1.1.1 Underapplication

The most frequently discussed forms of underapplication opacity are produced by counterfeeding rule orders, defined in 1. Counterfeeding rule orders produce two types of underapplication opacity: counterfeeding on focus and counterfeeding on environment.

In counterfeeding on focus, the two rules involved both apply to the same segment (e.g. chain shifts). This type of interaction is relatively easy to analyze in parallel OT, and there have been many proposals in the literature which invoke faithfulness constraints to block a second process from applying to a particular segment (e.g. Kirchner 1996; Gnanadesikan 1997; Moreton & Smolensky 2002; Jesney 2005).

In counterfeeding on environment, the two rules involved do not apply to the same segment. Therefore, creating a faithfulness constraint to prevent the second process from applying is more complicated. Sympathy Theory (McCarthy 1999), targeted constraints (Wilson 2001), local constraint conjunction (Moreton & Smolensky 2002), and OT with Candidate Chains (McCarthy 2007a) have been proposed as possible solutions which can account for both types of counterfeeding. Our proposed analysis accounts for both types of counterfeeding in parallel OT and HS. We further discuss how our analysis relates to these and other previous approaches in §4.2.

Our proposal is intended to provide analyses for counterfeeding on focus and counterfeeding on environment. Types of underapplication opacity that we do not aim to account for include: class/level restrictions, optionality, exceptionality (see Baković 2011a for discussion of these phenomena as underapplication), and fed counterfeeding (see §2.4 for discussion of why our proposal does not account for fed counterfeeding).

1.1.2 Overapplication

The most commonly discussed type of overapplication opacity is produced by counterbleeding rule orders, where a rule appears to have applied even though its structural description is not met.

Counterbleeding is difficult to analyze in OT because two processes must apply when applying only one would be sufficient to satisfy the relevant markedness constraints. To illustrate, suppose that applying process A satisfies some markedness constraint M_A, and applying process B satisfies some markedness constraint M_B. If applying process B removes the structure that triggers process A, then it also removes the violation of M_A. Thus, applying process B alone is sufficient to satisfy both M_A and M_B, and applying process A in addition would only incur another, gratuitous violation of a faithfulness constraint (this explanation adapted from Jarosz 2014).

In OT, local constraint conjunction (Smolensky 1995; Ito & Mester 2003) has been used to analyze overapplication effects (e.g. Łubowicz 2002). Other approaches make use of output-output faithfulness (Burzio 1994; Benua 1997). We discuss further details of these approaches and how our analysis compares with them in §4.2.

There is usually no distinction made in the literature between counterbleeding on environment and counterbleeding on focus, and the two types may be logically equivalent (Baković 2011a). To our knowledge, only one example of counterbleeding on focus has been reported (Kiparsky 1968; Baković 2011a). Baković (2007) proposes categories (and OT analyses) for additional overapplication interactions including self-destructive feeding, gratuitous feeding, and cross-derivational feeding, which are not problematic for OT grammars. Our analysis is only intended to account for cases of overapplication which involve a “gratuitous violation of a faithfulness constraint” (Baković 2007) and we consider counterbleeding on environment as an example case.

2.2.1 Counterfeeding on focus

An example of counterfeeding on focus is found in Hijazi Bedouin Arabic (HB Arabic; Al-Mozainy 1981).⁴ There are two relevant processes: vowel raising and syncope. Underlying short /a/ raises to [i] in non-final open syllables, and high vowels delete in non-final open syllables. This interaction can be analyzed with a sequence of rules. The rules and their definitions are given in 4.

This interaction can be analyzed in a rule-based framework by ordering the syncope rule before the raising rule, as shown in the rule derivation in 5. High vowels which were raised from low vowels are not subject to deletion because that rule applied earlier in the derivation. This can be seen with the output [difaʕ], which contains the environment for syncope, but the rule has not applied.

A standard OT analysis using the constraints given in 6 cannot capture the counterfeeding pattern. High vowel syncope is motivated by a markedness constraint against high vowels in open syllables (*iCV) ranked above MAX, as shown in 7. However, attempting to account for vowel raising in the same way, by ranking the relevant markedness constraint (*aCV) above IDENT[low], predicts /dafaʕ/ → *[dfaʕ], not the attested [difaʕ], as illustrated in the tableau in 8. This is because the intended output [difaʕ] still violates the constraint motivating syncope (*iCV), which must be ranked above IDENT[low]. The two tableaux in 7 and 8 present a ranking contradiction: the crucial ranking for [∫arbat] to win is *iCV ≫ MAX, but the crucial ranking for [difaʕ] to win would be MAX ≫ *iCV.

