2.1 Structure of the clause

Across all examples, I assume that the structure in (4) is present even when not all elements are pronounced:

The heads Z, Y, and X are null functional elements. The specifiers for these phrases are the destination of movement used to derive various word orders in section 3. I will not speculate about the identity of these heads here, as it is not relevant for the question at hand.⁶

2.2 Headedness

One popular approach to linearization is the Linear Corespondance Axiom (LCA) of Kayne (1994). The LCA relies on asymmetrical c-command to determine order, stating that if a syntactic node X asymmetrically c-commands another node Y, then all terminal nodes that X dominates precede the terminal nodes that Y dominates. The result of this algorithm is that all languages should be basically head-initial, and any other surface order must be derived through movement. Unlike restrictive approaches to linearization like the LCA, I hold that phrases may be either head-final or head-initial underlyingly – heads may be linearized either to the left or right of their complements. To clarify, I assume that structures are generated with no discernible word order (i.e. as sets), and that word order is established by setting a parameter that determines whether the resulting string will be head final or head initial. This parameter linearizes structures either as entirely head initial or entirely head final, so the linearization parameter produces purely harmonic structures. This is similar in some respects to the more permissive approach to linearization put forward in Abels & Neeleman (2012) or Sheehan (2013), however in that system each phrase must be specified as either head-final or head-initial, and headedness is determined on a phrase-by-phrase basis. Unlike the Abels and Neeleman approach, I take underlying headedness to be harmonic and movement to be necessary for disharmonic patterns.

Headedness here is defined over extended projections (Grimshaw 1991). Biberauer et al. (2014) observe that the FOFC does not hold across extended projections:⁷ word orders like the one in (2d) are not typologically marked where β and α are, for example, a verb and a determiner. By taking headedness parameters to target extended projections, the disharmonic orders that cross them are not predicted to exhibit the FOFC. The parameter relevant for controlling headedness in the typology here is given in (5):

This parameter is sufficient to produce the harmonic head-initial and head-final orders:

Headedness (VP) by definition cannot control the order of any components within the NP. I take this to be controlled by a Headedness (NP) parameter, not included here as it would explode the amount of weakly-equivalent languages in a way that is symmetric across the four conditions (and therefore would not affect the results).

2.3 Movement

Following Grohmann (2003; 2011); Abels (2003); Abels & Neeleman (2012), I assume that movement is sensitive to anti-locality. Specifically relevant here will be the idea that complement movement into a local specifier is insufficiently distant movement. This is contra approaches that derive the FOFC from the LCA (e.g. Biberauer et al. (2014)), which utilize anti-locality-violating roll-up movement.

The assumption of anti-locality is related to the decision to include heads X, Y, and Z in the syntactic structures used here. The specifiers of these heads act as the landing sites for movement, while anti-locality ensures that complement movement to more local specifiers need not be considered.⁸ The decision to include X, Y and Z, and to observe anti-locality in the training data was done in order to optimize for a small number of parameters. It may be possible to get rid of these assumptions without affecting the results, if additional parameters controlling movement were added, but I will not consider this further.

In addition to adopting anti-locality, all movement considered here is leftward, as is likewise assumed in Abels & Neeleman (2012). The asymmetry between leftward and rightward movement has been linked to the FOFC before, as in Cecchetto (2013); Zeijlstra (2016); Clem (Forthcoming). It is widely acknowledged that leftward movement is more common than rightward movement, and that rightward movement, when it occurs, is more constrained (Ross 1967). While rightward movement and specifiers are impossible in the current model, this is not a claim that rightward movement doesn’t exist as a phenomenon, just that it is constrained in such a way that is not relevant for the question at hand (for example, if its source is prosodic rather than syntactic, as proposed in Potsdam & Edmiston (2016) for Malagasy).

3.1 Predicted languages

Languages predicted by the parameters set out here will be henceforth named with their parameter settings. For example, the head-final language with no movement is called 1000, where the setting of each parameter appears in the order they were introduced, as in Table 1.

