We use model theory to rigorously evaluate Q-Theory as proposed in Shih and Inkelas 2019 as an alternative to Autosegmental Phonology. We find that Q-Theory is remarkably similar to Autosegmental Phonology, contra some of Shih and Inkelas’s claims. In particular, Q-Theory does not eschew the association relation, in Q-Theory the tone-bearing unit is the vowel, and Q-Theory and Autosegmental Phonology are equivalent in terms of the constraints they can express. However, this formal analysis clarifies the truly novel contribution of Q-Theory, which is the empirical claim that all segments are tripartite.

## 1 Introduction

An important question when evaluating theories of phonological representation is whether two theories differ substantially in terms of their empirical predictions. This is related to the question of whether one theory states important generalizations in simpler terms than another. Answering these questions is not easy, as it requires a precise definition of representations in each theory and how the grammar operates over these representations. However, there are two related frameworks from mathematics and the theory of computation that provide tools for solving these problems: *model theory* (Libkin 2004) allows us to rigorously study phonological structure, and *mathematical logic* (Enderton 1972) provides constraint definition languages (Potts and Pullum 2002, de Lacy 2011, Jardine and Heinz 2016) that specify a range of constraints that operate over structures. These techniques have been applied to phonology before (Bird 1995, Scobbie, Coleman, and Bird 1996, Potts and Pullum 2002, Graf 2010a,b, Jardine 2014), in particular with respect to comparing different theories of syllabic structure (Strother-Garcia and Heinz 2017, Strother-Garcia 2019).

In this article, we use model theory to evaluate Q-Theory as proposed in Shih and Inkelas (SI) 2019 as an alternative to Autosegmental Phonology (AP; Goldsmith 1976, Clements 1977). Q-Theory representations (QRs; see also Inkelas and Shih 2013, Shih and Inkelas 2014) capture classically “autosegmental” tonal behavior, such as contours, by splitting each segment into exactly three subsegments. SI then analyze agreement and disagreement through surface correspondence, in an effort to incorporate tone into conceptions of phonological phenomena as mediated by surface correspondence (Hansson 2001, 2010, Rose and Walker 2004, Rhodes 2012, Bennett 2013).

To give an example, SI propose the QR in (1a) for the Basaá word [hólôl] ‘ripen’ (Dimmendaal 1988, Hyman 2003), in which the first vowel is a level high tone and the second vowel is a falling tone. Each [o] vowel segment, or *Q*, is split into three subvowels, or *q*s, each of which carries a tone. Indices on the *q*s represent correspondence between them. (The *q*s of the consonants have been abbreviated.)

In (1a), the first *q* of the second vowel is H(igh)-toned while the other two vowels are L(ow)-toned; this thus represents the falling contour of the second vowel. Furthermore, the last *q* of the first vowel and the first *q* of the second are in correspondence (and both H-toned). This indicates that the falling contour on the second vowel is the result of partial agreement with the H-toned first vowel—a fact that in AP would be represented with spreading of a H tone from the first vowel to the second, as depicted in (1b).

In comparing QRs to autosegmental representations (ARs), SI claim that (a) QRs dispense with the need for an association relation, thus simplifying the representation; (b) the *q* serves as the tone-bearing unit (TBU), thus solving issues with ARs in determining what the relevant TBU is; and (c) QRs capture patterns that ARs cannot. In rigorously comparing QRs to ARs, we present several findings that weaken these arguments. The first is that, contrary to SI’s assertions, QRs do not dispose of the association relation in ARs.^{1} Briefly, this is because there must be some relation connecting *Q*s to their composite *q*s in order for constraints to, for example, refer to the featural content of a particular *Q*. To see this, consider (2), a visual depiction of the model-theoretic representation of (1). (As above, the representation of consonants has been abbreviated.)

In (2), *q*s are depicted as the nodes labeled H and L, arrows indicate the order on the string of *Q*s and the string of *q*s, curved lines between *q*s depict correspondence, and straight lines depict the relation between *Q*s and their respective *q*s. As this visual representation makes clear (and as we will show formally in section 2), this relation is virtually identical to AP’s association relation between TBUs and autosegments.

It is then possible to show that the similarities between the two representations run even deeper. SI assume that correspondence implies identity,^{2} which allows us to define a *transformation* from QRs to ARs based on an equivalence between chains of corresponding *q*s and autosegments. An example of this transformation is given in (3).

This transformation clarifies several aspects of QRs. First, QRs are equivalent to ARs in which the segment is the TBU. That is, *q*s are equivalent to autosegments, whereas *Q*s are equivalent to the vowels to which autosegments are associated. In other words, while SI claim that in QRs “*q* is the tone-bearing unit” (2019:152), our results show that it is *Q*s that are equivalent to the AP notion of a TBU. Thus, instead of solving the autosegmental problem of determining the correct TBU (SI 2019:155–156), QRs revert to the original autosegmental conception of the segment as the TBU (as in, e.g., Leben 1973, Goldsmith 1976). We discuss below how this weakens SI’s claim that QRs better capture tone-consonant interaction.

Furthermore, a transformation between structures also implies a translation between constraints over these structures, and we establish that this shows two aspects of QRs to be equivalent to ARs. First, *Q*-correspondence—that is, correspondence between *Q* segments—cannot be derived from *q*-correspondence between *q* subsegments (or vice versa). SI argue that QRs are superior to ARs in their ability to capture patterns in which entire contours assimilate to each other. With ARs, assimilation of entire contours as units cannot be captured with autosegmental spreading, and thus must be captured with a separate mechanism checking the identity of TBUs. SI criticize ARs for this dual model of assimilation, but QRs also employ a dual model of assimilation: one based on *q*-correspondence and one based on *Q*-correspondence, which we show to be two distinct mechanisms. We illustrate this concretely with SI’s analysis of Changzhi contour assimilation (Hou 1983, Duanmu 1994), which forms one of SI’s main empirical examples of tone patterns that they claim are better captured with QRs. We show that *Q*-correspondence is essentially analogous to vowel agreement in ARs.

