Bedouin Arabic multiple opacity with indexed constraints in Parallel OT

Phonological opacity (Kiparsky 1973) is a phenomenon that has sparked many debates in theoretical phonology, including the question of whether the phonological component of language uses extrinsic ordering of rules or processes (e.g., McCarthy 2007, Jarosz 2014). Here, I will consider the case of Bedouin Arabic multiple opacity (McCarthy 2007), which McCarthy argues to be a case in favor of serial computation and extrinsic ordering of processes. As will be reviewed in section 3.2, McCarthy argues that parallel representational analyses in terms of Turbidity (Goldrick 2001) or Colored Containment (Van Oostendorp 2008) are impossible, while intrinsic ordering of processes through morphology-phonology interaction in Stratal OT (Bermúdez-Otero 1999) is also argued to be an unsatisfactory explanation, which only leaves a derivational account with extrinsic ordering (see McCarthy 2007, Jarosz 2014). In this paper, I will offer an account of Bedouin Arabic multiple opacity that, on the contrary, does not require any extrinsic ordering or serial computation in phonology (even if it is compatible with the notion). It is a representational account (analogous to Van Oostendorp 2008, Boersma 2007). that uses multi-level surface representations. However, the levels in this model are formed by various diacritics (indices, Pater 2000), assumed to be discovered by the learner in the course of acquisition. Such diacritics/indices are normally found in analyses of exceptionality (see section 4.1). However, following a proposal I have made in earlier work (Nazarov 2019; see section 4), they can be extended to other phenomena, including opacity. This will be the basis of a Parallel OT analysis of Bedouin Arabic, as will be presented in section 5: the three processes involved in the interaction will be presented as essentially non-interacting due to the activity of indices/diacritics (see also section 3.3). Crucially, the surface realization of segments with particular diacritics is regulated by the grammar, so that just those alternations that occur in the language are allowed by the grammar. This shows that extrinsic ordering is not necessary to account for this particular case. The rest of this paper will be built up as follows. Section 2 will give an overview of the problem of opacity and its connection to exceptions. Section 3 will then discuss the Bedouin Arabic data and how they are claimed to necessitate extrinsic ordering; a short preview of the alternative analysis will also be given. In section 4, I will explain the mechanism needed to make the alternative analysis work in OT. The latter will then be presented in section 5. Finally, section 6 will discuss the consequences of this analysis for our understanding of opacity and phonological grammar, and will offer some concluding remarks.