The derivations in (7), (9), (11), and (13), along with the two harmonic structures in (a.) and (b.), are the set of structures giving minimal derivations for all the word orders in (2). However, they are not the only way to derive the attested patterns, as many string vacuous movements are also possible. The full set of predicted languages is given in Table 2, sorted by their surface word order pattern.

As Table 2 shows, the head-initial, head-final, and initial-over-final patterns hold of a number of languages beyond the minimal derivations. This is because our parameters predict that string-vacuous movements will be possible for each of these patterns. Unlike the first three orders, the marked final-over-initial pattern (*FOFC) can only be achieved through the derivations in (9) and (13), which correspond to the languages 0010 and 1110 respectively. No vacuous movement is possible while maintaining this word order. Additionally, these parameters predict two additional word order patterns, each expressed by only a single language. I have called the first Head-initial+VP fronting. This pattern corresponds to the language 0110, and generates the word orders Aux-O, V-O, and V-Aux-O. It obtains when remnant movement of the verb phrase occurs in a head initial language. The second pattern is Head-final + AuxP fronting, and is expressed only by 1111. The word orders generated by this pattern are O-Aux, O-V, and O-Aux-V. This pattern is caused when all available movements occur in a head-final language.

4.1 Hidden structure

Natural language data with hierarchical structure presents language learners with a problem: they must deduce a hidden structure of each datum they encounter from its surface representation alone. The reason that this presents a challenge is because surface strings may be consistent with a number of underlying representations, and therefore a number of different parameter settings. For example, the following structures all result in a harmonic head-initial word order auxiliary, verb, noun:

The string auxiliary, verb, noun is well-formed in languages 0000, 0001, and 0011. Movement obscures the underlying structures, and therefore the correct parameter settings.

An additional difficulty is that a single string may be derived by two or more languages that do not have similar parameter settings. In other words, the grammars that generate a particular language may not be contiguous in the parameter space. The language 0110 for instance, has more parameter settings in common with the final-over-initial language 1110 then it does with the head-initial language 0001, but produces surface strings that have more in common with head-initial languages then FOFC violators.

Ambiguity and the absence of smoothness are two factors that can make a particular pattern challenging to acquire. The harder it is to set parameters correctly, the more difficult a language is to learn. Given the number of languages even a small parameter system generates, it is difficult to intuit where and how the systems proposed to account for syntactic phenomena will present the greatest challenges to learners. To investigate what role such factors might play in the typology of word order, the learning tasks here are conducted using a computational learner as a proxy for a human learner. Although only a small number of parameters (and correspondingly small typology) will be considered, it is assumed that a much larger set of parameters are learned during acquisition.

4.3 Learning tasks

In both learning tasks, the learner started with unordered sets representing syntactic hierarchy and learned from training data comprised of surface strings with no constituency information. The model corresponds to a slice of learning that takes place after lexical categories (POS tags) have already been acquired. This is a common starting-point in work on syntactic parameter learning (Gibson & Wexler 1994; Yang 2002; Gould 2015; 2016; Sakas et al. 2018; Prickett et al. 2019). Similarly, the unordered constituency (section 2.1) is taken to be a part of UG. Learning is focused on the acquisition of order, and not lexical category or unordered hierarchy. The training data did not include any languages with variable word order. Additionally, the learner was only exposed to strings, and had no access to information about each datum’s hidden structure beyond hierarchy (i.e. no movement or headedness). Because of this, nothing in the data the learner sees differentiates the languages within a word order pattern, and so the learner did not make distinctions between them (see section 5).

In both tasks the learner was trained on every language for 80 passes through the data (unless otherwise noted) using a learning rate of 0.1, and averages were taken across 40 runs per language.⁹ Learning was deterministic (i.e. the learner acquired a single parameter setting for each pattern) in all cases except one (the 3-token final-over-initial order; section 5.4), and averages were taken across runs to reduce noise. In the 3-token task, there were three data points per language an {Auxiliary, Object} sequence, a {Verb, Object} sequence, and an {Auxiliary, Verb, Object} sequence. The 3-token typology is given in Table 3.

