2.1 A brief overview of MDL

For our sketch of how MDL accomplishes this balance, it will be convenient to think of both grammars and their encoding of the data as sitting in computer memory according to a given encoding scheme. We will write $|G|$ for the length of the grammar $G$ as measured in bits. The encoding of the data $D$ using $G$ will be written as $D:G$ , and the tightness of fit will be the length of this encoding, which we will write as $|D:G|$ . Using this notation, MDL can be stated as follows:Footnote ^[3]

The balancing of grammar economy and tightness of fit has made MDL – and the closely related Bayesian approach to learning – helpful across a range of grammar induction tasks, in works such as Horning (Reference Horning1969), Berwick (Reference Berwick1982), Ellison (Reference Ellison1994), Rissanen & Ristad (Reference Rissanen and Ristad1994), Stolcke (Reference Stolcke1994), Grünwald (Reference Grünwald, Wermter and Riloff1996), de Marcken (Reference de Marcken1996), Brent (Reference Brent1999), Clark (Reference Clark2001), and Goldsmith (Reference Goldsmith2001), among others.

In the context of language learning, MDL can be thought of as follows. Universal Grammar provides a template for specifying grammars. In a sense, it is a programming language in which various programs – equivalently, grammars – can be written. The programming language says exactly how programs are written, and this specification is provided ahead of the actual data that are encountered and independently of them. $|G|$ , from this perspective, is simply the storage space required for representing the grammar/program $G$ according to the specifications of the programming language. (What can be stated in a grammar and how much it costs can of course vary greatly between programming languages. One programming language, for example, might allow context-sensitive rewrite rules to be stated and might make vowels costlier than consonants. Another programming language will allow markedness and faithfulness constraints to be stated and will make consonants costlier than vowels. And so on.) While grammars are stated in terms of an a priori programming language, they also interact with the particular data $D$ that the child happens to see. In particular, a grammar can be used to parse $D$ , which can be thought of as a sequence of instructions to $G$ that result in the generation of $D$ . This sequence of instructions is what we referred to above as $D:G$ . The MDL metric says that the best grammar $G$ given the data $D$ is the one that minimizes the overall storage space of the grammar itself (stated according to the innate programming language) and of the sequence of instructions through which $G$ generates $D$ . In the remainder of the present section, we will try to make the above discussion concrete by going through two relevant linguistic examples.

2.2 Two examples: learning alternations and phonotactics using MDL

The MDL view predicts that the child will invest in grammatical statements only when the cost of the investment (in terms of increase in $|G|$ ) will be offset by the increase in tightness of fit to the data (in terms of decrease in $|D:G|$ ). As an illustration, consider the pattern of optional L(iquid)-deletion in French, discussed by Dell (Reference Dell1981) and Rasin et al. (Reference Rasin, Berger, Lan, Katzir, Hucklebridge and Nelson2018a). In French, L-deletion applies optionally in certain contexts – specifically, word-finally and following obstruents. For example, the French-learning child might hear both and ‘table’ and both and ‘tree’ (but only ‘train station’). With some simplification, the relevant process can be described as follows:

A learner minimizing $|G|$ will favor collapsing pairs such as and onto a single UR and deriving the two surface forms using a process of optional $L$ -deletion. This is so since, with sufficiently many alternating surface pairs, the savings obtained by storing just one UR for each will outweigh the costs of adding the relevant rule to the grammar. To illustrate the trade-off between grammatical statements and lexicon size, consider the two competing grammars in (4): G1 has simply memorized the data without making any rule-based generalization, while G2 generates pairs like and from a single UR using the optional L-deletion rule.

