress. In addition to the external bias toward alternating stress, we include a external bias toward stressing syllables containing the feature /S/ at a rate of 5%. We do not need to specify that any of the additional features are part of a coda, because we do not overtly include vowels or onsets.Footnote ¹³, Footnote ¹⁴

Table II

Low-frequency and high-frequency sonorant-coda lexicons.

40% of the words in this lexicon are just like those in the previous simulation, containing no coda of any kind (row 1 in each table). Another 40% contain an obstruent coda (rows 2 and 3), and the remaining 20% contain a sonorant coda (rows 4 and 5). All words contain initial and final edge features (I and F respectively), so just as before, feedback error will promote development of a consistent alignment of a stress value to an edge. Since the sonorant codas are split across the lexicon between syllables 1 and 2, for 10% of the lexicon a constant error bias towards stressing syllables with sonorant codas will act to disrupt any edge-aligned pattern. While the edge-aligned pattern can be stabilised by associations that are present in every word, the only opportunity for a sonorant-coda stressing pattern to be stabilised by feedback is if a consistent association between the feature /S/ and /Stress: +1/ can develop in the 20% of words that contain this feature. This association would also create an incipient association between /C/ and /Stress: +1/ within the words containing a obstruent coda, but because these words are in the minority of all words containing /C/, positive feedback is unlikely to gain a foothold to induce the larger number of words containing an obstruent coda to follow suit. And out of 20 independent runs of this simulation, at 1000 cycles eighteen showed one of the four possible lexicon-wide consistent edge alignments, and two were caught between patterns, indicating that a positive feedback driven association of stress to an edge in all words can overcome a strong external bias toward stressing sonorant codas in a few words.

Given the role of consistency of patterning in promoting positive feedback in this model, this result is not surprising. The constant bias to stress syllables with sonorant codas applies to few words in the lexicon, and these few words share many features with other words that are not affected by this constant bias, meaning that words containing sonorant codas will always be pulled away toward the majority pattern by positive feedback over other associations. Naturally, what we should try next is a lexicon with a greater proportion of words with sonorant codas relative to those with obstruent codas, as shown in the final column of Table II, where the number of lexical entries in rows 2 and 3 has been halved, and that in rows 4 and 5 doubled. As a result, twice as many words now have sonorant codas as obstruent codas, instead of vice versa.

When twenty independent runs of this simulation were run out to 1000 cycles, three different patterns emerged, plus a residue of inconsistently patterned lexicons in transition between states. I will consider each of these patterns in turn.

Four out of the 20 lexicons showed a pattern of stressing all syllables with an /S/ feature, while the syllables without an /S/ feature showed a consistent alignment of a stress value to an edge. Clearly, increasing the number of syllables with sonorant codas in the lexicon to 40%, in conjunction with a strong constant bias toward stressing these syllables, allowed the development of a sufficiently strong association between /S/ and /Stress: +1/ to maintain a separate pattern from the rest of the lexicon, which itself went ahead and aligned consistently to an edge. These outcomes parallel those languages that stress CVV and CV[+son] but not CV[−son]. Interestingly, however, this is not the majority pattern, for another eleven out of the 20 lexicons showed a pattern of stressing all syllables with a coda, without regard to whether that coda was sonorant or obstruent, with the remaining 40% of the words consistently showing one of the four edge alignments. This occurs for two reasons. Recall that of the syllables containing /C/ in the high-frequency sonorant lexicon, 67% also contain /S/, and that at any given time about half of the remaining syllables containing /C/ will also be stressed, because stress is alternating. Consequently, as soon as all sonorant codas are stressed, more than 80% of all syllables containing /C/ are stressed, creating a strong basis for positive feedback to promote the development of stress in all syllables containing /C/. Secondly, as soon as all syllables containing sonorant codas become stressed, the edge-alignment pattern becomes weakened, because half of the words containing sonorant codas will violate any established edge alignment. The combination of weaker positive feedback maintaining alignment to an edge, combined with the strong incidental association of /C/ with /Stress: +1/ allows development of a lexicon in which all syllables with a coda are stressed, even though the actual bias promoting this pattern only applies to a subset of those syllables. Finally, two of the 20 lexicons showed lexicon-wide edge-alignment of stress, and three did not show a clear pattern.

The wide range of patterns arising in this simulation illustrates several issues in a model of a rich lexicon evolving through positive feedback. First, positive feedback that generalises beyond the source pattern must make use of categories, whether present innately or abstracted from the input data (Maye et al. Reference Maye, Werker and Gerken2002, Saffran & Thiessen Reference Saffran and Thiessen2003, reviewed in Gerken Reference Gerken2006). In this case, the fact that the simulation can generalise to codas from sonorant codas requires, as asserted by many featural systems, that a separate category of ‘consonant’ be available which can serve as the basis for generalising positive feedback. Second, the type frequency of categories and category associations in the lexicon matters. Positive feedback is definitionally autocatalytic, such that the ability of a positive feedback mechanism to influence a process is constrained by the ability of an event to catalyse a subsequent event. If an external bias only affects a small proportion of a set of lexical items linked by similarity in a number of features, noise in production–perception operating over the entire set of lexical items will lower the probability that this local bias can initiate a set-wide change. As clear from the simulation above, and from Gordon's empirical findings on the influence of the coda inventory on weight systems, relative frequency of the target for a bias within a similarity set has a significant role in influencing the effect of the bias. As a consequence, this model predicts that a strong bias against a low-frequency combination of features may be less likely to initiate an active phonological pattern than a weaker bias against a higher-frequency combination.

Once entrenched, patterns in the lexicon can be self-sustaining through continuous positive feedback. Given that positive feedback also promotes coarse-grained over fine-grained patterns, this provides a potential explanation for observed divergent allophonic context-dependencies. As an illustration, let us return briefly to the example of context-dependent lateral allophones. Let us assume, for the sake of argument, that there are external biases favouring plain (vs. velarised) laterals in the syllable onset, and that there are independent external biases favouring plain laterals before front vowels. As illustrated above, positive feedback is likely to favour the development of a simple contextual condition for plain allophones of /l/. This means that, all else being equal, a plain allophone is more likely to either appear in onset position (as in English) or preceding a front vowel (as in Georgian), and less likely to appear in a context defined by an interaction of the two independent conditions. Standard Optimality Theory accounts for divergent patterns in different languages by positing the existence and language-specific ranking of innate context-specific constraints. In contrast, the model presented here accounts for such divergent patterns through the operation of positive feedback over external biases targeting distinct contexts.