In the second task, there are only two data points per language, an {Auxiliary, Object} sequence, and a {Verb, Object} sequence. No language in this training data included the ternary {Auxiliary, Verb, Object} sequence. The 2-token typology is given in Table 4.

Since the only thing that distinguishes the +fronting word order patterns from the harmonic patterns is the ternary token, these languages are absorbed into the harmonic word order patterns in the 2-token task.

5.1 Harmonic head-final (3-token)

Parameter learning curves for languages with a harmonic head-final surface order can be seen in Figure 2. As with the by-language and by-pattern curves, these are taken as an average over 40 reps. The parameter values in this particular graph are for language 1000, but as noted in section 5, the same parameter values are consistently learned for all languages with a single surface pattern.

Figure 2

Head-final parameter curves.

The first parameter to be learned is Headedness (P1) This parameter is set to 1 by about 60 updates. At about 120 updates, VP-SpecYP (P3) reaches 0. After this, the learner does not converge on a value for the parameters NP-SpecXP (P2) or AuxP-SpecZP (P4). Given that harmonic head-final languages were the easiest to acquire (as demonstrated in Figure 1), this suggests that the settings of these parameters does not matter for the purposes of acquiring head-final languages, at least once the first and third parameters are set.¹⁰ This is confirmed by looking at parameter curves for a single run (analogous to a single learner; Figure 3).

Figure 3

Head-final parameter curves (1 run).

What this means is that the learner can pick any setting for P2 and P4, and any combination of those two settings, and still produce the correct output. A closer look at the data confirms this. Table 5 shows to what extent the individual tokens in training data for the harmonic head-final languages are correlated with a setting of one for each individual parameter.

This is calculated by looking at how many times a particular token appears in a language with that parameter set to one (across all language patterns), and dividing that number by the total number of the languages the token occurs in. For example, the token O-V occurs in 10 languages (five head-final, five initial-over-final). In seven of these languages, the first parameter is set to 1. Therefore, the extent to which O-V favors setting P1 to 1 is .7.

The easiest parameter for the learner to set is Headedness (P1) because the ternary token disambiguously prefers 1 (i.e. O-V-Aux never appears in a language where P1=0). VP-SpecYP (P3) is also relatively easy to set, because all three tokens push the learner towards a 0 setting, although none are disambiguating. The tokens are not united in which direction they push the learner for P2 and P4. O-Aux and O-V-Aux push the learner towards 0, while O-V pushes the learner towards 1. One might anticipate that in this case, both parameters will eventually be set to 0, since the majority of tokens favor this setting. However, it is important to note that the preferences in Table 5 are inherent to the training data and only hold prior to any learning. Once some learning has occurred, certain languages become more heavily weighted than others, and the parameter settings most likely to produce a match may change. For example, once two parameters have been set for the head-final data (i.e. P1 and P3), this helps eliminate some potential ‘nearby’ languages which are otherwise compatible with the training data:

Because of this, the likelihood of the learner producing the correct outputs with particular settings of the remaining parameters changes. In this case, once P1 has been set to 1 and P3 has been set to 0, the settings of P2 and P4 no longer matter – every combination of settings will product a match with the training data. The learner therefore need not select a single setting for these two parameters in order to successfully learn a harmonic head-final language.

In summary: despite ambiguity in the data (i.e. many parameter settings compatible with the observed tokens), the harmonic head-final pattern is relatively easy for the learner because only two parameters need to be set for it to be successful, and one of these parameters is disambiguated by the ternary token. The language ultimately learned for the head-final pattern was 1 ~0~, where ~ indicates a parameter which did not need to be consistently set to either 1 or 0. To clarify, this means that the grammar the learner acquired was (covertly) stochastic. Whenever the learner goes to use a ~ setting, it picks either 1 or 0 at random, and since any setting produces the right output, there is no variation in the tokens it produces (i.e. no surface evidence that the grammar is stochastic).

5.2 Harmonic head-initial (3-token)

The harmonic head-initial pattern was the second easiest language (i.e. had the second steepest curve). The average parameter curves are given in Figure 4.