There are various conceivable ways of measuring grammar size. In this paper, we will assume – in the spirit of the evaluation metric of early generative grammar – that grammar size grows with the number of symbols used to state URs in the lexicon as well as the processes (either rules or constraints). So, for example, the cost of the L-deletion rule of G2 above would be the cost of the six symbols $L$ , $\emptyset$ , $-$ , son, , and $\#$ , and the cost of storing of URs like in the lexicon would be the cost of the relevant three symbols (one symbol for each segment). More precisely, within MDL, grammar size is measured in terms of bits. We can think of the calculation of the size of the grammar as follows. We first obtain a string representation of the grammar. For G1 and G2 above, the string representations would be along the lines of the following strings (with various delimiters separating URs, rules, and the different components of the grammar; the three dots are placeholders for the rest of the lexicon):

String representations are then converted into bit strings, according to an encoding scheme that assigns a binary code to each symbol that can be used in writing grammar-string representations. For example, if the segment $t$ is assigned the code 0010 and the segment $a$ is assigned the code 1011, the beginning of the bit strings for both grammars would start with the bits 0010 1011. Rasin et al. (Reference Rasin, Berger, Lan, Katzir, Hucklebridge and Nelson2018a) discuss specific encoding schemes, but here we will not commit to one encoding scheme in order to keep the discussion general. What will matter for the discussion is the assumption that the number of bits required to specify a symbol grows with the number of symbols the choice is made from. For example, if the possible set of symbols for writing URs contains 7 symbols including $t$ , the cost of specifying $t$ in the lexicon would be, say, 3 bits, but if the possible set of symbols for writing URs contains 10 symbols including $t$ , the cost of specifying the choice of $t$ from among that set would be higher, say, 4 bits. In other words, a larger alphabet for the lexicon results in a higher cost for each underlying segment, a point that will play a central role in our discussion in Section 4. After the string representation of a grammar is converted to a bit string, the size of the grammar is taken to be the length of the resulting bit string. In our current French example, the bit string for G1 will be longer than the bit string for G2, provided that the data include enough collapsible pairs such as and , which justifies the addition of the L-deletion rule to G2.

This kind of balancing within the $|G|$ component – adding a phonological process to the grammar despite its length, since the addition makes it possible to attain savings within the lexicon that outweigh the added complexity – is familiar from early work in generative phonology (in particular, Halle Reference Halle1962). It will also play a central role in our discussion in Section 4. However, as noted by Dell (Reference Dell1981), if the learner only minimizes $|G|$ (as in the evaluation metric of early generative phonology but differently from MDL, where it is balanced by $|D:G|$ ), the learner will overgeneralize, preferring the incorrect but shorter (5), which permits L-deletion regardless of context, to the correct but more complex (2).

Minimizing the second component of the MDL metric, $|D:G|$ , helps address this problem by ensuring that the grammar is restrictive. As mentioned above, $D:G$ lists a sequence of instructions for $G$ that result in the generation of $D$ . In the case of rule-based phonology, this sequence includes the instruction to pick a specific UR from the lexicon, along with instructions for whether any applicable optional rule applies. A grammar that is more restrictive will allow the input data to be specified using fewer instructions than a less restrictive one and will thus typically result in a shorter $D:G$ . For example, an instruction to make a binary choice (such as applying or not applying an optional rule whose structural description is met) will cost 1 bit. In the present case, a grammar using the less restrictive (5) will require a specification within $D:G$ for each underlying liquid stating whether it is deleted or not. Thus, for example, encoding the surface form using (5) will require first specifying the choice of the UR from the lexicon and then – since (5) optionally deletes liquids anywhere – paying an additional bit to specify that optional deletion does not apply to the final segment. With the more restrictive (2), on the other hand, only liquids that are in the appropriate context (word-finally and following an obstruent) require such a specification. This means that under (2), encoding the surface form does not require an extra bit for specifying that deletion does not apply to the final segment, leading to a shorter $D:G$ (see Rasin et al. Reference Rasin, Berger, Lan, Katzir, Hucklebridge and Nelson2018a for a more detailed discussion of measuring $|D:G|$ ).

Like minimizing $|G|$ alone, minimizing $|D:G|$ alone is problematic. Specifically, while helpful in avoiding overly general grammars, minimizing $|D:G|$ alone gives rise to a problem of under-generalization: it leads the learner to overfit the data, regardless of how complex the grammar becomes. For example, if the learner has heard only [sabl] but not yet its $L$ -elided variant (both for the UR ‘sand’), we might expect them to generalize beyond the data and accept the L-deleted . A learner minimizing $|D:G|$ alone, however, will incorrectly prefer to treat [sab] as ungrammatical: by doing so, when the UR is selected, no further statement as to whether the final $L$ is deleted will be needed, which in turn leads to a shorter $D:G$ .