Second, this translation between constraints shows that intervowel *q*-correspondence and intravowel *q*-correspondence are equivalent to constraints enforcing sharing of an autosegment and banning contours, respectively. Thus, the analyses using these constraints throughout SI’s article, including their analyses of Basaá and Mende, are no less possible using ARs.

In fact, while we focus on a small number of constraints, this transformation guarantees that for *any* surface constraint in Q-Theory, there is an equivalent constraint in AP that behaves exactly the same way and is no less complex. This result, which is based on established logical and model-theoretic techniques (Enderton 1972, Courcelle 1994, Courcelle, Engelfriet, and Nivat 2012), is presented in detail in Danis and Jardine 2019. To summarize, Danis and Jardine give *first-order translations* from QRs to ARs and vice versa. First-order logic is a weak predicate logic that is still able to express most, if not all, phonological constraints (Scobbie, Coleman, and Bird 1996, Potts and Pullum 2002) and thus seems an appropriate upper bound for the complexity of phonological generalizations (Rogers et al. 2013). Danis and Jardine’s translations show that, for any constraint that can be written in the first-order logic of ARs, an equivalent constraint can be written in the logic of QRs, and vice versa. Thus, the two representations are equally expressive with respect to the kinds of constraints that one would write for phonological analyses. The purpose of this article is to detail the consequences of this formal result for phonological theory.

While these results contradict some of SI’s assertions, we do not claim that ARs and QRs are entirely identical, and we do not reject the utility of QRs. Our results show that ARs and QRs are equivalent with respect to their expressivity, but there are ways in which the two representational theories are conceptually distinct. Chief among them is the QR axiom that all segments are made up of exactly three parts. This claim is not usually implemented in ARs, and thus QRs allow phonologists to ask questions about phonology that they might not pose otherwise. Rather than negating this distinction, the formal results we present highlight it as the true difference between QRs and ARs. Our article thus follows in the spirit of Kornai and Pullum (1990), who formally analyze X-bar theory in order to distinguish its true novel theoretical contributions from specious differences with context-free grammars. We also find that interesting questions remain regarding the consequences of different assumptions about Q-Theory and correspondence, which we discuss in detail below. This thus highlights the value to phonological theory of the rigorous model-theoretic analysis of phonological structure.

It is worth noting that this article focuses on constraints governing surface correspondence in QRs and the equivalent markedness constraints in ARs. This is because it is the nature of these surface constraints that forms the bulk of SI’s arguments. Additionally, this article focuses on the structural similarities between ARs and QRs. We thus leave an exploration of the relevant input-output correspondence constraints to future work. The contribution of this article can be viewed as follows: *if* there is a difference in expressivity between Q-Theory and AP, then it lies in input-output faithfulness, and not in constraints on surface well-formedness (as SI argue).

This article is structured as follows. Section 2 introduces model theory and uses it to define each representational theory. Section 3 shows how a transformation from one representation to the other can be defined. Section 4 then gives examples of how constraints can be translated from one theory to the other. Section 5 discusses the consequences and scope of these results, as well as the remaining questions, and section 6 concludes.

## 2 Defining the Representations

### 2.1 Model Theory

Model theory is the mathematical study of structure (Enderton 1972, Libkin 2004) and can be used to rigorously define phonological constraints and phonological theories (Bird 1995, Potts and Pullum 2002, Graf 2010a,b). In intuitive terms, a *model* of a structure explicitly describes the composite elements of the structure and the relationships between these elements. Formally, a *relational model* is a tuple

with a *domain D* and a finite set of *k* relations over *D*. Here we only need to consider models where each *R _{i}* is a

*binary*relation

*R*⊆

_{i}*D × D*or a

*unary*relation

*R*⊆

_{i}*D*. A

*signature*is a fixed set of relations {

*R*

_{1},

*R*

_{2}, . . . ,

*R*}. For example, strings of

_{k}*a*s and

*b*s (e.g.,

*ab, ba, babab, bbbb*, etc.) can be represented by relational models of the signature in (4).

(4) {⨞,

P,_{a}P}_{b}

In a string model, *D* is the set of positions in a string, ⨞ is a binary *successor* relation indicating the order on positions, and *P _{a}* and

*P*indicate the sets of positions occupied by an

_{b}*a*and a

*b*, respectively. For example, the string

*abba*can be represented by the model given in (5).

The string *abba* contains four elements, enumerated by the domain set *D* = {1, 2, 3, 4}. The relation ⨞ in (5a) explicitly shows that element 1 is succeeded by element 2, element 1 is succeeded by element 3, element 2 is succeeded by element 3, and so forth. The unary relation *P _{a}* represents the set of elements labeled

*a*, namely, 1 and 4. Likewise,

*P*represents the set of elements labeled

_{b}*b*, namely, 2 and 3. The figure in (5b) depicts this model visually, with the elements in

*D*depicted as circles, ⨞ depicted as arrows, and the elements belonging to

*P*and

_{a}*P*marked with the labels

_{b}*a*and

*b*, respectively.

We can represent any string of *a*s and *b*s with a similar model using the signature {⨞, *P _{a}*,

*P*}. The converse, however, is only true if we assume some axioms on the model: a model

_{b}*D*;⨞,

*P*,

_{a}*P*represents a string if and only if ⨞ is a well-formed successor relation and the sets of

_{b}*P*and

_{a}*P*partition

_{b}*D*into distinct sets.

In this way, models allow us to explicitly define classes of structures, and ultimately to define constraints that compute over these structures (see also Potts and Pullum 2002, Graf 2010a,b). In this article, we use model theory to explicitly define and compare ARs and QRs.

### 2.2 Autosegmental Theory

SI (2019:137) characterize Autosegmental Phonology (AP) as having these two defining properties:

(6)

“[F]eatures exist autonomously, each on its own independent tier, organized by a central timing skeleton.”

“[T]he association between elements on featural tiers and elements on the timing tier can be one-to-one, one-to-many, many-to-one, or even zero-to-one in the case of floating features or featurally underspecified timing units.”