Figure 4

Head-initial parameter curves.

Here the learner quickly converges on a 0 setting for Headedness (P1) and NP-SpecXP (P2). For Headedness, this happens by about 50 updates, followed by NP-SpecXP at about 80. The other two parameters are much slower, with AuxP-SpecZP(P4) eventually converging on 1, and VP-SpecYP(P3) plateauing. In fact, although the learner gets very close to converging very early on, it actually takes around 500 updates for the AuxP-SpecZP(P4) to be completely set to 1. This can be seen in Figure 5.

Figure 5

Head-initial curves (500 updates).

Although it takes approximately 500 updates for P4 to be set to 1, the learner is close enough by around 150 updates that the fact that it hasn’t completely converged is not easily discernible from Figure 1. Again, looking more closely at the data sheds light on why the parameters are learned in this order and at this rate. The correlation of each head-initial token with parameter settings of 1 is given in Table 6.

The ternary token disambiguates the first two parameters, and this accounts for why they are set quickly. P2 takes slightly longer to set than P1 on account of the minority report of the Aux-O token which, unlike V-O and Aux-V-O, pushes the learning towards 1 instead of 0 for P2. After the first two parameters are set, the following languages are possible:

Unlike in the head-final case, setting two parameters is not sufficient for learning the head-initial pattern, as there is a nearby language (0010) which will not produce the right output. This “attractive neighbor” needs to be rejected before learning can be successful, and so additional parameters must be set. Specifically one additional parameter needs to be set. The two possible solutions for the learner are to a) set P3 to 0, in which case the setting of P4 does not matter, or b) to set P4 to 1, in which case the setting of P3 does not matter. The reason that P4 is set first is because 2/3 of the head-initial tokens prefer setting this parameter to 1, so that setting gets an initial boost before other learning occurs (as can be seen in (27)). The head-initial tokens do not push P3 towards 0 as strongly, and so P3 plateaus after P4 is set.

As noted above, despite a relatively steep learning curve, the learner is fairly slow to completely converge on parameter settings for this word order pattern. The reason for this is that, before P4 completely converges (with P1 & 2 already set), the learner is already very accurate. Given the languages that are highly weighted at that point (i.e. the languages in 27), the learner has a 75% chance of correctly producing Aux-V-O and Aux-O tokens, and 100% chance on V-O.

In combination with how close P4 & P3 have been pushed towards 1 and 0 respectively, this means that the learner is hardly ever wrong. Responsibility for matches is therefore diffused between both parameter settings, and it takes a long time for one to become more highly weighted than the other. This is one reason why the time it takes for the learner to fully set parameters for a language pattern is not used here as a complete measure of how learnable that pattern is. The head-initial pattern has a relatively steep learning curve, and the learner quickly achieves high accuracy, despite taking many updates to totally converge on all parameter settings.

Ultimately, the parameter values learned for the harmonic head-initial pattern were 00~1. Although the first two parameter settings are disambiguated by the ternary form, this pattern was still a bit more difficult than the head-final harmonic pattern. The reason for this is that more parameters needed to be set correctly in order for learning to be successful. The head-initial pattern requires the correct setting of three parameters, while the head-final pattern only required two.

5.3 Disharmonic initial-over-final (3-token)

The initial-over-final pattern (a.k.a the unmarked disharmonic pattern) was only slightly harder to learn than the two harmonic patterns, as in Figure 1. The parameter curves for initial-over-final can be seen in Figure 6.

Figure 6

Initial-over-final parameter curves.

AuxP-SpecZP (P4) and NP-SpecXP (P2) are both set by about 90 updates. Headedness (P1) is the next to be set, more slowly, to 0, and at that point VP-SpecYP (P3) plateaus. Before any learning occurs, the preference of tokens for setting each parameter to 1 are in Table 7.