By minimizing the sum of $|G|$ and $|D:G|$ , MDL aims at an intermediate level of generalization: the $|G|$ component biases the learner toward relatively general grammars, while the $|D:G|$ component protects the learner from overly general grammars. In the case of French $L$ -deletion, MDL will avoid the excesses of both $|G|$ minimization (which leads to free $L$ -deletion, as in the short (5)) and $|D:G|$ minimization (which leads the overfitting memorization of as non-alternating). Instead, it will lead to the correct (2): this grammar is only slightly more complex (and less general) than (5) but fits the data much better; and it fits the data only slightly less well than a grammar that rules out but is much less complex than it. The three grammars and their MDL costs are illustrated schematically in Figure 1.

Figure 1 Schematic illustration of three hypotheses. (The order of URs in the lexicon and of tokens in $D:G$ are unrelated.) Introducing a naive lexicon (top), in which and [tab] have distinct URs results in a complex grammar. Capturing optional $L$ -deletion with (5) allows the grammar to be simplified (middle): the complexity of the rule is outweighed by the savings of eliminating unnecessary URs. Moreover, since there are now fewer URs than with the naive lexicon, each UR can be specified more succinctly. However, an additional bit is needed for specifying the actual surface form of each occurrence of $L$ in a UR (for each surface token of that UR). Finally, restricting the context of L-deletion, using (2), allows us to limit the extra bit to just those URs that require it (bottom): e.g., /tabl/ but not .

In addition to supporting the learning of patterns involving alternations, like L-deletion in French, MDL can also ensure the learning of phonotactic patterns. Here, the trade-off between the size of the lexicon and the size of the component where processes are specified also plays an important role. This trade-off will be particularly important in Section 4, where we discuss the possible choices and their implications in detail. For now, though, let us illustrate this with a well-known example from English, due to Chomsky & Halle (Reference Chomsky and Halle1965) and originally stated in a learning framework that is different from MDL (specifically, the simplicity metric, where only $|G|$ is minimized). We wish to emphasize that we use Chomsky & Halle’s case only as an informal motivational example and will present the case only in very rough outline, leaving important questions unaddressed.

As noted by Halle (Reference Halle1962) and Chomsky & Halle (Reference Chomsky and Halle1965), speakers judge some nonce forms as nonexistent but possible – that is, as accidental gaps – and other nonce forms as nonexistent and impossible – that is, as systematic gaps. In English, for example, is an accidental gap while is a systematic gap. With respect to , it is generally true that if an English word begins with a stop, the following consonant cannot be nasal: it must be a liquid. The lexicon of English can be simplified by removing the specification of the [liquid] feature from many consonants:Footnote ^[4] for example, the word could be represented without a specification that /r/ is a liquid. Instead, this feature could be specified using the following rule:

Adding (6) to the grammar will make the rule component more complex, but this added complexity will be easily offset by the savings in the lexicon, which are obtained by eliminating the [liquid] specification from the second consonant of every English word starting with a stop-consonant cluster. A learner that minimizes both the size of the lexicon and the size of the rules as part of $|G|$ will add (6) to the grammar and correctly treat as ungrammatical.Footnote ^[5] For words like , as Chomsky and Halle note, the situation is different, since excluding such words by means of a rule would require a rule whose complexity outweighs the savings in the lexicon. For instance, since brick is an English word, a rule that excludes will have to be very specific and ensure that a liquid is never in this specific environment (while leaving other consonants unaffected):

Rule (7) is quite complex, but it only allows eliminating a single feature from the lexicon: the [lateral] feature of the second consonant in the word brick. By minimizing both the size of the lexicon and the size of the rules, a learner that minimizes $|G|$ will not add (7) to the grammar and will correctly treat as an accidental gap. Note that in this example there are two phonotactic patterns in the data that could in principle be captured by the grammar: the absence of word-initial sequences like #bn and the absence of the phonotactic configuration . With a $|G|$ -minimizing learner, only the phonotactic pattern that leads to an improvement in terms of $|G|$ – namely, the ban on #bn – is captured by the grammar.