Nazarov
(1) Definition of opacity following McCarthy (1999) A process P = A → B / C_D is opaque iff the application of some other process, Q, leads to: a. instances of A → B outside of C_D (i.e. P overapplies), or b. instances of CAD on the surface (i.e., P underapplies).
Patterns of the overapplication kind include Raising in Canadian English (Chomsky 1956, Chambers 1973, a process that changes diphthongs /aɪ,aʊ/ to [ʌɪ,ʌʊ] before voiceless consonants, as in (2a). Another process, Flapping, turns /t,d/ into a voiced flap, [ɾ], in certain intervocalic environments, as in (2b). Crucially, in instances of /aɪtV/ and /aʊtV/ (in relevant prosodic environments), Raising changes the diphthong to [ʌɪ/ʌʊ] before voiceless /t/, but Flapping changes /t/ to a voiced [ɾ], yielding instances of /aɪ/ → [ʌɪ] outside the proper environment: before a voiced instead of a voiceless consonant. This, by clause (1a), is a form of overapplication opacity. (See Nazarov 2019 for an account of Canadian Raising in the same framework as pursued here.) (2) Underapplication is exemplified by both interactions in Bedouin Arabic that will be discussed here (see section 3.1 for a fuller description). Briefly, Raising in Bedouin Arabic changes /a/ to a high vowel ([i] or [u] -see McCarthy 2007:189-190 for the exact conditioning) when it occurs before a CV sequence, see (3a). In addition, final obstruent-sonorant sequences are broken up by an epenthetic vowel, as in (3b). Finally, there is a Syncope process, which deletes high vowels before a CV sequence, as in (3b). However, as can be seen in (3ab), the surface configuration that should trigger Syncope, [iCV], is, in fact, attested in surface forms when Raising or Epenthesis applies, which is an instance of underapplication by clause (1a). In addition, Raising also underapplies when Epenthesis is applied: the surface configuration [aCV] is created by Epenthesis when the configuration /CaCR#/ (R = sonorant) occurs -but Raising does not apply in these cases, as shown in (3d). Thus, there is a nested underapplication interaction here: Syncope underapplies because of Raising, while Raising itself underapplies due to Epenthesis.
(3) Raising in Bedouin Arabic: underapplication a. Raising b. Epenthesis c. Syncope (underapplies in a.,b.) /samiʕt/ → [simiʕt] 'I heard' /libn/ → [libin] 'clay' /kitib/ → ktib 'itM was written' d. Raising underapplies /ɡabl/ → [ɡabil] 'before' As shown by Idsardi (2000) and others, over-and underapplication patterns of this kind generally cannot be accounted for in Parallel OT (although see Baković 2011 for particular kinds of opacity that can be analyzed in Parallel OT). The problem is that, in overapplication patterns, there is no Markedness constraint that can motivate the overapplication just when another process applies, while in underapplication patterns, there is no Faithfulness constraint that can stop the opaque process from applying whenever another process applies. One solution to this problem is introducing extrinsic ordering of stepwise optimization steps in serial forms of OT (OT-CC, McCarthy 2007; Harmonic Serialism with Serial Markedness Reduction, Jarosz 2014).
Extrinsic ordering of phonological processes was central in SPE (Chomsky & Halle 1968) and has been applied in generative analyses of opaque interactions since at least Chomsky (1954). It may be defined as follows: (4) Extrinsic ordering is a language-specific, obligatory ordering of two processes in a derivation that is unpredictable from anything else in that derivation (forms, rules, interactions with morphology…).
Rule ordering (Chomsky & Halle 1968 While extrinsic ordering is a very effective way of accounting for opacity, it is an open question whether its power is truly needed in phonological grammars is a hotly debated topic. McCarthy (2007) provides arguments in favor of this necessity on the basis of the Bedouin Arabic data that will be re-examined in this paper -see section 3.2 for a summary of McCarthy's arguments. While other arguments for extrinsic ordering exist, I will focus on Bedouin Arabic only, and show that an account without extrinsic ordering is possible.

Connection between opacity and exceptionality
As could be seen in (1) above, taken from McCarthy (1999), opacity may be defined in terms of over-and underapplication. However, over-and underapplication also applies to exceptionality: a process with exceptions either applies outside its usual context, or fails to apply when the right context is present. In fact, if the interacting process Q in (1) is replaced by "lexical items", we obtain a definition of exceptions: 2 (6) Definition of exceptions A process P = A → B / C_D is opaque iff there are lexical items that exhibit: a. instances of A → B outside of C_D (i.e. P overapplies), or b. instances of CAD on the surface (i.e., P underapplies).
This parallel -that both opacity and exceptions are both types of over-or underapplication -is the basis of the type of account pursued here. According to the parallel definitions in (1) and (6), opacity is a type of exceptionality where one process, Q, creates exceptions to another process, P -as opposed to the lexicon creating exceptions to the process. This link is further strengthened by work that views opacity either as being always exceptional (e.g., Sanders 2003), or sees opacity as arising from language-specific, non-universal constraints (Pater 2014). The major theories of exceptionality in OT are indexed constraints (Kraska-Szlenk 1995, Pater 2000 and cophonology theory (Inkelas & Zoll 2007). Here, indexed constraints (see section 4.1) will be extended to accommodate opaque mappings. The intuition is that, while an exceptionful process is conditioned by indices/diacritics that have a phonologically arbitrary distribution (i.e., any morpheme may be marked as exceptionally (not) undergoing a process), an opaque process is conditioned by indices that have a phonologically systematic distribution (e.g., only morphemes that have an underlying (low) vowel in a Nazarov particular position may be marked as undergoing the process). As will be explained in section 4, this is done by extending indices to be binary features on specific segments. The exact way in which this helps in accounting for Bedouin Arabic will be explained in section 5.

Bedouin Arabic: data and previous analyses
3.1 Bedouin Arabic data The data discussed here are taken from McCarthy (2007:Chapter 4.3), whose description is based on Al-Mozainy (1981) along with additional data points. In the multiple opaque interaction that I will focus on, three processes are important: Syncope, Raising, and Epenthesis. I omit interactions with stress, height harmony, consonant place, and metathesis. Syncope, (7a), deletes short high vowels in a non-final open syllable (in Bedouin Arabic, this is equivalent to 'before CV'). However, Raising, (7b), turns short low vowels into short high vowels in the same environment (before CV). Finally, Epenthesis, (7c), inserts [i] to break up a word-final and sonorant-final consonant cluster.