(7)	HB Arabic counterfeeding: high vowel deletion (transparent) in standard parallel OT

(8)	HB Arabic counterfeeding: vowel raising (opaque) in standard parallel OT5

Adding a contextual faithfulness constraint of the type we define in §2.1 can resolve the ranking paradox, allowing for the analysis of this counterfeeding pattern in parallel OT. The necessary constraint prevents underlying low vowels from raising to high (defined in 9).

In the case of vowel raising, shown in 10, ranking *aCV above IDENT[low] motivates raising an underlying low vowel to high, while ranking the contextual faithfulness constraint above *iCV prevents underlying low vowels from deleting.

(10)	HB Arabic counterfeeding: vowel raising (opaque) with contextual faithfulness

(11)	HB Arabic counterfeeding: syncope (transparent) with contextual faithfulness

The high-ranked contextual faithfulness constraint prevents the problematic candidate (3.) from winning by specifying faithfulness to a particular property in the input. In this case, the underlying low vowel receives a violation for deletion. This constraint only applies to underlying [+low] segments (because the context is input defined), therefore the analysis of syncope in 11 remains the same: *iCV must be ranked above MAX. The underlying high vowel in the transparent case 11 is unaffected by the same constraint, because the domain of faithfulness is restricted to underlying low vowels.

2.2.2 Counterfeeding on environment

Counterfeeding on environment is similar to counterfeeding on focus, except the two interacting rules overlap in context instead of focus. An example of counterfeeding on environment comes from Lomongo (Hulstaert 1961). Intervocalic voiced obstruents delete, and hiatus violations are repaired with glide formation. However, hiatuses derived from intervocalic voiced obstruent deletion do not undergo glide formation. The rules and their definitions are given in 12. We use the feature [vocalic] to distinguish vowels and glides.

This pattern can be analyzed in rule-based frameworks by ordering the glide formation rule before the intervocalic deletion rule. As illustrated by input /o-bina/ in 13, even though intervocalic deletion creates an environment in which glide formation could apply, the output surfaces as [oina] because the glide formation rule is ordered earlier in the derivation.

As with counterfeeding on focus, contextual faithfulness constraints can be used to analyze this interaction. The relevant constraints are given in 14 and include the contextual faithfulness constraint ID[voc]/_VCEOBS, which prevents changes in the feature [vocalic] for segments which occur before voiced obstruents ([+voi,-son] segments, abbreviated by VCEOBS) in the input. Essentially, this constraint prevents glide formation for candidates which contain the appropriate context for obstruent deletion in the UR.

(15)	Lomongo transparent interaction: /o-isa/ → w-isa

(16)	Lomongo counterfeeding interaction: /o-bina/ → o-ina

The tableau in 15 shows transparent glide formation for input /o-isa/. Glide deletion is motivated by ranking *HIATUS above ID[voc], yielding output [wisa]. The tableau in 16 shows the opaque counterfeeding interaction for input /o-bina/. Intervocalic obstruent deletion is motivated by ranking the markedness constraint demanding deletion (*VCEOBS/V_V) above MAX, the relevant faithfulness constraint, and *HIATUS. This rules out the faithful candidate [obina], because it does not delete the intervocalic obstruent.

The candidate which is problematic for standard parallel OT is candidate (3.), [wina]. This candidate undergoes both intervocalic deletion and glide formation, thereby satisfying both relevant markedness constraints (*VCEOBS/V_V and *HIATUS). However, when ranked above these markedness constraints, the contextual faithfulness constraint rules out this candidate. This constraint essentially prevents gliding for segments which become prevocalic due to the deletion of a voiced obstruent. This is done by including the specified input context–the context specified in the constraint is the same context in which the intervocalic deletion would occur.

This analysis captures the counterfeeding interaction and does not interfere with the transparent interaction, shown in 15. The high ranking contextual faithfulness constraint is vacuously satisfied when the context and/or focus is not present in the input.

3.1.1 Why overapplication is analyzeable in HS with contextual faithfulness

We have demonstrated an analysis of overapplication opacity using contextual faithfulness constraints. The addition of these constraints to an HS system allows us to capture certain rule-ordering effects which cannot be captured with contextual faithfulness constraints in parallel OT. In this section, we discuss why this analysis works and what features of HS make the overapplication analysis possible.

For many rule ordering effects, HS provides a way of ordering processes by constraint ranking. HS uses the principle of gradualness, which requires that the output of each stage of the derivation be more harmonic relative to the previous output (McCarthy 2010). Because of this, higher ranked constraints will be satisfied first. McCarthy (2008) uses this feature of HS to account for metrically conditioned syncope, using constraint ranking to provide an intrinsic ordering where stress assignment applies before syncope. When constraints assigning stress are ranked higher than the constraint demanding syncope, stress assignment occurs earlier in the derivation and will condition syncope. Similar approaches are used by Elfner (2016) for the analysis of metrically conditioned epenthesis, and Pruitt (2010) for metrically conditioned shortening.