2.3 Motivation for testing the predictions of MDL

MDL is certainly not the only learning framework that has been proposed for phonology.Footnote ^[6] As mentioned above, our goal in this paper is not to argue against alternative learning approaches. Rather, we will briefly explain (in the remainder of this section) why we find MDL promising, so as to motivate our exploration below of its implications for the representational question of language-specific statements in the lexicon.

Our first reason for finding MDL promising is that it has provided working distributional learners for phonology – that is, implemented algorithms that take raw surface data and induce a phonology and a lexicon. The MDL learner of Rasin & Katzir (Reference Rasin and Katzir2016), for example, takes unanalyzed surface forms and induces a lexicon, often with abstract URs that differ from the surface forms, along with a set of markedness and faithfulness constraints and their ranking.Footnote ^[7] The MDL learner of Rasin et al. (Reference Rasin, Berger, Lan, Katzir, Hucklebridge and Nelson2018a) and Rasin et al. (Reference Rasin, Berger, Lan and Katzir2018b) accomplishes a similar task but within rule-based phonology: it takes unanalyzed surface forms and induces a lexicon and a set of ordered context-sensitive rewrite rules. The learner has been shown to handle a range of linguistically relevant phenomena such as opacity (including both counterfeeding and counterbleeding), optionality, and the long-distance dependencies of vowel harmony.

Our second reason to focus on MDL is that it has been supported empirically by a range of lab experiments on generalization. In a variety of learning tasks in the lab, ranging from word learning (Xu & Tenenbaum Reference Xu and Tenenbaum2007) through causal reasoning (Sobel et al. Reference Sobel, Tenenbaum and Gopnik2004) to sensorimotor control (Körding & Wolpert Reference Körding and Wolpert2006) and visual scene chunking (Orbán et al. Reference Orbán, Fiser, Aslin and Lengyel2008), among many other tasks, human subjects have been argued to balance between the specificity of a hypothesis, corresponding to $|D:G|$ , and its independent plausibility, corresponding to $|G|$ . If humans indeed use this way of learning across different domains, it seems sensible to consider their use of the same in phonology.

Our third reason to focus on MDL is methodological. As mentioned above, we can think of MDL in terms of an innately given programming language in which grammars ( $G$ ’s) are stated, grammars that can then be used to parse the input data $D$ . As noted in Katzir Reference Katzir2014, this perspective highlights how much of the basis for MDL learning is available to the child without further stipulation. The grammars are real cognitive objects: they are stored according to the specifications of the programming language and take up space. Similarly, if the child remembers $D$ as parsed by $G$ – the sequence of instructions for generating $D$ using $G$ that constitutes an understanding of $D$ using $G$ – then they also store $D:G$ in memory, and that also takes up space. The overall storage space for the two specifications, $|G|+|D:G|$ is exactly the quantity used by MDL. All that is needed is a way to compare neighboring grammars in terms of this quantity and search for the one that minimizes it. If the above is indeed correct, MDL constitutes a simple, non-stipulative approach to learning, which is then a sensible starting point to an investigation of learning (and can of course be modified when empirical reasons are encountered to add stipulations).

4.1 Constraints acquired

For the first few configurations that we will consider, suppose that the child, using MDL, needs to acquire the constraints, with each additional constraint costing a positive number of bits (along the lines of our discussion in Section 1), and suppose further that and each costs some fixed number of bits to store in the lexicon. Using the perspective of an a priori programming language discussed above, we can say that the programming language determines costs for and that do not depend on the position of the segment in a given UR and that hold in any grammar that does not make further statements about the format in which URs are stored (statements that are possible if CURs are allowed but are not possible under ROTB).

The exact form of the argument depends on which of the two segments is costlier, if either. We will consider the three different possibilities in turn, followed by an examination of costs that vary between contexts. In this subsection, we assume that features are fully specified in the lexicon.

4.1.1 Globally fixed costs for and

Suppose that the cost of storing an instance of in the lexicon is greater than the cost of storing an instance of . Assume that the child encounters Dutch words such as ‘ice cream’ and ‘a small ice cream’. We first consider the situation of the child on the assumption that ROTB holds and show that in this case MDL chooses a grammar that suffers from the ERP as stated in (12).