Unlike the harmonic patterns, the initial-over-final pattern has no parameter setting that is completely disambiguated by one or more tokens. However, all three tokens push the learner to a 1 setting for P4, and 2/3 tokens push the learner towards 1 for P2, while the third is at chance. Both of these parameters are initially pushed towards one. At about 20 updates, the curves for P2 and P4 diverge, and P2 is set first, despite the fact that all tokens initially push P4 towards 1. To determine the source of this divergence, it is necessary to look at the approximate weight of each parameter at 20 updates. Table 8 gives the (approximate) probability that each parameter setting is 1 at 20 updates.

Based on this, the approximate probability of each possible initial-over-final language at that point is as in Table 9.

The combined weight of initial-over-final languages where P2 is set to 1 (0.4) has at this point become greater than the combined weight of languages where P4 is set to 1 (0.35). As learning goes forward this gets exacerbated, giving P2 a boost towards 1 that results in it being set before P4. Therefore, P1 and P2 are the first parameters to be correctly set. As in the head-initial pattern, the languages remaining after the first two parameters are set are not all initial-over-final, but include a neighbouring language:

As in the head-initial case, there are now two possible solutions for the learner to set the final parameters. P3 could be set to 0, in which case the setting of P4 would not matter, or P4 could be set to 1, in which case the setting of P3 no longer matters. P4 is set first, because it was pushed towards 1 by all tokens initially, unlike P3, which was only pushed towards zero by one token.

The parameter settings the learner converges on for the initial-over-final pattern are 01~1. First P1 & P2 are set, and then P4 is pushed slowly towards 1. Once P4 is set, the setting of P3 no longer matters for the pattern to be successfully learned. This is similar to the way the head-initial pattern was learned, with three parameter settings being necessary for successful learning. However, unlike the head-initial language, no parameters are disambiguated by the ternary form for the initial-over-final pattern.

5.4 Disharmonic final-over-initial (marked) (3-token)

The final-over-initial pattern (i.e. the marked disharmonic order) was the most difficult for the learner to successfully acquire. Average parameter learning curves for the example language 0010 are given in Figure 7.

Figure 7

Final-over-initial parameter curves.

The first parameter to be learned was AuxP-SpecZP (P4), which was set to 0 by approximately 60 updates. This was followed by VP-SpecYP (P3) being completely set to 1 by 80 updates. After these parameters are successfully valued, Headedness (P1) and NP-SpecXP (P2) appear to drift slightly towards 0 before appearing to plateau. Unlike in previous cases, however, no plateauing actually occurs for any parameters in the final-over-initial pattern. This becomes evident when looking at parameter curves for individual reps.

Figures 8 and 9 show that what is actually happening in Figure 7 is that the learner is not consistently landing on one parameter setting for this word order pattern. Instead, It converges on 1110 on just under 40% of runs, and on 0010 for the remaining (approx.) 60%. Because figure 7 shows curves over an average of reps, this gives the illusion of plateauing.

Figure 8

Final-over-initial parameter curves for 1 rep (A).

Figure 9

Final-over-initial parameter curves for 1 rep (B).

Looking at the starting preference of the final-over-initial tokens for setting parameters to 1, it is easy to see why parameters 3 and 4 are easy for the learner to set. Both of these parameters are disambiguated by the ternary form, as seen in Table 10.

V-O-Aux is only a possible output if P3 is set to 1 and P4 to 0, it does not occur in any languages that deviate from these settings. Once these two parameters are set correctly, the following languages remain:

There are several sources of difficulty for the learner here. First, setting one additional parameter after P3 and P4 are set correctly will not make the final parameter setting irrelevant. All four parameters must be correctly set for the final-over-initial pattern to be learned. This contrasts with the other word-order patterns, which only required two or three parameters to be set before the language could be successfully learned. This is because the final-over-initial order has more nearby competitors than the other word order patterns, which had at most one neighbour to rule out.