These rule ordering effects can be understood as a type of feeding interaction, where one process conditions the application of another. The works cited above use HS’s gradualness to analyze feeding interactions which are otherwise problematic with parallel evaluation. The constraint ranking in HS enforces a particular order of application for the phonological processes. The ranking of contextual faithfulness constraints also enforces a particular order of application for processes, though the appropriate definition of the relevant context is also crucial.

Overapplication opacity (and underapplication, though the focus in this section is on the overapplication analysis) cannot be analyzed in HS simply by ranking constraints according to a desired process ordering. Taking the HB Arabic case as an example, palatalization applies before vowel deletion. However, ranking the markedness constraint demanding palatalization above the markedness constraint demanding deletion does not suffice for the analysis. The tableau in 24 shows an attempted analysis in standard HS. At step 1, the desired winner is candidate (1.), the candidate which palatalizes [k] preceding [i]. At step 2, vowel deletion should apply. candidate (2.) at step 1 instead deletes [i] immediately, which simultaneously satisfies both relevant markedness constraints. In order to rule out this problematic candidate and force palatalization to happen first, MAX must outrank *iCV. This would result in a ranking paradox because *iCV must outrank MAX in order for deletion to occur later in the derivation, and in the transparent case, shown in 25. This is the same problem which occurs in parallel OT (see §2.3).

(24)	HB Arabic counterbleeding in standard HS
	Step 1: /ħa:kim-in/→ [ħa:kjmin]

(25)	Transparent vowel deletion in standard HS
	/∫aribat/ → [∫arbat]

Rule ordering effects like counterbleeding opacity present a problem for HS and parallel OT because there is no intrinsic way to order satisfaction of markedness constraints when simultaneous satisfaction of both relevant markedness constraints is an option.⁷ Ranking the markedness constraint which should be satisfied first over the one that should be satisfied later does not suffice to analyze overapplication opacity because there is one change which will satisfy both relevant markedness constraints simultaneously. This is the case in both parallel and serial evaluation systems.

Contextual faithfulness constraints effectively order application of processes by preventing the one change that would satisfy both relevant markedness constraints simultaneously. They do this by providing extra faithfulness in a particular context, essentially blocking a process from applying until another process has applied. In HS, the output at one stage of the derivation becomes the input at the next stage of the derivation, which provides the crucial mechanism allowing for the analysis of overapplication.

In the HB Arabic case, MAX_IO(i)/k_CV prevents vowel deletion in the context [k_CV]. Ranking this constraint above the relevant markedness constraints prevents deletion (the process which would satisfy both markedness constraints simultaneously) from happening at step 1. With that constraint preventing deletion, palatalization can instead apply to satisfy *ki at step 1. The output of step 1 then becomes the input to step 2. This input now has a palatalized [k^j] preceding the vowel, so the context specified by MAX_IO(i)/k_CV no longer exists. When the context in a contextual faithfulness constraint is not present in the input, the constraint is vacuously satisfied. At step 2, vowel deletion can occur to satisfy *iCV as *iCV is ranked above regular MAX.

This analysis is possible because HS uses an intermediate form as the input to next stage of the derivation. This allows contextual constraints to be rendered inactive⁸ after a particular process applies and the specified context no longer exists. In the case of overapplication, we prevent the problematic change from happening at step 1, but allow that same process to happen at step 2 by defining a context which is changed by application of palatalization.

Contextual faithfulness constraints allow for analysis of overapplication opacity in HS but cannot analyze all observed rule ordering effects. In order to analyze underapplication effects, we must add a distinction between faithfulness to the original underlying representation and faithfulness to the input of the current stage of the derivation. This is explored in the following section.

4.1.1 Factorial typology with contextual faithfulness

In this section, we show what would happen if these constraints were included in the factorial typology of the universal CON. We calculated two example factorial typologies within HS using OT-Help (Staubs et al. 2010), one using the constraint set from the HB Arabic counterfeeding analysis (see §3.2) which includes a FAITH_UO constraint, and one using the constraint set from the HB Arabic counterbleeding analysis (see §3.1) which includes a FAITH_IO constraint.

We show the two typology calculations separately so it is clear which predicted languages arise from the use of the FAITH_UO constraint versus the FAITH_IO constraint. The FAITH_UO constraint only adds the desired underapplication pattern. The FAITH_IO constraint adds the desired overapplication pattern, as well as an additional underapplication pattern which bears formal similarities to do something except when blocking (Baković 2011a).