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Individual processes involved in opaque interaction a. Syncope As previewed in section 2, both Syncope and Raising underapply in this complex interaction. Underapplication of Syncope is caused by both Raising and Epenthesis, as shown in (8a) and (8b), respectively. Raising creates novel instances of short high vowels before CV, which is the environment in which Syncope deletes short high vowels -but Syncope does not apply here, as in (8a). Epenthesis creates novel instances of CV, before which short high vowels may occur -Syncope should apply to such vowels, but does not, as in (8b). Epenthesis also causes underapplication of Syncope, as shown in (8c), through the same mechanism as for Syncope: Epenthesis makes new open syllables, which may contain instances of short [a] that do not undergo Raising. Section 3.2 will summarize McCarthy's (2007) arguments that representational and Stratal OT accounts of these interactions are not possible; section 3.3 will then give a brief (pre-OT) preview of the mechanics of the analysis proposed in this paper.
(8) Underapplication interactions among the three processes a. Syncope and Raising: /katab/ → kitab, *ktab (Syncope does not apply) b. Syncope and Epenthesis: /libn/ → libin, *lbin (Syncope does not apply) c. Raising and Epenthesis: /ɡabl/ → ɡabil, *ɡibil (Raising does not apply) McCarthy (2007) argues that this interaction of three processes cannot be handled by existing representational, non-ordering approaches (Goldrick 2001, van Oostendorp 2008 or by Stratal OT (Bermúdez-Otero 1999), which presupposes morphologically intrinsic ordering when dealing with opacity, and that an extrinsic ordering approach best captures opacity. The argument in the case of representational approaches is that the number of levels needed to describe multiple opacity is not available in representational approaches like Goldrick's (2001) (and van Oostendorp's 2008): there is only one intermediate level, which leads to problems with multiple opacity problems, such as the Bedouin Arabic case; McCarthy considers various other non-ordering approaches, and finds them to have similar problems. Boersma's (2007) Bidirectional Phonetics and Phonology approach to opacity (contemporary to the pressing of McCarthy's book) can, indeed, handle multiple opacity, since it has more than one intermediate representation. However, it does require that the transparent process be driven entirely by phonetic constraints, since the 'shallowest' level of representation is the phonetic surface form. It is not entirely clear whether the Bedouin Arabic Epenthesis process, which has to be represented at the phonetic Nazarov level, can be motivated by entirely phonetic considerations, since it is bounded by the word and does not take place or continuancy into account. However, this is an important area for future research.

Arguments for necessity of extrinsic ordering
In the case of Stratal OT (Bermúdez-Otero 1999), the argument is more subtle. In Stratal OT, opaque interactions arise because different morphosyntactic domains are associated with different OT grammars. Processes that apply at the stem level preceded processes that apply at the word level, which precede those that apply at the phrase level. This means that a word-level process could make a stem-level process opaque, while a phrase-level process should be able to make a stem-level process opaque: a case of intrinsic ordering of processes. McCarthy's (2007) objection to a Stratal OT analysis of Bedouin Arabic lies in the fact that the order of processes necessary for opaque interaction (Syncope < Raising < Epenthesis) does not correspond to their morphosyntactic domains. Specifically, McCarthy (2007:196-198) shows that Syncope has a greater morphosyntactic domain (it applies across words) than Raising (which does not apply across words), which predicts that Raising is at the word level, which Syncope is at the phrase level, and Raising applies before Syncope. This latter leads to the wrong prediction: /katab/ → kitab → *[ktab], leading McCarthy to conclude that ordering of processes cannot be entirely intrinsic.

Preview of analysis
The analysis of the Bedouin Arabic data is based on the idea that every segment carries a value for two non-phonetic features (indices; see section 4): [±L] ("behaves like a low vowel") and [±V] ("behaves like a non-epenthetic vowel"). The assignment of these two features in the lexicon is arbitrary -the constraint ranking ensures that only the attested alternations emerge (see section 5.3). In addition, the phonological component is not allowed to manipulate these features (see section 4).
In the proposed analysis, Syncope only applies to [-L] vowels, while Raising only applies to [+L] vowels. Both processes take place before a single C followed by a [+V] vowel, as shown in (9ab) below. By convention (see section 4), all epenthetic vowels receive the unmarked (minus) value of any non-phonetic feature, as shown in (9c).