Since MDL prefers an economical lexicon, it will push the child toward segmenting words like and into morphemes and toward storing repeating morphemes using a single UR in the lexicon (e.g., either or for ‘ice cream’). Since instances of are costly to store in the lexicon, it will be preferable in terms of MDL to store all stridents as (e.g., prefer to for ‘ice cream’) and invest in constraints that trigger palatalization of before (e.g., the markedness constraint $^{\ast }$ , ranked above the appropriate faithfulness constraints). Adding the constraints will cost a few bits, but this cost will be outweighed by the savings from not having to store multiple entries for a single morpheme and from not having to store any instances of the costlier in the lexicon. Two competing hypotheses are illustrated in (13), with G2 being preferred by a ROTB child that follows MDL. In these hypotheses, the lexicon is broken down to the alphabet (used to write URs in the lexicon) and the URs. By choosing G2 over G1, the child has successfully learned that pre-palatal stridents are systematically palatalized in the language.Footnote ^[14] For example, the child will now correctly rule out forms such as , with an alveolar before .

Unfortunately for ROTB, G2 is also the extent of the child’s acquisition of the pattern under MDL. In particular, the child will not learn to block forms such as , with [ʃ] in ‘elsewhere’ environments. The reason is that, for a ROTB child, such forms must be blocked through the input–output mapping, for example through $^{\ast }$ ʃ or a similar markedness constraint that penalizes [ʃ], as in G3 in (14) below. And there is simply no MDL justification for acquiring this kind of constraint. Under G2, all stridents are already stored in the lexicon as /s/ and are mapped faithfully to the surface. Consequently, a markedness constraint such as $^{\ast }$ ʃ in G3 will be of no use in deriving the observed forms in the input data from the lexicon, and the cost of adding such a constraint to the grammar will not be justified. A ROTB child, then, will become an adult who knows only half of the distributional pattern of stridents in Dutch: having learned G2, such an adult will correctly rule out but fail to rule out (which, due to ROTB, can be stored as is and then mapped faithfully to the surface given the acquired constraints). In other words, a ROTB child will arrive at a grammar, G2, that suffers from the ERP for [ʃ] in non-pre-[j] positions. Note that markedness constraints such as $^{\ast }$ ʃ in the present case are analogous to rule (7) (which blocked , as discussed in Section 1), which had no benefit in terms of MDL and thus was not added to the grammar, leaving a phonotactic pattern in the data as accidental.

If ROTB does not hold and CURs are allowed, the learning process can succeed in full and arrive at a grammar that does not suffer from the ERP. The first step – choosing G2 – is similar to the one with ROTB: the child will invest in a constraint like $^{\ast }$ and then store all stridents as ; again, the result will allow the child to correctly rule out forms like . But with the possibility of stating CURs, a crucial second step becomes available. The first step involved the extensional removal of all instances of from the lexicon. The child can now conclude that this was no accident, and that should be eliminated intensionally from the very alphabet in which the lexicon is written. That is, the child can reach the following conclusion (restating (8) above):

Let us first see why (15) is justified in terms of MDL. All things being equal, removing a possible segment from the underlying inventory makes the lexicon easier to encode. This is because of the assumption, mentioned in Section 1, that specifying a choice from a smaller set requires fewer bits. If the alphabet of Dutch includes , every segment in the lexicon (e.g., ɛ, ɪ, and in the UR ) will have to be specified from among the alphabet that includes . If, however, the alphabet of Dutch does not include , specifying the same segments (e.g., ɛ, ɪ, and s in the UR ) will be made from among the smaller alphabet and require fewer bits. As a result, the entire lexicon will be cheaper to encode with a smaller alphabet, thus providing MDL justification for adopting (15). The resulting grammar, now with (15), is given in (16).

We can now see why, in a world that allows CURs, the child can go beyond what was possible under ROTB and acquire the second part of the pattern of distribution of stridents. The reason is that with a grammar like G4, that has a CUR like (15), the child will now correctly rule out surface forms like : is now no longer a possible UR; and given the constraints that have been induced, no other UR is a potential source for this putative surface form. In other words, the impossibility of even stating in the lexicon, with its MDL justification, means that the learner has correctly learned to block illicit palatalization.