Table 1 shows the factorial typology calculated using the data and constraints from the HB Arabic counterfeeding example of underapplication opacity (see 6 and 29). There are five languages predicted in this typology; for each, we give the optimal outputs for inputs /dafaʕ/ and /∫aribat/ (which demonstrate whether the raising and deletion processes are applied), the ranking which yields that pattern, and a descriptive label. Languages 1–4 are all present in the typology without the addition of the contextual faithfulness constraint. The only language added to the typology with our proposed constraint is Language 5, the counterfeeding pattern.

Although McCarthy (2007a) was concerned about the use of contextually defined faithfulness constraints for counterfeeding resulting in an overly rich faithfulness theory, the inclusion of FAITH_UO contextual faithfulness constraints for opacity does not necessarily cause unwanted typological predictions. In this example, the only pattern added to the typology is the counterfeeding pattern.

The table in 2 shows the factorial typology calculated using the data and constraints from the HB Arabic example of overapplication opacity (see 18 and 20). There are eight language types produced in this typology; for each, we give the optimal outputs for inputs /∫aribat/, /ħa:kim/, and /ħa:kimin/ (which demonstrate whether the palatalization and vowel deletion processes are applied), the ranking which yields that pattern, and a descriptive label. Languages 1–6 are all present in the typology without the addition of the contextual faithfulness constraint.

The languages added to the typology with our proposed constraint are Languages 7–8. Language 7 is counterbleeding, the attested pattern in HB Arabic. Language 8 is an instance of underapplication in which vowel deletion repairs violations of *ki and *iCV, unless in the context of k_CV, where deletion is blocked.

The underapplication pattern in Language 8 could formally be considered do something except when (DSEW) blocking because deletion applies except in a particular context (Prince & Smolensky 1993/2004; Baković 2011b). In Language 8, deletion applies to repair *iCV and *ki except in the context of k_CV. This meets the formal criterion for DSEW blocking. However, this language differs from actual (observed) cases of DSEW blocking, which are typically motivated by a general phonotactic constraint in the language (Prince & Smolensky 1993/2004; Baković 2011b). This indicates that actual DSEW blocking is likely motivated by a highly ranked markedness constraint, not a highly ranked contextual faithfulness constraint.¹¹

For example, Baković (2011b: 12) cites Kisseberth (1970) as the earliest argument for DSEW blocking, using data from Yawelmani Yokuts (Newman 1944; Kuroda 1967; Kisseberth 1969): short vowels delete between consonants except when that deletion would result in a tautosyllabic consonant cluster. In this case, the blocking is phonotactically motivated (avoidance of a particular marked structure in the language). The blocking in our Language 8 is unlike this observed case of DSEW blocking in that the Language 8 blocking appears to be phonotactically unmotivated. Deleting [i] in the context k_CV does not result in a structure which is phonotactically illegal in the language. Although the pattern in Language 8 is not similar in motivation, it is formally similar to observed cases of DSEW blocking in that deletion applies except in a particular context.

4.1.2 Learning and constraint induction

The constraints we propose are highly specific and include a lot of information: a specified focus, a specified context, and reference to either the most recent input or the UR. Another approach could be that these constraints are induced by learners on a language specific basis when presented with opaque data. In this case, no instances of the phonologically unmotivated blocking pattern (examined in the previous section) would arise as the contextual faithfulness constraints would not enter the factorial typology.

Constraint induction has been used in other domains for language specific constraints, such as morphologically indexed constraints, where lexically indexed constraints are induced to resolve ranking inconsistencies (Pater 2010). In this section, we discuss some relevant background on learning in HS, and show how the contextual faithfulness constraints needed for a given case of opacity can be derived by composing preexisting constraints. This section is not intended to be a detailed proposal for a learning algorithm, but rather a discussion of key ideas that could support the development of such an algorithm in future work.

The key components of a learning algorithm which can induce contextual faithfulness constraints would be (1) the capacity to re-rank constraints as needed, (2) the ability to detect the need for inducing a new constraint, and (3) the ability to compose the new contextual faithfulness constraint. One major challenge to actually implementing such a learning algorithm is that there has been relatively little work on learning in HS.

One of the most commonly assumed algorithms for learning a constraint ranking in parallel OT is Recursive Constraint Demotion (RCD; Tesar & Smolensky 1998). Extending RCD to HS is not straightforward, however, because HS allows for multi-step derivations from the UR to the surface form. Learning constraint rankings motivated only at an intermediate step of a derivation presents a hidden structure problem for which there is no unified solution (for various approaches see Staubs & Pater 2012; Tessier 2013; Tessier & Jesney 2014; Jarosz 2016).