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Sketch of extended indexation analysis of Bedouin Arabic a. Syncope: The addition of these indices to each of these processes makes it so that they no longer interact. Syncope and Raising will never be able to apply to the same vowel: Syncope only applies to [-L] vowels, while Raising only applies to [+L] vowels, and the phonological component may not change a [-L] vowel into a [+L] vowel. This automatically entails that high vowels generated by Raising (which are [+L]) cannot undergo Syncope, leading to underapplication of Syncope. The fact that Raising and Syncope must apply before a [+V] vowel means that they can never apply before an epenthetic vowel, since the latter is always [-V]. This means that any ordering of the rules will yield the correct outcome, as shown in (10): the order Epenthesis < Raising < Syncope leads to the incorrect outcome for unindexed rules, but for the indexed rules in (9) ---------This non-interaction between the three processes is key to its being compatible with Parallel OT and a lack of extrinsic ordering. However, this non-interaction is dependent on the appropriate distribution of [±L] and [±V], which is not guaranteed, since indices are assumed to be randomly distributed throughout the lexicon (see also section 4). Having low, [-L] vowels in the lexicon would lead to exceptions to Raising: some words might have low vowels that do not undergo Raising despite being before CV. Similarly, having high, [+L] vowels in the lexicon would lead to exceptions to Syncope. Finally, having non-epenthetic vowels Nazarov with [-V] would lead to exceptions to either process. These exceptions will be worked out in section 5.3.
To rule out the possibility of such exceptions, redundancy rules of the type in (11) can be employed: whenever the rules in (9) do not apply, [+L] vowels should surface as low, [-L] vowels should surface as high, and [-V] segments should surface as consonants. This can easily be formulated in Parallel OT (see section 5.3) and ensures that the analysis of opacity in terms of indexed constraints is restrictive. I will now turn to explaining the representational and constraint framework that allows for this type of analysis.

Theoretical framework: Extended indexation of OT constraints
Indexed constraints (Kraska-Szlenk 1995, Pater 2000, originally intended to represent exceptionality effects, are copies or clones of existing universal constraints that are relativized to a set of words or morphemes: While indexation to words/morphemes works well to account for exceptions, it is not fine-grained enough to account for opacity, as shown in Nazarov (2019). The core proposal for the current re-analysis of Bedouin Arabic opacity is extended indexation (see also Nazarov 2019): indices are binary features, for which every segment has a value, as shown in (13)

. Extended indexation: /i[+i]t[-i]e[-i]/
This proposal is well-rooted in the literature. Firstly, SPE (Chomsky & Halle 1968) used a very similar device to represent exceptions: these were non-phonetic features of the type [-rule n] or of the type [+A] that were specified for an individual segment and could not be modified by phonological rules. Secondly, the representations used here are the union of two previous proposals for extending indexation in OT. Becker (2009) proposes to view indexation as binary: for every indexed constraint Ci, there is a constraint Cj, indexed to all the words/morphemes in the lexicon that are not in i. For instance, if we have *aii indexed to /-i/, /-i/, and /-ite/, there is also *aij indexed to all morphemes that are not /-i/, /-i/, and /-ite/. In addition, Temkin-Martínez (2010) and Round (2017) propose that indexation should be specified on individual segments, rather than attached to words/morphemes. If these proposals are combined, then, for every constraint that has indexed versions and for every segment in the lexicon, the segment is specified for which indexed version of the constraint it refers to. This is exactly the proposal of extended indexation.
Following the original indexation proposal, it is assumed that indices must remain constant between input and output: the phonological component may not change indexation. Whenever a constraint *[+voice][+i] outranks Faithfulness, the grammar will change the underlying voicing of segments indexed [+i], because the indices cannot be changed. In addition, if a segment deletes, its index will be left behind in the same segmental position: a[+i] → [+i]. As will be shown in section 5.3, this is a crucial component in allowing restrictive accounts of phonological opacity.