A second problem is that the learner is not strongly pushed towards a particular setting on the first two parameters. For the first parameter, there is no obvious starting preference in the final-over-initial languages for a setting of either 1 or 0, so the learner is initially pulled in different directions. The second parameter is slightly more likely to have a setting of 0, and the P2 parameter curve does initially drift towards 0 in Figure 7, but this effect is not strong enough to consistently cause the learner to pick a 0010 setting. Adding to this problem is the fact that the settings of the first two parameters are dependent. For example, setting P1 to 0 obliges the learner to also set P2 to 0 in order to be successful. This means that, in addition to being pulled in different directions for the individual settings of P1 and P2, the learner can be pulled in different directions by a ‘disagreement’ between P1 and P2.

Something notable about the parameter curves for the final-over-initial language is that the source of difficulty when learning this pattern does not appear to come from how long it takes the individual parameters to be set. Compare, for example, the relative number of updates for all parameters to be learned in Figure 6 vs. Figure 7. The difference here is that, although the initial-over-final and head-initial took longer for all parameters to be set, the learner quickly arrived at a point where it was choosing between languages which always or nearly always resulted in the correct outputs. On the other hand, while it took fewer updates for all the final-over-initial parameters to be set, the learner was much more likely to be wrong at each stage of learning, as the ratio of correct languages to nearby languages is much smaller for the final-over-initial order than for the other patterns. This is why the final-over-initial pattern has the flattest learning curve in Figure 1, and is the most difficult for the learner to acquire.

7.1 Relating results to the model

The syntactic system that generated the training data starts with an spine of form {_ZP Z, {_YP Y, {_AuxP Aux, {_XP X, {_VP V, {_NP N}}}}}. This unlinearized input is taken to be a part of UG in this model.¹² As stated in section 2, the null heads X, Y, Z, along with the assumptions of anti-locality, were adopted to optimize for a small parameter space, in order to make the results more transparent. It is plausible that, with additional parameters, null heads and anti-locality could be done away with without significantly impacting the results. Further speculation on what the mechanics of such a parameter system might look like are outside of the scope of this paper, but it would presumably need to produce an identical or highly similar typology.

The approach to linearization and the absence of rightward movement are less innocent, and have a significant effect on the results. First let us consider the impact of movement: One important factor contributing to the difficulty of the final-over-initial order was the fact that there were only two possible languages (parameter settings) compatible with this data, and they were sufficiently different from each other that the EDPL struggled to decide on the correct settings for each parameter. The learner got caught between two dissimilar, but equally good choices. Had there been additional languages, this would have smoothed out the parameter space by changing the correlation between final-over-initial tokens and parameter settings. For example, if the final-over-initial pattern had been generated by 1110, 0010, and 1010, then it would have been easy for the learner to set P1 to 1, since 2/3 languages would have this setting. Likewise P2 could have easily been set to 0, for the same reason. However, since there were only two languages, 1110 and 0010, there was no ‘tie-breaker’ to guide the learner to one setting over the other. The reason that this didn’t happen in the case of the initial-over-final order (i.e. the other disharmonic pattern), is that there are more weakly-equivalent initial-over-final languages. The presence of more contiguous, weakly-equivalent languages allowed the learner to determine that some settings were more likely to produce matches than others, so that no parameter ended up ‘tied’ between 1 and 0.

This asymmetry between final-over-initial and initial-over-final languages is reducible to the asymmetry between leftward and rightward movement. Although both disharmonic patterns have two minimal derivations (section 3), further string-vacuous leftward movements create a number of additional weakly-equivalent initial-over-final languages. Leftward movement does not create additional weakly-equivalent languages for the final-over-initial pattern, as no leftward movement applied after those in the minimal derivations is string vacuous. Any additional movement applied to a final-over-initial structure results in a different word order. In order to introduce vacuous movement that could produce more weakly-equivalent final-over-initial languages, rightward movement would need to be introduced to the system.

This brings us to headedness, and the question of whether the asymmetry between leftward and rightward movement is sufficient to reproduce the attested typology under any system of linearization. I will not attempt to provide a fully comprehensive answer to this question here, but will instead focus on showing how assuming base-generation of harmonic extended projections contributes to the results. The first contribution is in fact a part of the justification for adopting this version of a linearizing parameter; defining headedness over extended projections effectively captures the fact that the FOFC does not appear to hold across them. For example, if determiners are added to the data set, along with a parameter for determining the order of D and NP, this will not affect the difficulty of any class of languages. The additional parameter needed to control the headedness of DP would double the size of the hypothesis space, but it would do so evenly across all patterns, so the relationship between data points and parameter settings would be unchanged.