For classical ARs, the timing tier consists of segments (or more properly, root nodes), with the signature below. We can follow standard formal definitions of ARs (Goldsmith 1976, Coleman and Local 1991, Kornai 1995, Jardine 2014). As we focus on tone, we make two assumptions: first, we focus on two-tier ARs in which the “features” are high (H) or low (L) tones; and second, the timing tier nodes are vowels. Treating vowels as the TBU is no longer standard (Yip 2002), but as we argue throughout the article (in particular, in section 3.3), this is the closest autosegmental equivalent to QRs.

In this signature, ⨞_{A} and

_{A}are binary relations referring to successor and autosegmental association, respectively.

*V*

_{A},

*H*

_{A}, and

*L*

_{A}are all unary relations that label nodes as either vowels or H or L tones. For simplicity, but without loss of generalization, we omit the consonants from the representation.

For example, in the model in figure 1, H_{1} is associated to both V_{3} and V_{4} (i.e., these pairs are members of

_{A}), L

_{2}is associated to V

_{4}, H

_{1}precedes L

_{2}, and V

_{3}precedes V

_{4}. This model also shows a one-to-many relation from tones to vowels (from H

_{1}) and a one-to-many relation from vowels to tones (from V

_{4}). However, additional axioms are required to ensure a well-formed AR, such as the No-Crossing Constraint (Goldsmith 1976, Hammond 1988). We refrain from detailing these axioms here; for a discussion of axiomatizations of ARs, see Jardine and Heinz 2015.

#### AR model for the Basaá word [hólôl] ‘ripen’

We further assume an axiom that a vowel can associate to at most three tonal autosegments. This is not often made explicit in the phonological literature, but formal work has shown how axioms limiting the length of vowel-to-tone associations can be naturally expressed in ARs (Yli-Jyrä 2013, Jardine 2014, Jardine and Heinz 2015).

### 2.3 Q-Theory

The foundational idea of Q-Theory is that representations consist of *Q* segments (*Q*s) subdivided into three *q* segments (*q*s) each. Featural content—in this case, tone—is then part of the *q*s. Another assumption of Q-Theory is that QRs also include surface correspondence relations relating *Q*s and *q*s. As *q*s carry featural information, agreement in tone is mediated through the *q*s.

However, the larger *Q* units are also essential to QRs. (“*Q* segments are strings of *q*’s that the grammar can refer to” (SI 2019:151); the constraints SI present (2019:153) indeed refer to *Q*s.) SI (2019:151) suggest that the *Q*s may be “emergent” and not fundamental elements of the grammar. However, much of the extant literature on Q-Theory makes extensive use of them (Inkelas and Shih 2013, Shih and Inkelas 2014). As an example, SI’s analysis of the contour patterns in Changzhi and Tianjin (SI 2019:160, 162) requires correspondence at the *Q* level, and necessarily not at the *q* level. (In fact, SI state that “reference to the segment as a whole (*Q*) is crucial for capturing contour behavior” (2019:152).) We thus must accept *Q*s as an essential part of QRs under the current version of the theory.

Thus, for every vowel, a QR has a *Q* element representing the vowel at the segmental level and a series of three ordered *q* elements representing the vowel’s featural information at the subsegmental level. This is depicted explicitly for the QR from (1a) in figure 2. (As with figure 1, we abstract away from consonant *Q*s.) Elements 7 and 8 are *Q*s, and elements 1 through 6 are *q*s.

#### QR model for the Basaá word [hólôl] ‘ripen’

Crucially, because the featural content is not carried by the *Q*s, yet identity between *Q*s is based on the featural content of their constituent *q*s (SI 2019:153), there must be some relation connecting a particular *Q* to its composite *q*s. In model-theoretic terms, the QR signature must include a binary relation that relates *Q*s to *q*s. Let us call this *A*_{Q}, which is depicted as straight lines between *Q*s and *q*s in figure 2. We need not necessarily call *A*_{Q} “association,” but as a binary relation associating elements on different tiers, it is essentially identical to the *A*_{A} relation in ARs.

Finally, QRs also contain a correspondence relation; let us call this *R*_{Q}. This is depicted as curved lines between *q*s in figure 2. Three things are worth mentioning about *R*_{Q}. First, while SI claim that their correspondence relation is nontransitive (2019:140), their correspondence constraints are sensitive to *correspondence chains* of transitively connected chains of corresponding segments. SI state, “[A] sequence of three identical consecutive segments *S* in a grammar requiring that identical segments correspond would satisfy that constraint as follows: *S*_{1}*S*_{1,2}*S*_{2}, where coindexation encodes correspondence” (2019:140). That is, in such a configuration the first *S*_{1} and third *S*_{2} satisfy any correspondence constraints. This directly contradicts the statement that correspondence is intransitive—it is because *S*_{1} corresponds to *S*_{1,2}, and *S*_{1,2} corresponds to *S*_{2}, that *S*_{1} and *S*_{2} satisfy the constraint, by transitivity. For simplicity, we encode this transitivity in the relation itself.^{3} However, in our depictions of correspondence (as in figure 2), for visual clarity we only depict consecutive correspondence relations.

The second important assumption about correspondence is that it implies identity: “Our operating assumption is that GEN does not even produce candidates in which elements obey CORR but violate the associated IDENT-XX constraint” (SI 2019:155). This is crucial for the transformations between structures discussed below.

Third, correspondence can operate either between *q*s or between *Q*s; see section 4.1.

Thus, in a QR we have

(8)

S_{QR}= {⨞_{Q}, R_{Q},A_{Q},Q_{Q},H_{Q},L_{Q}}.

In *S*_{QR}, ⨞_{Q} is the ordering relation on the *q* tier and the *Q* tier (which functions identically to the ⨞_{A} relation in ARs), *R*_{Q} is the correspondence relation, and *A*_{Q} is the “association” relation between *q*s and *Q*s. There are also three unary relations: *Q*_{Q} identifying *Q*s, and *H*_{Q} and *L*_{Q} identifying H-toned and L-toned *q*s, respectively (as only *q*s carry featural information, these featural relations are enough to distinguish the *q*s from *Q*s).