Note that while the CUR in (15) and a markedness constraint like $^{\ast }$ ʃ are similar in their complexity and their role in the adult grammar, they have a different status from the perspective of MDL, which causes an MDL learner to acquire the CUR but not the markedness constraint. The key difference from the perspective of MDL is that the CUR allows for a shorter specification of the lexicon. A surface constraint, on the other hand, only affects the input–output mapping and does not contribute to savings in the lexicon. In this case, and in many others, this difference will result in a CUR being acquired (if UG allows it to be represented) while a mapping statement such as a markedness constraint will not be acquired, even when the two are very similar.

We conclude that, under the representational choice of constraints that need to be acquired and of , the ability to state CURs allows for the full distributional pattern of stridents to be acquired, while the adoption of ROTB leads to a failure in learning half of the pattern.

4.1.2 Globally fixed costs for /s/ and

Suppose now that . In this case, the Dutch-learning child will be able to avoid the ERP even with ROTB. They can do so by storing throughout the lexicon and acquiring constraints that will ban [ʃ] in ‘elsewhere’ (that is, non-pre-palatal) environments: before a vowel, before a non-palatal consonant, word-finally, and so on. In this case, it will be costly to state constraints that ban [ʃ] directly in all of these contexts, and it will make more sense for the child to learn to ban [ʃ] in general and allow it only before [j]. That is, the child will acquire a low-ranking $^{\ast }$ ʃ to prevent underlying from surfacing faithfully and a high-ranking $^{\ast }$ to ensure that stridents surface correctly before . On this scenario, the learner will have correctly acquired the full pattern without requiring a CUR, thus avoiding the ERP and allowing ROTB to be maintained.Footnote ^[16]

While this scenario provides a way to learn the distribution of stridents without CURs, it is ultimately unsuccessful because the cost assignment makes the mirror image of the Dutch pattern – a pattern with stridents that are alveolar in some specific environments but are palatalized elsewhere – unlearnable on the assumption of ROTB. Consider Bengali, where the default sibilant is [ʃ], and [s] occurs only in word-initial consonant clusters and word-medially before dental stops (Evers et al. Reference Evers, Reetz and Lahiri1998). For example, the nonce forms , with an [ʃ] before a velar consonant, and , with an before a dental consonant, are both accidental gaps, while the nonce form , with an before a velar consonant, is a systematic gap.Footnote ^[17] An appropriate constraint ranking for the Bengali pattern (ignoring both optionality and word-initial clusters) that does not rely on CURs would be the following:

And paralleling the discussion of the Dutch pattern with the earlier cost assignment of , the present cost assignment of will prevent the full Bengali pattern from being acquired. In particular, it will lead a ROTB child to a grammar that suffers from the ERP for in pre-[j] positions. Given the present cost assignment, a ROTB child will store all stridents as and then acquire a markedness constraint forcing stridents to surface as in the relevant environments. The same reasoning used earlier will prevent the learner from acquiring a constraint that enforces [ʃ] elsewhere (in this case, since all stridents are already stored as in the lexicon), which will result in an inability to rule out nonce forms with in ‘elsewhere’ environments (e.g., before velar consonants, as in , contrary to fact. On our current assumptions that constraints are acquired and that is less costly than , succeeding in learning Bengali requires abandoning ROTB and using CURs (here, a CUR that removes from the alphabet of the lexicon).

4.2 Constraints given in advance

If the constraints are not acquired but rather given to the learner in advance, as is commonly assumed in the OT literature, a slightly more complex situation arises. We now turn to this case (keeping our other assumptions from Section 4.1 unchanged), building on the argument in Rasin & Katzir Reference Rasin, Katzir, Bui and Özyıldız2015 that, unless a preference for markedness over faithfulness is incorporated, an MDL learner would still need to abandon ROTB and adopt CURs to avoid the ERP. Suppose that the learner is given the two relevant markedness constraints for the Dutch pattern: $^{\ast }$ sj, which penalizes alveolar pre-palatal stops; and $^{\ast }$ ʃ, which penalizes in general. Suppose further, as in Section 4.1.1, that .