Despite these difficulties, one major benefit of using RCD is that it provides a mechanism for detecting ranking inconsistencies, which is a crucial piece needed in detecting when it is necessary to induce a new constraint. Jarosz (2014) demonstrates that cases of potential feeding/bleeding display characteristic properties in HS, leading to characteristic types of ranking inconsistencies for opaque outputs. These properties could be used by a learning algorithm to diagnose opacity and inform the construction of a new contextual faithfulness constraint. RCD will rank constraints successfully for cases of potential bleeding/feeding when they result in transparent interactions. RCD will fail for these cases when they result in opaque interactions. The crucial properties of potential bleeding and potential feeding are formally different, so this information can be used to inform the learner about whether to induce a FAITH_UO constraint or a FAITH_IO constraint when RCD fails. We illustrate these characteristic properties using two of the examples analyzed here.

In the HB Arabic counterbleeding example, the two relevant processes are palatalization and deletion. Jarosz (2014: 3) uses the same HB Arabic example to illustrate the characteristic ranking inconsistency posed by counterbleeding in HS, which we summarize here. The ranking requirement for deletion can be modeled as M_A ≫ F_A (*iCV ≫ MAX). The ranking requirement for palatalization can be modeled as M_B ≫ F_B (*ki ≫ IDENT[back]). When implemented in HS, application of process A (deletion) simultaneously causes satisfaction of the markedness constraint involved in process B (palatalization), M_B. Thus, “the essential characteristic of a potential bleeding interaction is that satisfaction of M_A (*iCV) results in the satisfaction of both M_A (*iCV) and M_B (*ki)” (Jarosz 2014: 3).

This essential characteristic can be used to identify cases of counterbleeding and trigger induction of a contextual faithfulness constraint. We illustrate this ranking problem in 24, reproduced here as 37. The desired winner is candidate (1.), the candidate which palatalizes at the first step of the derivation (indicated with the arrow). The incorrect winner based on the current constraint ranking is candidate (2.), the candidate which deletes at the first step (indicated with the bomb symbol). The faithful candidate is candidate (3.). The crucial characteristic of potential bleeding is that satisfaction of M_A (done by candidate 2.) results in simultaneous satisfaction of both M_A and M_B. The ranking inconsistency is caused by the fact that the desired candidate only satisfies M_B.

(37)	HB Arabic counterbleeding in standard HS
	Step 1: /ħa:kim-in/→ħa:kijmin

These properties of the candidates would be present when RCD fails. We assume here (along with previous work, summarized above) that the learner would have access to inputs, outputs, candidates and violation profiles. Given the violation profiles of the three candidates we provide in 37, the learner can identify which constraints are M_A and M_B–these are the two markedness constraints violated by the faithful candidate (3.) but satisfied by the problematic candidate (2.). Of these two constraints, M_A is the constraint which is satisfied by the problematic candidate but violated by the intended winner, candidate (1.). When presented with this particular characteristic ranking inconsistency, the learner would be triggered to induce an IO contextual faithfulness constraint.

The ranking problem of counterfeeding is different. We again summarize Jarosz (2014: 3) on characteristic ranking inconsistencies in HS, applying this analysis to our example of counterfeeding in Lomongo. There are two relevant processes: A (intervocalic deletion) and B (glide formation). Unlike the counterbleeding case, standard HS can model the first step of the derivation where process A (deletion) applies with the ranking M_A ≫ F_A (*VCDOBS/V_V ≫ MAX).¹² The next step of the derivation presents the ranking inconsistency: there is no way to prevent process B from applying (which would require the ranking F_B ≫ M_B) as the ranking M_B ≫ F_B is independently needed to account for transparent instances of process B. The characteristic problem of counterfeeding is that application of process A creates violations of M_B, which are subsequently not resolved even though process B is attested elsewhere in the language.

This characteristic inconsistency can be used to identify cases of counterfeeding and trigger induction of a UO contextual faithfulness constraint. The ranking problem presented by the Lomongo counterfeeding example is shown in 38. Once the ranking inconsistency has been detected, the learner can use properties of the candidates and their violation profiles to identify this as a counterfeeding-related inconsistency. In step 2 of the derivation, where RCD would fail, we show the three candidates the learner would need to correctly identify the inconsistency and involved constraints (whose properties are necessary for induction).