## 3 From One Representation to Another

Visually, the similarities between figures 1 and 2 are clear: if we merge all of the *q*s in correspondence chains in figure 2 (and relabel the *Q*s as Vs), then we obtain the AR in figure 1.^{4} In this section, we describe a step-by-step process that transforms one structure into the other and show how this implies a translation of constraints from one theory to the other. That this transformation is of limited computational complexity—and thus preserves the expressivity of the constraints in each theory—is shown in Danis and Jardine 2019; here, we give an informal presentation of the procedure.

First, we note that the identity of vowels in each theory is based on the string of units to which they are associated—autosegments in ARs, and *q*s in QRs. Thus, for example, an AR representing a vowel associated to a single H tone is equivalent to a QR with a vowel *Q* associated to three corresponding H-toned *q*s, as shown in (9a). Similarly, a vowel associated to a HL sequence in an AR corresponds to a QR with a vowel *Q* associated to a H-toned *q* and two corresponding L-toned *q*s. The correspondence between *q*s is based on our assumption of an equivalence between autosegments and correspondence chains of *q*s; we will describe this in more detail momentarily.

In general, this method guarantees a way to convert an AR into a QR (and vice versa) on the basis of tone strings to which each vowel is associated. As illustrated above, implementing the equivalence between autosegments and *q*s is a matter of assigning correspondence relations in the QR according to the number of autosegments. That is, for each tone *t* associated to a vowel in the AR, there is a correspondence chain *c* in the QR. In the following section, we give a step-by-step method for transforming ARs into QRs, and then QRs into ARs, based on this idea. Throughout, we show that this is possible due to SI’s assumption that correspondence implies identity.

### 3.1 From Autosegmental Representations to Q-Theory Representations

As illustrated in (9), we can translate individual vowels in an AR to vowel *Q*s in a QR on the basis of the string of autosegments associated to each vowel. Such a translation is carried out in full in (10). Correspondence between *q*s in (10) is indicated with indices.

Two assumptions are required for this transformation. The first is that correspondence implies identity in the QRs, as also assumed by SI. Without this, we could not equate *q*-correspondence chains to autosegments. Going from ARs to QRs, the chart in (10) assumes that for each autosegment we can create a string of corresponding, like-toned *q*s. Going from QRs to ARs, (10) assumes that a chain of like-toned *q*s can be collapsed into a single autosegment. If correspondence did not imply identity, then *q*-correspondence chains could include both H-toned and L-toned *q*s, and there would be no way to equate these with a single tonal autosegment.

The second assumption is that there is a bound on the number of tones associated to vowels in the ARs, because of the bound on the possible contrasts in QRs. SI propose the bound on *q*s for empirical reasons: contours (whether tonal or segmental) never appear to be more than three units long. As has already been stated, this is not an unreasonable constraint to put on ARs; for example, Jardine and Heinz (2015) show how this can be done in a natural way.

Note that not all possible correspondences among elements in the *q* strings are attested in the table. Some represent contrasts that appear to have no extensional consequences: for instance, a string H_{2}H_{2}H_{1} of H *q*s in which the first two *q*s are in correspondence but the third is not. Such distinctions are not crucial to any of SI’s analyses; indeed, they may not play any role in any grammar. Candidates consisting of identical segments but not in surface correspondence are all harmonically bounded in the systems investigated by Bennett and DelBusso (2018), supporting Bennett’s (2013:33) claim that “while there are many candidates that differ only in their surface correspondence structure, the majority of these are irrelevant for any given interaction.” Thus, while it is true that QRs can make distinctions that ARs cannot, these distinctions do not have any extensional consequences, and therefore we ignore them for the purposes of this translation.

However, there are some contrasts that do have empirical consequences: SI entertain the possibility of distinctions between *q* strings such as HLL versus HHL (as reported in Dinka; Remijsen 2013, SI 2019). This could be relaxed by allowing ARs that violate the Obligatory Contour Principle (OCP); thus, the Dinka contrast could similarly be captured as the difference between a vowel associated to a HL string and a vowel associated to a HHL string. However, as SI’s analyses do not make use of the full range of these possibilities, the chart in (10) is sufficient for our purposes.

Given this chart, we can posit a step-by-step process that transforms an AR into a QR. This is given in (11).

(11)

Algorithm for transforming ARs into QRs

For each V in the AR, place a Q in the QR associated to a string of

qs according to the chart in (10).For each pair of Vs

V_{1},V_{2}in the AR associated to the same autosegment, for their equivalentsQ_{1}andQ_{2}in the QR, draw a correspondence chain between theirqs.For each pair of Vs

V_{1},V_{2}associated to identical strings of autosegments in the AR, draw a correspondence relation between their equivalentsQ_{1}andQ_{2}in the QR.

Step (11.1) simply creates a string of *Q*s associated to *q*s on the basis of the chart in (10). This step is illustrated in (12) using two different ARs as inputs.

On the left in (12a) is the AR model for the Basaá word [hólôl] ‘ripen’. Given the instructions in step (11.1), the algorithm creates two *Q*s. Because the first V in the AR is associated to a single H, the first *Q* is associated to a string H_{1}H_{1,2}H_{2} of H-toned *q*s belonging to a single correspondence chain, as specified in the autosegment/*q*-string translation chart in (10). As the second V in the AR is associated to a HL string, the second *Q* is associated to a string H_{1}L_{2}L_{2} of *q*s, the first being a H-toned *q* and the second two being L-toned *q*s in correspondence.

On the left in (12b) is an AR that shows “plateauing” of a H tone across three vowels. Following step (11.1) and the autosegment/*q*-string translation chart in (10), this maps to three *Q*s, each associated to a string H_{1}H_{1,2}H_{2} of corresponding *q*s.