As in the setting with acquired constraints, the constraint $^{\ast }$ poses no special problem for an MDL learner following ROTB. Ranking this markedness constraint above the relevant faithfulness constraints will serve the MDL purpose of enabling the elimination of from all URs. As for $^{\ast }$ ʃ, the learner is now assumed to be given this constraint in advance; differently from the case of a learner that needs to acquire the constraints, the presence of $^{\ast }$ ʃ will no longer incur costs in the present setting. However, the constraint still offers no MDL advantage. Consequently, the learner will not benefit from ranking this constraint above any faithfulness constraints, such as Ident[ant], that penalizes modifications of the feature anterior. We would thus expect speakers to vary in the relative ranking of $^{\ast }$ ʃ and Ident[ant]. But this means, on ROTB, that speakers of Dutch should differ in whether they accept forms such as as possible, contrary to fact.Footnote ^[19] In other words, for an MDL learner following ROTB that is given the constraints in advance, the problem lies not with the possibility of attaining the appropriate constraint ranking but rather with ensuring that this ranking is attained systematically, for all speakers, and not just occasionally.

It is at this point that a preference for $M\gg F$ becomes relevant.Footnote ^[20] In the settings discussed in Section 4.1, with binary features and acquired constraints, $M\gg F$ does not solve the problem for ROTB, and adopting CURs would be needed to ensure the learning of the distribution of stridents. With constraints that are given in advance, on the other hand, $M\gg F$ enables successful acquisition: as we just saw, the challenge in this case is not justifying the constraints (which, in the current setting, are already provided) but rather ensuring that the markedness constraints outrank the faithfulness constraints; by stipulation, a preference for $M\gg F$ addresses this challenge.Footnote ^[21] The combination of $M\gg F$ with constraints that are given in advance, then, is one way to preserve ROTB in the face of the learnability challenge (in effect, by giving the child knowledge of the pattern as part of the initial state). We now turn to a less stipulative response available within rule-based phonology.

4.3 Special case: rule-based phonology with underspecification

So far the discussion in this section assumed that features are fully specified. This assumption contributed to the fact that a ROTB child would always store part of the distribution of stridents faithfully, which in turn made it superfluous to acquire that part of the distribution within the input–output mapping, thus leading to the challenge to ROTB. We saw two responses to this challenge: allowing language-specific CURs (and thereby rejecting ROTB); and endowing the child with prior knowledge of the pattern (in the shape of constraints that are given in advance combined with $M\gg F$ ). A third response suggests itself if underspecification is allowed (for general discussion of underspecification in generative phonology, see Archangeli Reference Archangeli1988 and Steriade Reference Steriade and Goldsmith1995). If storing an underspecified value for anteriority is less costly than either of the specified values, the learner might prefer storing all stridents unfaithfully as underspecified for anteriority and invest in the markedness constraints $^{\ast }$ and $^{\ast }$ ʃ, along with a high-ranking markedness constraint that blocks underspecified values from surfacing. This response still requires something like $M\gg F$ to ensure that $^{\ast }$ and $^{\ast }$ ʃ outrank faithfulness (since otherwise inappropriate stridents in nonce forms could be accommodated, as discussed earlier), but otherwise this seems like a way to allow ROTB to be maintained without giving full prior knowledge to the learner (since the markedness constraints are now acquired). However, as we now show, the help that underspecification offers ROTB is considerably more limited than might initially appear to be the case: within OT, capturing certain simple cases of systematic gaps will still require both innate constraints and $M\gg F$ , which means that underspecification leaves the challenge to ROTB without change; and within rule-based phonology, underspecification will enable general learning while maintaining ROTB, but only under specific assumptions. As before, abandoning ROTB and adopting language-specific CURs allows the learner to succeed straightforwardly.

The problem with the use of underspecification by an MDL learner on the assumption of ROTB is that, while underspecification indeed makes a correct grammar (with underspecified URs and an investment in the requisite markedness constraints) more economical than the kinds of incorrect grammars considered earlier, it sometimes makes a new kind of incorrect grammar more economical still. As a concrete illustration, consider a case of four consonants such as the velar obstruents , , [x], and , which are identical with respect to all features but two (voice, which distinguishes the voiced and from the voiceless and , and continuant, which distinguishes the continuants and from the stops and ). Consider now a language that has exactly three of those four consonants – for example, German, which has , , and , but not . To correctly rule out surface forms with , the German-learning ROTB child will need to learn a high-ranking markedness constraint such as $^{\ast }$ ɣ. Earlier, with binary features, a ROTB child might have avoided positing such a constraint (depending on feature costs): in analogy with our discussion of Dutch and Bengali, an incorrect grammar such as the following $G_{0}$ , which stores voicing faithfully (and which has no need for $^{\ast }$ ɣ), could have been optimal.