Outside of explaining why the FOFC does not cross extended projections, defining headedness as underlyingly harmonic also introduces an important asymmetry between the harmonic and disharmonic languages. The ability to be base-generated increases the number of harmonic languages in a way that is contiguous, and therefore makes them easier to learn. If headedness was decided for each XP individually, as proposed in Abels & Neeleman (2012), this asymmetry would be lost, and the number of disharmonic languages would be greatly increased. Without this difference between harmonic and disharmonic languages, the remaining asymmetry between leftward and rightward movement is insufficiently strong for capturing the typological distribution of word order patterns. This is shown in Figure 13, which gives the parameter curves for a simulation identical to the 3-token task, with the exception that there were two separate parameters determining the headedness of AuxP and VP.

Figure 13

3-token learning curves with separate headedness parameters for AuxP and VP.

As this figure shows, when any order can be base-generated, the resulting curves bear no resemblance to the word order typology described in the literature. Therefore the asymmetry between the harmonic and disharmonic languages conspires with the absence of rightward movement to produce the hypothesis space that will lead the learner to mimic the frequency of word order patterns in natural language typology. The EDPL in particular does this because it is sensitive to the smoothness of the parameter space, which is why it struggles more when the languages within a word order pattern have discontiguous settings, as in the case of the final-over-initial pattern. More specifically, the EDPL is sensitive to the contribution of individual parameter settings to a match or mismatch. This is what leads it to get stuck between 1110 and 0010 for the final-over initial languages, as discussed above. Learners that are similarly sensitive to smoothness could conceivably be used to achieve comparable results.

7.2 Richness of the data

What I have done in this paper is to show that typologically marked orders like V-O-Aux, which violate the FOFC, may have a starting bias linked to how difficult it is for a learner to locate an appropriate grammar for them in the hypothesis space of languages. To have a more definitive answer about whether the FOFC stems from learning, the next step for this project is to model how this bias might be translated across generations. Although beyond the scope of the work presented here, iterated learning will stress-test the results by showing whether the bias is amplified as predicted, and also what word order(s) is(are) predicted to be learned in place of FOFC violators.

Additionally, the data used in this model is necessarily a vast oversimplification of the challenge that actual language learners are presented with, as is the standard in computational modelling work of this kind. Because of this, it is natural to wonder whether the results would be preserved with a richer data set that more accurately reflected the kind of input human learners receive. It is currently impractical if not impossible to model acquisition of the full extent of natural language data a human child is exposed to, of course, but a reviewer raises the question of a few areas relevant to the FOFC that one could imagine might affect the results. First is the question of how the learner might handle scrambling languages, which have variable word order but still seem sensitive to the FOFC. While there is no way to definitively answer this question without testing it, but depending on how scrambling is implemented, it is very possible that the results will be reserved (for instance, one might expect a similar hypothesis space to emerge if scrambling is implemented by making movement operations either fixed or optional). This is a natural next step for the project, although it is beyond the scope of the present work. There is also the (twofold) question of adverbs, which a) exhibit FOFC-like effects in V-Adv-Aux sequences, and b) seem to disambiguate movement where it might be string vacuous in other cases. In the first case, since the model learns sequences of stings without really “knowing” what a word is, a minimally different typology with V-Adv-Aux tokens would produce the same results. It may be necessary to make different syntactic assumptions about the hierarchical configurations that generates V-Adv-Aux sequences, but since the learner never has access to this information anyways, it would be not be expected to effect the results of the model unless the assumptions generated a wildly different distribution of word orders. This is likewise true of any other FOFC-like configuration (the head-final filter, etc). In terms of adverbs ability to provide cues for learning word orders, adverbs position varies greatly across and even within languages and this brings into question the amount of information they can really provide to a learner trying to acquire word order. It is not clear to me that they are predictable enough to be disambiguating. That being said, the question of how the model presented here behaves with a richer data set is an interesting one that should be investigated in future, especially in light of research in the Iterated Learning Framework (Kirby 1999; Smith et al. 2003; Kirby et al. 2004) that shows how weak biases (like the one put forward here) may be amplified into strong generalizations when transmitted across generations.