The next step, (11.2), completes the implementation of the idea that autosegments are equivalent to *q*-correspondence chains, which is how to equate autosegments associated to multiple vowels with *q*-correspondence chains that span across multiple vowel *Q*s. This is simple: for any two vowels *V*_{1} and *V*_{2} associated to the same autosegment in an AR, ensure that there is a correspondence chain between the *q*s of the equivalent *Q*_{1} and *Q*_{2} in the QR. In our diagrams, this amounts to connecting the final *q* of *Q*_{1} and the initial *q* of *Q*_{2} for each successive pair of *Q*s whose equivalent vowels share a tone in the AR.^{5} This is illustrated in (13) for both our contour and plateau examples. (New correspondence relations are highlighted in bold.)

In (13a) and (13b), the first and second vowels in the ARs are associated to the same H autosegment. Thus, in their QRs, an additional *q*-correspondence relation is drawn between them. In (13), this also happens between the second and third vowel *Q*s, as the vowels in the AR also share a H autosegment. Thus, we have created a single *q*-correspondence chain in the QR for each autosegment in the AR. Note that the corresponding *q*s are all identical because they derive from the same autosegment—thus, this transformation is only possible if correspondence implies identity in the QR.^{6}

Finally, step (11.3) addresses correspondence between *Q*s. As SI explain, the tone value of a *Q* is based on its component *q*s. (More on this in section 4.1.) As correspondence of *Q*s is based on this value, starting from an AR, we can create correspondence between vowel *Q*s based on the strings of autosegments associated to their equivalent Vs in the AR. Thus, step (11.3) draws correspondence relations between *Q*s in the QR whose equivalent Vs are associated to matching strings of autosegments. This is illustrated with our running example in (14). (New correspondence relations are highlighted in bold.)

In (14a), the two vowels in the AR are associated to a H string and a HL string of autosegments, respectively. Thus, their corresponding *Q*s in the QR do *not* correspond. In contrast, in (14b), the first and second vowels in the AR are both associated to a matching string of autosegments—namely, H—and so their equivalent *Q*s correspond. The same goes for the second and third vowels and (though not depicted) the first and third vowels (again, we are considering transitive correspondence—see section 2.3). Note that this results in QRs in which identity implies correspondence—it is impossible to generate structures in which *Q*s are identical but do not correspond. As we have noted already for *q*s in the discussion following (10), this contrast does not play any role in the grammar and thus omitting it has no effect on our goal of showing that the two theories are equally expressive.

A third example emphasizes that *Q*-correspondence is based on the autosegmental *strings* and not the sharing of autosegments. In (15), two H-toned vowels are separated by a third, L-toned vowel; step (11.3) of our transformation would draw a correspondence relation between the *Q*s equivalent to these two H-toned vowels. (The crucial correspondence relation is highlighted in bold.)

Note that, in a sense, this reduces *Q*-correspondence to agreement. This appears to be a consequence of the assumption that correspondence implies identity, and so is entirely compatible with the analyses given by SI. For more discussion, see section 4.1.

Thus, the algorithm in (11) produces a QR from any AR whose contours are no more than three autosegments long. Again, this is based on the idea that correspondence chains of *q*s are equivalent to autosegments—an assumption made possible by the fact that corresponding elements in the QRs must be identical.

## 3.2 From Q-Theory Representations to Autosegmental Representations

We now give the algorithm for the reverse operation: creating an AR from a QR. Following the notion that *q*-correspondence chains are equivalent to autosegments, we simply “merge” members of a *q*-correspondence chain into a single autosegment. This is outlined explicitly in (16).

(16)

Algorithm for transforming QRs into ARs

For each Q in the QR, create a V in the AR.

For each correspondence chain of

qs:

Draw an autosegment of the same tone value in the AR.

Associate that autosegment to each

V_{1},V_{2}, . . . ,Vfor each_{n}Q_{1},Q_{2}, . . . ,Qthat was associated to some_{n}qin the correspondence chain.

Step (16.1) is simple: vowel *Q*s in the QR are equivalent to Vs in the AR. Step (16.2) is split into two substeps, which are best illustrated separately. The first, which creates an autosegment for each correspondence chain, is illustrated in (17) with our running contour and plateau examples.

On the left in (17a) is the QR of the Basaá word [hólôl] ‘ripen’. There are two *q*-correspondence chains here: one consisting of a series of H-toned *q*s, followed by another consisting of a series of L-toned *q*s. Thus, in the AR, we draw one H autosegment equivalent to the chain of H-toned *q*s, followed by one L autosegment equivalent to the chain of L-toned *q*s. On the left in (17b) is a QR of a plateau of H-toned vowels. As all *q*s correspond, step (16.2i) creates a single H tone in the equivalent AR.^{7}

Step (16.2ii) generates association lines in the AR based on associations in the QR, as illustrated in (18).

Step (16.2ii) states that for each correspondence chain, for every *Q _{i}* that is associated to some

*q*in that chain,

_{i}*Q*’s equivalent

_{i}*V*in the AR is associated to the autosegment

_{i}*a*equivalent to that correspondence chain. For example, in (18a), both vowel

_{i}*Q*s are associated to

*q*s in the chain of corresponding H-toned

*q*s. Thus, both vowels in the AR are associated to that chain’s corresponding H autosegment. In contrast, only the second vowel

*Q*is associated to

*q*s in the chain of corresponding L-toned

*q*s, so only the second vowel in the AR is associated to the equivalent L. In (18b), all

*Q*s are associated to some

*q*in the chain of corresponding H-toned

*q*s, and so all vowels in the AR are associated to the same tonal autosegment.

This illustrates how the two steps in (16) can generate an AR from any QR. Note that the algorithm does not need to make any reference to correspondence between *Q*s. As we will show in section 4.1, this is due to the fact that correspondence can be replaced with tonal identity in the vowels. Thus, the function played by *Q*-correspondence can be assumed by existing parts of the representation (namely, the association relation linking *Q*s/vowels to *q*s/tones).

## 3.3 Locality of Correspondence and Consonants

One last point with respect to the correspondence between *q*s bears clarifying. It may be that long-distance correspondence of *q*s is necessary for correspondence chains of V *q*s to “skip” the *q*s of Cs that are not specified for tone. This appears equivalent, in ARs, to Cs not projecting tones on the tonal tier. In terms of our transformation between ARs and QRs, this amounts to not assigning *q*s unspecified for tone in the QR to any tone in the AR. This is illustrated in (19), which explicitly adds a toneless consonant to the representation.