Individual processes The analysis of the Bedouin Arabic data (see section 3) in terms of indexed
constraints is based on the introduction (during the acquisition stage) of two indices: [±L] and [±V]. While the names of these indices are arbitrary, I have chosen mnemonic names for expository purposes: [+L] stands Nazarov for "behaves like a low vowel (undergoes Raising but not Syncope)" and [+V] stands for "behaves like a (non-epenthetic) vowel". This means that [-L] vowels behave like high vowels (undergo Syncope but not Raising), while [-V] vowels do not behave like non-epenthetic vowels (they are either consonants or epenthetic vowels). The table in (14a) summarizes how these indices are intended to map onto types of segments in the data (long vowels are excluded), and the examples in (14b) show this categorization on the data points shown in section 3.1 (+ values of indices are indicated with gray shading).

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Illustration of indices in this analysis a. Based on this representational world (which will be formalized further in section 5.3), we can formulate three Markedness constraints that will motivate the three processes involved: Syncope, Raising, and Epenthesis, while expressing their underapplication contexts, as shown in (15) As can be seen, the constraint that motivates Syncope, the process that underapplies due to two processes, has two indexed segment positions: one to prevent feeding by Raising (only [+L] vowels undergo Raising), and one to prevent feeding by Epenthesis (only [-V] vowels can be epenthetic). The constraint that motivates Syncope has one indexed position to prevent feeding by Epenthesis (again, because only [-V] vowels can be epenthetic). Finally, the pro-Epenthesis constraint has no indexed positions, since Epenthesis applies transparently. Section 5.2 will show how this helps account for opacity. For Syncope to be motivated by the constraint in (15a), the latter must dominate MAX(V) (1 violation mark for every deleted vowel). Vowels in closed syllables must be protected by MAX(V)/CLOSEDSYLL (1 violation mark for every deleted vowel in a closed syllable). Additionally, DEP must be above Max(V) to prevent insertion of an additional consonant to close the open syllable. This is shown in (16), where the winning candidate, b., has a Max(V) violation that weighs less strongly than the *V[-L]CV[+V] violation of candidate a. In addition, candidate c., where deletion takes place in the final closed syllable, is harmonically bounded. Finally, candidate d., with an additional [t] to close the first syllable, can be ruled out by DEP.

Interaction between processes
The interaction between Syncope and Raising is one of underapplication (or counterfeeding): the surface context for Syncope is present, but it does not apply. This behavior actually falls out from the already established ranking: as shown in (20)   With the adjustments made here, the account laid out in 5.2 becomes a truly general account of multiple opacity: any input given to the grammar, regardless of whether its indexation follows the pattern in (14), will yield an output that conforms to the multiple opaque interaction described in section 3.

Discussion and concluding remarks
As shown in section 5, an account of Bedouin Arabic multiple opacity without extrinsic ordering is, indeed, possible, as long as the grammar provides for the possibility of extended indexation (Nazarov 2019). This allows for an account that acknowledges opacity's connections to exceptionality (see section 2.2) while retaining a systematic, non-exceptional account of the opaque mapping described by McCarthy (2007).
One important implication of an extended indexation account of opacity is that indexed constraints are not present universally, but must be induction in the course of acquisition, since they refer to morphemes that are language-specific. Algorithms exist for inducing such constraints (e.g., Pater 2010, Becker 2009, Coetzee 2009). The necessity of such an induction step for finding indexed constraints, which inherently comes with a chance that this induction step might have gone wrong, lead to the prediction that any pattern that is impossible without indexed constraints will occur less often typologically. This prediction is made based on the idea that acquisition mistakes will accumulate over time, and it is less likely for once-innovated patterns that require indexed constraints to be robust over time (see also Staubs 2014). In addition, it is predicted that opaque processes will take longer to learn, since they require an additional step compared to similar transparent processes. The intuition that opaque interactions are rarer compared to transparent processes is still to be tested in rigorous typological work, and the learnability predictions are still to be tested in acquisition studies, but computational studies have yielded results that also predict this (Jarosz 2016).
Much additional work is still needed, both on the details of accounting for additional Bedouin Arabic data (see fn 3 in section 5.2) and on further testing of the extended indexation model of opacity. In addition, an account of Bedouin Arabic in terms of Boersma's (2007) framework should also be considered. Further implications of extended indexation for opacity should also be explored, including implemented learning simulations to test whether the indexation assumed in the current account of Bedouin Arabic can be plausibly discovered by a learner. Finally, the combination of extended indexation with Harmonic Serialism (McCarthy 2008) should be explored to allow for serial accounts of opacity without the necessity of extrinsic ordering.