With underspecification, faithful storage of this kind is no longer optimal. In particular, storing the attested as underspecified for voicing in the lexicon will provide an incentive to derive the voicelessness of [x] through the input–output mapping using $^{\ast }$ ɣ, which, in turn, would correctly prevent URs with underlying /ɣ/ from surfacing faithfully, as in the grammar $G_{1}$ in (19). Here we use [0voice] to refer to an underspecified value for voicing, which could mean either that a feature value for voicing is not listed in the lexicon at all or that [0voice] acts like a third value that is listed and could be referred to by the grammar. The constraint $^{\ast }$ [0voice] penalizes underspecified values on either of these two options.

The challenge to ROTB with underspecification and acquired constraints is that the correct $G_{1}$ has a simpler but incorrect alternative grammar $G_{2}$ , which stores not only but also as underspecified for voicing, and which maps only underspecified velars to voiceless:

$G_{2}$ is simpler than $G_{1}$ for two reasons. First, its constraint ranking is slightly simpler since it replaces the specific constraint $^{\ast }$ ɣ ( $\text{=}^{\ast }[\text{velar},\text{+}\text{cont.},\text{+}\text{voice}]$ ) with the more general constraint $^{\ast }[\text{+}\text{voice}]$ . Second, and much more significantly, its lexicon is more economical since it stores more features as underspecified than $G_{1}$ does. As opposed to $G_{1}$ , the simpler $G_{2}$ does not rule out , because the faithfulness constraint ident[voice] preserves underlying instances of . And crucially, the existence of prevents a preference for $M\gg F$ from being helpful: a ranking with the markedness constraint $^{\ast }[\text{+}\text{voice}]$ above ident[voice] would rule out both and and will thus fail to generate the data. In other words, a ROTB child with underspecification and acquired constraints will converge on $G_{2}$ and fail to capture the absence of . Avoiding this problem requires a combination of an innate $^{\ast }$ ɣ and $M\gg F$ , which in turn means that within OT, underspecification offers ROTB no learnability advantage over full specification.

The problem with using underspecification to keep ROTB can be replicated within rule-based phonology, where $G_{0}^{\prime }$ , $G_{1}^{\prime }$ , and $G_{2}^{\prime }$ serve as counterparts of $G_{0}$ , $G_{1}$ , and $G_{2}$ :

As with OT, the incorrect rule-based solution of $G_{2}^{\prime }$ is more economical than the correct one in $G_{1}^{\prime }$ . Differently from OT, however, rule-based phonology offers a way to avoid the problem: the simple but incorrect $G_{2}^{\prime }$ crucially relies on referring to underspecified values within a rule, while the more complex but correct $G_{1}^{\prime }$ does not. (This contrasts with the case of the constraint-based $G_{1}$ and $G_{2}$ above, neither of which required referring to underspecified values in their constraints: for those grammars, $^{\ast }[0\text{voice}]$ simply penalizes feature vectors that are underspecified for voicing.) So if it is impossible to refer to underspecified values within a rule (but still cheaper to state underspecified values in the lexicon, so that the rule in (21) will be learned at all), then the incorrect $G_{2}^{\prime }$ will not be statable, and the correct $G_{1}^{\prime }$ will be acquired. The following summarizes the two assumptions that we relied on here to preserve ROTB in the present setting (by benefiting from underspecification on the one hand and avoiding the trap of $G_{2}^{\prime }$ on the other hand):

Within rule-based phonology, then, and assuming the possibility of underspecification along with the specific statements in (24), CURs are not necessary for an MDL learner to acquire systematic gaps (as in the Dutch, Bengali, and German patterns above) even if knowledge of the pattern is not given to the learner in advance.