In the QR in (19), the middle consonant *Q* is unspecified for tone, indicated by the Ø label on its *q*s, and the correspondence relation between the toned-vowel *q*s “skips” these toneless *q*s. This requires a slight modification to our rule positing correspondence between *q*s of adjacent vowels, but it amounts to redefining “adjacent” vowels as those for which only Cs intervene. If we additionally posit that toneless Cs in the QR project no autosegment in the AR, then we have captured this equivalence and maintained the notion that autosegments in the AR correspond to correspondence chains of *q*s in the QR.

We should point out that SI posit that it *is* possible for Cs to bear tone: in their analysis of depressor consonants in Siswati, Cs are given L-toned *q*s. Given the transformation we have specified, this is equivalent in ARs to consonants that project a L tone. This was the standard analysis of depressor consonants in AP (Cassimjee and Kisseberth 1992). Thus, while SI argue that QRs better capture tone-consonant interactions, in fact their argument calls for a return to the segment as the TBU, as this more accurately captures tone-consonant interactions.

## 4 A Translation between Constraints

Here, we apply the transformation outlined in the previous section to the constraints of Q-Theory to compare how tone patterns are analyzed over QRs as opposed to ARs. We present two findings, both of which significantly weaken SI’s arguments against ARs. First, the *Q*-correspondence at the core of SI’s analysis of contour assimilation is a mechanism distinct from *q*-correspondence, contrary to SI’s argument that only ARs must posit a separate assimilation mechanism for these patterns. Second, *q*-correspondence is identical to autosegmental spreading in ARs, and thus SI’s constraints on “intersyllable similarity” and “intrasyllable similarity” are equivalent to the AR concepts of sharing of autosegments and bans against contours, respectively.

### 4.1 Vowel Identity and Q-Correspondence

We first explore constraints concerning *Q*s in Q-Theory, which SI argue more insightfully capture assimilation and dissimilation patterns in tone languages in which entire contours behave as units. However, these constraints are identical to agreement constraints between vowels in ARs. Furthermore, we find that *Q*s cannot be “emergent” from *q*s, as SI claim (2019:151). This means that *Q*-correspondence is necessarily distinct from *q*-correspondence. Thus, in Q-Theory assimilation at the segmental level and assimilation at the subsegmental level are captured by two distinct mechanisms, a situation for which SI criticize AP.

One type of pattern that SI claim is more directly captured in Q-Theory is reduplicative contour assimilation, as in Changzhi (Hou 1983, cited in Duanmu 1994). In Changzhi, the diminutive suffix /-tə^{ʔ}_{535}/ takes on the tone of its root; thus, /təu_{53}-tə^{ʔ}_{535} / ‘bean-DIM’ is realized as [təu_{53}-tə^{ʔ}_{53}]. SI analyze this as a CORR-VV constraint, which forces adjacent vowels to correspond and thus be identical. Thus, [təu_{53}-tə^{ʔ}_{53} ] surfaces because *[təu_{53}-tə^{ʔ}_{535} ] is ill-formed. This is shown with QRs in figure 3, with the contours 53 and 535 represented as HL and HLH, respectively, in keeping with the notation used throughout this article.

#### QRs for the (a) identical [təu_{53}-tə^{ʔ}_{53}] and (b) nonidentical *[təu_{53}-tə^{ʔ}_{535} ] in Changzhi

In figure 3, (b) does not satisfy the CORR-VV requirement because the two adjacent *Q*s are not in correspondence. In contrast, (a) does satisfy the CORR-VV constraint, and its concomitant identity requirement, because the two vowels are in correspondence, and the *Q*s are associated to identical strings of *q*s. In general, a constraint on *Q* identity requires that for *Q*_{1} and *Q*_{2} to be in correspondence, if the *q*s associated to *Q*_{1} form a string *w* of tones, then the *q*s associated to *Q*_{2} must also form a string *w*.

This notion of identity does not change in autosegmental terms. To show this, we apply our transformation to the representations in figure 3. We can conceive of vowel identity in ARs as identical to *Q* identity in QRs: two vowels are identical if they are associated to identical strings of tones. This obtains the distinction in figure 4, as in figure 4a both vowels are associated to HL tone strings, but in figure 4b the first vowel is associated to a HL tone string while the second is associated to a HLH tone string.

#### ARs for the (a) identical [təu_{53}-tə^{ʔ}_{53}] and (b) nonidentical *[təu_{53}-tə^{ʔ}_{535}] in Changzhi

SI criticize a vowel-identity mechanism for assimilation in ARs, as then AP does not have a unified theory of assimilation (SI 2019:159–160). Their reasoning is that in AP, assimilation is operationalized as spreading, but in these tone contour cases we must resort to tone copying. However, Q-Theory also has two distinct mechanisms for accomplishing assimilation: *Q*-correspondence and *q*-correspondence. To see that *Q*-correspondence (and thus identity) is not derived from *q*-correspondence, imagine that we attempt to derive *Q* identity from *q*-correspondence. To ensure the identity of the *q* strings, we must check that the first *q* of *Q*_{1} is the same as the first *q* of *Q*_{2}, that the second *q* of *Q*_{1} is the same as the second *q* of *Q*_{2}, and so on with the third *q*s. This implies a one-to-one correspondence between *q*s as depicted in figure 5a.

#### A QR showing identity of *Q*s through *q*-correspondence (a), and an equivalent AR with line crossing (b)

SI posit no such *q*-correspondence; rather, correspondence between *Q*s is predicated on the fact that the *Q*s are adjacent, and the identity between the two *Q*s is determined through the tonal identity of their respective *q* strings. Thus, *Q*-correspondence is a mechanism independent from *q*-correspondence.

Furthermore, if we apply our transformation to figure 5a and merge all corresponding *q*s, we obtain a line-crossing AR in figure 5b. SI point out that such an analysis of contour in ARs is impossible. However, our transformation shows that an equivalent correspondence relation between *q*s is similarly impossible in Q-Theory. Thus, the two theories agree that contour agreement and spreading agreement are handled by distinct mechanisms.