(38)	Lomongo counterfeeding interaction in standard HS
	Step 1: /o-bina/ → oina
	Step 2: oina → [oina]

Candidate (1.) is the intended winner, which is also the faithful candidate, indicated with the arrow. Candidate (2.) is the problematic candidate which applies process B and incorrectly wins under the current ranking (indicated with the bomb symbol). Candidate (3.) is the candidate which removes the application of process A (the original UR). The crucial property of counterfeeding is that resolving M_A creates violations of M_B, and those violations must be left unresolved. The fact that candidate (1.) is the intended winner but candidate (2.) wins under the current ranking shows that the inconsistency is caused by the fact that M_B must be left unresolved. When this inconsistency is present, and the learner has independent evidence of M_B ≫ F_B (from the transparent case), the learner would be triggered to induce a UO contextual faithfulness constraint.

Once the inconsistencies have been identified, the induction of a new contextual faithfulness constraint can be accomplished by combining properties of the existing involved constraints. In cases of counterbleeding opacity, the IO contextual faithfulness constraint can be constructed by combining properties of F_A with properties of M_A and M_B (these constraints would have been identified during the inconsistency detection). F_A provides the type of faithfulness for the constructed contextual faithfulness constraint. M_A and M_B provide the focus and context of faithfulness. The contextual overlap between the two markedness constraints becomes the focus, and the surrounding material becomes the context.

As an example, we show how induction would proceed for the HB Arabic counterbleeding interaction (see §3.1 for the analysis). In this example, deletion of [i] in open syllables should follow palatalization of [k] before [i]. The detection of a ranking inconsistency associated with counterbleeding would trigger the learner to induce a contextual FAITH_IO constraint and allow the learner to identify the involved constraints (M_A and M_B, F_A and F_B). In this case, MAX is F_A, so the contextual faithfulness constraint will be a MAX_IO constraint.

The two markedness constraints which must ultimately be satisfied are *iCV (M_A), which is satisfied by applying deletion, and *ki (M_B), which is satisfied by applying palatalization. The markedness constraints have a contextual overlap with the segment [i], so the focus of the new contextual faithfulness constraint will be [i]: MAX_IO (i). The remaining material from the markedness constraints (k_ from *ki and _CV from *iCV) is combined to form the context, yielding the final constructed contextual faithfulness constraint: MAX(i)/k_CV. This process is schematized in 39.

(39)	Building a contextual faithfulness constraint for counterbleeding
	Step 1.	Identify FA:
		– MAX
	Step 2.	Find the contextual overlap between MA and MB:13
	Step 3.	The contextual overlap becomes the focus of the contextual faithfulness constraint:
		– MAX(i)
		The material surrounding the overlap becomes the context for the faithfulness constraint:
		– MAX(i)/k_CV

For cases of counterfeeding opacity, the learner would need to induce a FAITH_UO constraint, also by combining properties of the existing faithfulness and markedness constraints involved. In this case, F_B provides the type of faithfulness constraint, and the focus of M_A provides the context of faithfulness. The learner would again be able to identify the relevant constraints during inconsistency detection by comparing the violation profiles of the intended winner, the problematic candidate, and the faithful candidate (as summarized above).

In the case of the Lomongo counterfeeding interaction (see 38), F_B is ID[vocalic], so the type of the constructed contextual faithfulness constraint will be ID[voc]_UO. M_A is *VCDOBS/V_V, the markedness constraint satisfied by the application of intervocalic voiced obstruent deletion. The focus of this constraint is VCDOBS (=[+voi, –son], see 14 for constraint definition), which becomes the context of new contextual faithfulness constraint. This yields the final constructed contextual faithfulness constraint: IDENT_UO (vocalic)/_[VCEOBS]. This process is schematized in 40.

In this section, we have discussed some key pieces needed to implement a learning algorithm for inducing contextual faithfulness constraints in cases of opacity: (1) and RCD-like algorithm for ranking constraints which can detect ranking inconsistencies implemented in HS, (2) a definition of the ranking inconsistencies which are characteristic of counterbleeding and counterfeeding opacity, and (3) a means of constructing the needed contextual faithfulness constraints by composing elements of independently-needed constraints. The main goal of the current paper is to showcase the analytical potential of contextual faithfulness constraints for analyzing opacity by providing analyses of several opaque interactions, and we leave the computational implementation of a learning algorithm for future work.

4.2.1 OT with Candidate Chains

OT with Candidate Chains (OT-CC; McCarthy 2007a) uses HS to analyze counterfeeding and counterbleeding. OT-CC essentially combines the OT framework with a derivational framework by storing faithfulness violations of intermediate forms during candidate evaluation. Candidates therefore consist of a chain of all intermediate output forms in addition to the final optimal output. After these chains are generated, EVAL has access to both the terminal link (the final, most harmonic member) of the chain and the path of improvement, given by the ordering of the locally unfaithful, intermediate mappings.