7.3 Broader connections

It is important to note that the source of the problem with final-over-initial pattern is the difficulty that the learner has trying to decide on a parameter setting for these languages. This means that it is not actually the *FOFC string which is challenging for the learner, but rather locating parameter settings that will produce that string in the hypothesis space. This is confirmed by the 2-token task, where the learner was never exposed to strings that violated the FOFC, but still struggled to acquire languages where such strings might be predicted to appear. The results of the 2-token task mirror the results of an adult learning study conducted in Culbertson et al. (2012). The purpose of their study was to investigate whether or not adult language learners are biased against learning typologically infrequent word order patterns in the nominal domain. Specifically, it dealt with Greenberg’s Universal 18, which observes that when adjectives precede a noun, numerals and demonstratives generally do as well (Greenberg 1963). This universal is sometimes considered to be a part of the FOFC, as the marked word order Adjective-Noun-Numeral can be modelled with a *FOFC structure. In the experiment, adult learners were exposed to miniature artificial languages that had an inconsistent mix of word order patterns (an idealization of change in progress), and then tested on what they had learned. Culbertson et al. found that when the artificial language had dominant harmonic and unmarked disharmonic orders, learners shifted that language towards the majority order (i.e. produced the majority order at a greater frequency than it was found in the training data). When the dominant order was the marked disharmonic pattern, learners shifted the language towards a harmonic order, instead of reinforcing the dominant order in the training data. Crucially, learners in this study only learned binary sequences of {Adjective, Noun} and {Number, Noun}. As in the 2-token learning task conducted here, the human learners in Culbertson et al. dispreferred learning the constructed languages which would predict a FOFC violation, without ever actually being exposed to a *FOFC structure.

Although the final-over-initial pattern was consistently the most difficult to learn, the EDPL never actually failed to converge on parameter settings for final-over-initial languages in either task. This result is compatible with the idea that the challenges *FOFC word orders pose to a learner may be a source of the rarity of those word orders. If the challenges presented by the final-over-initial pattern do not necessarily make it impossible for *FOFC orders to be learned, then this allows for the existence of exceptions to the FOFC, which have been documented in the literature (Bhatt & Dayal 2007; Biberauer et al. 2014; Erlewine 2017; Sheehan et al. 2017). Additionally, the relative order of the other word-order patterns likewise parallels the typology, with the exception of the ranking of head-initial and initial-over-final languages in the 2-token task. In both tasks, all languages were easier than the final-over-initial pattern, and in the 3-token task, harmonic languages were also easier to learn than disharmonic, even when the disharmonic language did not violate the FOFC.

A.1 Head-initial + VP fronting & Head-final + AuxP fronting

Figure 14

Head-initial + VP fronting parameter curves.

In the head-initial + VP fronting pattern, all parameters are set quickly, and this is because there is only one possible language. Since ternary tokens are unique to a language pattern, the ternary form disambiguates every parameter setting in patterns containing only one language. This can be seen in Table 16.

This is likewise true for the head-final + AuxP fronting word order pattern, which consists of a single language comprised of the tokens O-Aux, O-V, and O-Aux-V. Parameter curves for this language are given in Figure 15.

Figure 15

Head-final + AuxP fronting parameter curves.

The learner is able to set each parameter quickly because there is only one language in this pattern, and therefore the ternary token disambiguates each parameter setting, as seen in Table 17.

B 2-token head-final and initial-over-final pattern

The parameter curves for the head-final pattern and initial-over-final pattern in the 2-token task are given in Figure 16 and 17 respectively:

Figure 16

Harmonic head-final parameter curves (2-token).

Figure 17

Initial-over-final parameter curves (2-token, 400 updates).