### 4.2 Translating between q Constraints and Autosegmental Constraints

We now further explore the equivalence of *q*-correspondence to autosegmental spreading, basing the discussion on SI’s analysis of Basaá. This is briefly illustrated in tableau (20) for the input /h(ó ô ó)l(ó ó ó)l/, where an all-H-toned *Q* is followed by an all-L-toned *Q*. Again, we focus here on surface well-formedness constraints, setting aside faithfulness (though see section 5 for discussion). The key constraint for this analysis is CORR-v:$:v, which states that vowel *q*s separated by a syllable boundary must be in correspondence. When CORR-v:$:v outranks CORR-v::v, the constraint that requires adjacent *q*s within a *Q* to correspond, the winning candidate has the last *q* of the first vowel and the first *q* of the second vowel in correspondence. Since correspondence entails identity, the two *q*s also agree, thus creating a contour tone.

In (20), candidate (a), which has a contour, satisfies CORR-v:$:v because the correspondence relation with index 3 bridges two *q*s across syllables. Candidate (a) thus violates CORR-v::v (and input-output faithfulness), because the vowel-internal *q*s now disagree, but it wins over the faithful candidate (b), which violates CORR-v:$:v.

Let us now look at these constraints more carefully. For brevity, we avoid introducing syllable structure directly into the representation and simply state that vowel *q*s are in different syllables if they are associated to different vowel *Q*s. Given this, CORR-v:$:v enforces a logical implication that states that for two adjacent *q*s, *q*_{1} and *q*_{2}, if they are associated to adjacent vowels, then they must be in correspondence. This is illustrated in figure 6.

#### Visual representation of the requirement of CORR-v:$:v

Figure 6 depicts the requirement of CORR-v:$:v—namely, when *q*_{1} is immediately followed by *q*_{2}, but *q*_{1} and *q*_{2} are associated to distinct *Q*s (*Q*_{1} and *Q*_{2} ), then *q*_{1} and *q*_{2} must be in correspondence. A structure in which they are not in correspondence is forbidden; note that the absence of the relation is crucial. To illustrate, we give the full models of the candidates from (20) in figure 7, with the structures targeted by CORR-v:$:v highlighted in bold.

#### Examples of CORR-v:$:v.

Applying our transformation to the structures in figure 6, we obtain the autosegmental constraint in figure 8. In figure 6, the structure that CORR-v:$:v requires includes two *q*s, *q*_{1} and *q*_{2}, in correspondence. Applying our transformation merges these into the single tonal autosegment *t*_{1}, yielding the required structure in figure 8. Likewise, in the structure that CORR-v:$:v identifies as marked, *q*_{1} and *q*_{2} are not in correspondence. Such a *q*_{1} and *q*_{2} are not merged by our transformation. Thus, transforming the marked structure yields an autosegmental equivalent that states that the marked structure yields *t*_{1} and *t*_{2} associated to adjacent vowels *V*_{1} and *V*_{2}, respectively, with *t*_{1}*not* associated to *V*_{2} . To illustrate, we give the autosegmental equivalents of candidates (20a) and (20b) in figure 9, with the relevant structures highlighted.

#### Autosegmental equivalent of CORR-v:$:v

#### Autosegmental equivalents of the candidates in (20), illustrating satisfaction and violation, respectively, of the autosegmental equivalent of CORR-v:$:v given in figure 8

As the figure illustrates, the autosegmental equivalent of CORR-v:$:v assigns a violation for adjacent vowels that do not share an autosegment. Thus, CORR-v:$:v is essentially equivalent to a constraint enforcing autosegmental spreading, such as SHARE (McCarthy 2010).

For completeness, we briefly explore CORR-v::v the same way. As shown in (20), CORR-v::v assigns a violation for an adjacent pair of *q*s, *q*_{1} and *q*_{2}, that belong to the same *Q* but are not in correspondence. Figure 10 depicts this visually the same way that figure 6 did for CORR-v:$:v. Applying our transformation to this constraint, we get an autosegmental constraint that states that vowels can be associated to a single tone (by merging the corresponding *q*_{1} and *q*_{2} in the left side of figure 10 into a single tonal autosegment *t*_{1} ), but not to more than one.

#### Visual representation of the requirement of CORR-v::v

#### Autosegmental equivalent of CORR-v::v

Thus, we have illustrated that our transformation between ARs and QRs also implies a translation between well-formedness constraints on spreading and contour creation, such that a structure violates a constraint in one theory if and only if its equivalent structure violates the equivalent constraint in the other theory. This is significant, as constraints referring to intersyllable identity and intrasyllable identity play a major role in SI’s analysis of Mende, which SI claim is superior to an autosegmental analysis. However, the results here show that such constraints are equally expressible with ARs.

## 5 Discussion

In arguing for QRs, SI claim that (a) QRs dispense with the need for an association relation; (b) the *q* serves as the TBU; and (c) QRs capture patterns that ARs cannot. By formally investigating these representations, we have shown that none of these claims hold: (a) QRs require a relation identical to association; (b) *Q*s are the closest equivalent to TBUs in ARs, and thus QRs are similar to ARs in older autosegmental theories in which vowels are the TBU; and (c) QRs and ARs are equivalent in terms of constraints they can express. We now discuss the implications and scope of this result.

The argument here focuses mainly on markedness, or surface, constraints. This is primarily because autosegmental relations are a theory of output structure, and it is this power to which SI explicitly compare QRs. A translation between input-output correspondence constraints would be similar, but more complex: as Potts and Pullum (2002) discuss, formalizing input-output correspondence requires including input structures as well as output structures in the representation. As the purpose of this article is to demonstrate the structural similarities between QRs and ARs, we focus on the surface correspondence constraints, but note that our results likely extend to input-output structures as well.