Markedness constraints evaluate the final member of the chain and faithfulness constraints evaluate the relationship between the first and last forms in the chain. OT-CC faithfulness constraints are similar to our UO constraints in that they demand faithfulness between the first/underlying form and the last/output form in the HS derivation. The candidate chains are evaluated by a new type of constraint, a precedence (PREC) constraint, which evaluates the complete derivations by specifying a preferred order of faithfulness violations. This effectively controls the order in which markedness violations are repaired.

One of the major critiques of OT-CC is that the large amount of new capacity added to basic HS only re-creates the derivational account. Kiparsky (2015: 9) writes that the additions of OT-CC make a systems which is “like stipulative rule ordering…only with constraint ranking dictating the order of application.” However, this approach accounts for a wide range of opaque phenomena, including counterfeeding, counterbleeding, and fed counterfeeding (Kavitskaya & Staroverov 2010). Our proposal accounts for counterfeeding and counterbleeding, but cannot account for fed counterfeeding (see §2.4 for an explanation of the unique analytical challenge posed by fed counterfeeding). Another benefit of OT-CC is that it has the potential to constrain instances of the too many repairs problem (e.g. Lombardi 2001; Steriade 2001; Wilson 2001). Unattested repairs could be prevented from surfacing in OT-CC due to a refined definition of gradualness combined with the use of PREC constraints (McCarthy 2006).

We argue that our approach is more economical in two major ways: additional representations and CON. OT-CC adds the new class of PREC constraints to evaluate candidate chains and a metaconstraint to control rankings of PREC constraints. While our approach does introduce new constraints, they are within the class of faithfulness constraints and are composed of focuses and environments from existing markedness constraints. We also do not require a ranking metaconstraint to mitigate typological effects. Even if contextual faithfulness constraints are included in the universal CON, they only add opaque patterns and a case of blocking which is formally similar to observed cases of do something except when blocking. As detailed in the previous section, the blocking pattern and an overly rich inventory of faithfulness constraints could be avoided if the constraints are induced on a language-specific basis, which we see as a promising avenue for future work.

Our approach is also more economical with regards to candidate representation. OT-CC stores intermediate representations during candidate evaluation to create a new type of representation, the candidate chain. The chain of intermediate forms must be stored and available throughout the derivation. Our analysis does not require an enriched theory of candidate representation through storage of any additional representations. We only require that the UR be referenced by FAITH_UO constraints. We assume that the UR would already be stored in the lexicon, so this form need not be additionally stored, only referenced. No additional stored forms or references are needed to evaluate FAITH_IO constraints.

4.2.2 Serial Markedness Reduction

Serial Markedness Reduction (SMR; Jarosz 2014) also uses HS, and introduces a new family of constraints that work within the derivation itself, evaluating input-output mappings instead of entire chains of candidates (as in OT-CC). These constraints evaluate the order in which markedness constraints are satisfied. This is similar in concept to OT-CC’s PREC constraints, which evaluate the order of faithfulness violations across candidates. However, in SMR, there is no requirement for storage of intermediate forms. Jarosz (2014) argues that this adds considerably less machinery to basic HS relative to OT-CC and also avoids the problems of global rule interactions admitted by OT-CC.

In SMR, the candidate representations are elaborated to include a list of markedness constraints satisfied at each step. This information is stored on the candidates themselves in the form of a list, called the Mseq. The candidates are evaluated against a standard CON with the addition of a new class of constraint, the serial markedness (SM) constraint. The SM constraint evaluates each candidate’s Mseq. We provide the general form for SM constraints in 41, where M1 and M2 represent two markedness constraints.

SMR adds considerably less technology to basic HS (relative to OT-CC), while maintaining the ability to account for many opaque phenomena. As in basic HS, introducing SMR only requires one EVAL loop per step of the derivation. It does not require a “second evaluation with an expanded set of constraints, as in OT-CC” (referring to PREC constraints) (Jarosz 2014: 6). This local evaluation, combined with the fact that SM constraints are vacuously satisfied unless both relevant markedness constraints are present in a candidate’s Mseq, mitigate the issues of global rule interaction admitted by OT-CC. OT-CC allows interactions which are not allowed by rule ordering, such as counterfeeding from the past (Wilson 2006). Jarosz argues that OT-CC “cannot predict an opaque interaction between processes that are ordered far apart in the derivation because process interaction is defined locally” (Jarosz 2014: 12). SMR is overall argued to be more economical and more realistic with respect to typological predictions relative to OT-CC.