We have focused here on the expressive power of QRs with respect to existing ARs, as well as critically evaluating SI’s arguments regarding the superiority of QRs over ARs. However, one clear empirical prediction made by QRs that is not obvious with ARs is that segments, *all* segments, are made up of three parts. As SI mention (2019:138), this idea has existed for some time with various manifestations, such as Akinlabi and Liberman’s (2001) tonal complexes, Steriade’s (1993, 1994) aperture theory, or articulatory phonology (e.g., Gafos 2002). However, QRs are the first such formalization of segments with an across-the-board abstract tripartite structure. One consequence of this is that contrasts in affricate types (such as Hungarian /ts/ vs. /tʃ/; SI 2019:150) are unified with contrasts in tone contours (such as Dinka HHL vs. HLL; SI 2019:150–151). Having this three-part segment structure hardwired into the theory allows linguists to ask questions they might not have posed otherwise, questions that may lead to new insights, especially when looking beyond just tone.

Thus, we do not wish to point out flaws in the absolute utility of QRs; we merely wish to call into question the argumentation SI invoke in comparing the power of these representations to the power of existing ARs. In fact, our work shows that the central contribution of Q-Theory is neither the abandonment of association nor its replacement with correspondence; rather, it is the statement that segments have a fundamentally tripartite structure. This follows the spirit of Kornai and Pullum (1990), who provide a mathematical definition of (one set of assumptions for) X-bar-theoretic phrase structure, showing that it is no more powerful or predictive than the unconstrained context-free grammars on which it is based. However, through this same method, Kornai and Pullum are able to focus on the parts of the theory that are novel or contentful, and build from there. We hope to do the same for Q-Theory.

Finally, what we have shown is that under a set of assumptions consistent with SI’s description of Q-Theory, its computational power is no greater (or more restricted) than that of classical Autosegmental Phonology. However, seemingly minute changes in the definition of the theory can alter its power in perhaps unexpected ways. For example, Danis and Jardine (2019) find that QRs with unbounded correspondence are likely much more powerful than ARs or than QRs with local correspondence, as we have defined them here. In brief, given an arbitrary binary correspondence relation, one can define constraints over this relation that are much more powerful than those that appear to be attested in the phonological literature. These changes are perhaps not apparent until mathematically defined, as they are here. Future work can thus follow the model of this article (and Danis and Jardine 2019) in formally evaluating different versions of surface correspondence.

This also applies to SI’s assumption of correspondence implying identity. As discussed in the analyses above, this assumption plays a key role in the translation between ARs and QRs—essentially, it means that corresponding *q*s in QRs are analogous to tonal autosegments in ARs. Future work can apply the techniques used here to rigorously study the consequences of relaxing this assumption, but we briefly state our expectations here. This assumption restricts the space of representations under QRs: elements cannot be in correspondence without also agreeing. Relaxing this assumption would allow a greater range of QR structures, including ones that do not have a direct AR analogue. In terms of expressivity of the theory, this means that one could define constraints in Q-Theory that one could not define in AP. However, it is not clear that any such constraints are necessary—SI’s analyses of a variety of tone patterns all assume that correspondence implies identity. Further work can rigorously examine exactly what constraints are allowed when correspondence does not necessarily imply identity, and whether or not surface correspondence allows constraints that are more powerful than necessary (as discussed above with regard to unbounded correspondence).

## 6 Conclusion

Model theory provides a method for rigorously evaluating theories of phonological representation. By doing so, we can clarify what aspects of two theories are truly distinct. We have shown that, formally, QRs are no more or less expressive or simple than ARs in capturing tone patterns. However, moving beyond tone, the tripartite structure of QRs provides a new avenue for investigating and unifying certain segmental phenomena, which do not have clear analogues in AP. Thus, rather than undermining QRs, investigating them in a mathematically rigorous way brings their actual contribution into focus.

## Notes

^{1} “[T]one contours and tone assimilation can be straightforwardly captured in [Association by Correspondence]+Q without recourse to the special representational machinery of autosegments and association lines” (SI 2019:155).

^{2} “Our operating assumption is that GEN does not even produce candidates in which elements obey CORR but violate the associated IDENT-XX constraint” (SI 2019:142).

^{3} If we did not encode this transitivity in the representation, the correspondence constraints themselves would have to be endowed with the ability to detect an unbroken chain of local, intransitive correspondence relations. This is demonstrably beyond the expressive power normally attributed to phonological constraints. To relate this to the notion of first-order definability mentioned in section 1, detecting an unbroken path through an intransitive relation is taking its *transitive closure*, which is well-known not to be first-order definable (see, e.g., Libkin 2004).

^{4} SI treat *q*s as feature bundles, so this description abstracts away from other features. To accommodate other features, we would “pull out” tonal features from corresponding *q*s and put them on a tonal tier in the autosegmental representation.

^{5} To see that this creates a transitive correspondence relation, note that shared association to a tone is transitive: if *V*_{1} and *V*_{2} share *T*_{1}, and *V*_{2} and *V*_{3} share *T*_{1}, then of course *V*_{1} and *V*_{3} also share *T*_{1}.

^{6} There is one more result of this transformation, which is that correspondence between *q*s is limited to adjacent *q*s. Strictly speaking, SI’s definitions appear to allow long-distance correspondence of *q*s. However, their analyses make no explicit use of it. There is one potential case in which it is necessary, with respect to consonants, and this is discussed in section 3.3. For now, as long-distance correspondence between *q*s does not play a crucial role in SI’s analyses, we assume for the rest of the article that *q*-correspondence is restricted to adjacent *q*s.

^{7} This follows the assumption of the chart in (10), in which adjacent, like-toned *q*s correspond. This essentially boils down to the OCP: if we relax this assumption and allow, for example, a string of H-toned *q*s that are not in correspondence, we then get a string of H autosegments in the AR. We forgo discussing the universality of the OCP, as this assumption bears little on our results.

We would like to thank Jeffrey Heinz, Paul de Lacy, and the members of the PhonX (the Rutgers phonetics and phonology reading group) for their thoughts and suggestions, and Stephanie Shih and Sharon Inkelas for discussing their theory with us. We would additionally like to thank three anonymous reviewers for their helpful comments. All errors are our own. Contributions by the third author were made before taking a position at Amazon.

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