Our approach bears some similarities to SMR, as contextual faithfulness constraints also work within the derivation and evaluate candidate mappings. Neither approach requires storage of a chain of intermediate steps. Both approaches also order the satisfaction of markedness constraints. SMR does this directly, by keeping track of what markedness constraints have been satisfied and evaluating the order of satisfaction with a new class of constraint.

In our approach, markedness constraint satisfaction does not have to be ordered directly, but instead emerges from the use of faithfulness constraints. Once the context has changed, the constraint is no longer active. This provides a way of ordering satisfaction of markedness constraints without explicitly demanding it. Our approach does not require the introduction of a new class of constraint (the SM constraint). Instead we require new constraints introduced into the class of faithfulness constraints, which is already present in basic HS and OT.

Our approach differs from both OT-CC and SMR in that it does not require an enriched theory of candidate representation, as long as we assume underlying lexical representations are already stored in speakers’ lexicons. Our constraints do require access to those underlying representations at each step of the derivation, but does not require access to information about any intermediate steps, or previous constraints satisfied. For these reasons, we argue that our approach is more economical.

4.2.3 Local constraint conjunction

Local constraint conjunction has been proposed as a richer faithfulness system for OT (Smolensky 1995; Ito & Mester 2003). The local conjunction of two constraints is violated when both conjoined constraints are violated. These constraints have been explored as a method of analyzing counterfeeding opacity, but cannot account for other cases of opacity (McCarthy 1999; Padgett 2002). However, a benefit of this approach is that it is also able to account for many other non-opaque phenomena including coda conditions (Smolensky 1993) and derived environment effects (Łubowicz 2002; Ito & Mester 2003).

Critiques of this approach have often involved typology, with the claim being that introducing the ability to conjoin constraints overpredicts observed patterns. Because local conjunction relies on the segmental proximity of violations in a specified domain, enlarging the domain (to the entire word for example) predicts interactions with triggers over a nonlocal context. Such interactions are not associated with process interaction and are an undesirable prediction of the system. Łubowicz (2005) showcases this issue and proposes that the domain of conjunction must be restricted to avoid predicting unattested patterns where, for example, palatalization triggered by a vowel context in one part of the word can trigger spirantization in another part of the word.

Analyzing opacity with contextual faithfulness constraints does not run the same risk of overprediction because they only reference a single segment as their focus, not an entire domain. This is not to say there is no risk for overprediction with contextual faithfulness—see §4.1 for discussion of typological predictions. McCarthy (2007a) argues that local conjunction account of counterfeeding opacity therefore relies solely on the proximity of the two violations, in some specified and crucially overlapping domain, obscuring the actual interaction of two processes. We argue that FAITH_UO constraints provide a more explanatory account by combining the pre-existing mechanism for producing contrast in OT, the interaction of markedness and faithfulness constraints, with the generalization that counterfeeding opacity is characterized by a special case of faithfulness in a particular context.

There is a formal similarity between our contextual faithfulness constraints and local constraint conjunction with respect to the potential for constraint induction. While contextual faithfulness constraints are only single (non-conjoined) constraints, they could be induced through the combination of pre-existing markedness constraints. However, because they are evaluated as a single non-conjoined constraint, they avoid the typological overprediction of conjoined constraints discussed by McCarthy (2007a).

4.2.4 Output-output faithfulness

Output-output faithfulness (OO-FAITH; Burzio 1994; Benua 1997) has also been used to analyze opacity. This approach bears many analytical similarities to our contextual faithfulness approach. In an OO.FAITH analysis of opacity, “opacity is conditioned by a faithfulness relation between the output and the output of another related form” (Goldrick 2001: 12). Both our approach and OO.FAITH provide extra faithfulness for a particular segment/feature in a particular context between the output and some other representation. For OO.FAITH, this other representation is a related output form, often a form where a crucial feature from the lexical form/UR surfaces faithfully. In our approach, we instead implement a direct faithfulness relationship between the lexical form and the output of the current derivation. Effectively both approaches demand additional faithfulness, but refer to different representations.

We argue that our approach provides a more economical way of obtaining this extra faithfulness, as we only require access to either the input of the current stage of the derivation or the underlying form (depending on whether the constraint is IO or UO). This does not require CON to have access to additional intermediate forms, output forms, or forms associated with other derivations. Our approach does not require storage of any additional forms that are not already stored in the lexicon. In addition, our approach provides a general solution to counterbleeding and counterfeeding opacity, while OO.FAITH is restricted to cases of opacity where there is a related morphological form where the UR surfaces faithfully. The OO.FAITH approach can only account for particular cases of counterbleeding and does not offer a general solution for opacity in OT.