2.1 The Copying-Weakening-Implication

This section presents empirical evidence for the claim that copying is correlated with a greater likelihood to undergo phonological reduction. This finding is formulated as the CWI in (2). The term ‘copying’ in the CWI refers to the phonological operation of duplicating phonological material which is taken to underlie all instances of reduplication, the empirical focus of this paper. ‘Weakening’ in (2) implies a greater likelihood to undergo phonological processes. In all examples discussed in this paper, these processes are deletion of whole segments or certain feature specifications. Both these processes are summarized as ‘reduction’ in the following.

The CWI’s statement that every copy operation weakens all the elements involved in the copying results in two restrictions on possible language types (2A+B). They describe an implicational universal that the presence of reduction in a given context implies that this reduction also applies in a context with more copying. This means that the CWI predicts that languages can have thresholds for reduction: reduction can be restricted to material that is at least copied once or even to material that is at least copied twice. The typology of languages predicted from these thresholds and the implicational restrictions (2A+B) are illustrated in (3) where ‘Reduction’ marks the existence of a phonological reduction process and ‘Faithful’ its absence. It compares the three contexts of 1) no copying, 2) one copy operation, and 3) two copy operations. Four language types are possible under the CWI (3A) whereas four others are excluded (3B). It is, for example, impossible that a language shows a reduction pattern in comparable phonological contexts in non-copied strings and in strings that are copied twice but fails to show this reduction in comparable phonological contexts for strings that are copied only once (*Lg 6).

(3)	Predicted typology: copying and weakening

In section 3, a theoretical account of reduplication is presented that predicts the four attested language patterns (3A) from the interaction of standard faithfulness and markedness constraints and systematically excludes the four patterns (3B).

Note that the languages in (3) only contrast the three contexts without, with one, and with two copy operations. In principle, much more thresholds bound to contexts with three, four, or even more copying operations are predicted by the CWI. However, instances of productive combination of two reduplicative processes are already rather rare in the languages of the world¹ and those combining even more reduplicative morphemes are even harder to find. Potential examples include the addition of more reduplicative morphemes to intensify the meaning (e.g. in Fungwa (Akinbo 2018), Tigre (Rose 2003), or Toqabaquita (Lichtenberk 1945)). None of these patterns show a reduction process that is bound to reduplication.

Another prediction of the CWI arises from its reference to ‘all elements involved in the copying’. This implies that reduction is more likely for the copied portion usually described as the ‘reduplicant’ and also for the copied portion within the ‘base’. It is important to emphasize this symmetric weakening of all copied elements since the classical phonological account to reduplication based on BR-correspondence (e.g. McCarthy & Prince 1995) only predicts a special status for the reduplicant that is more prone to reduction, in contrast to the CWI.

This distinction into reduplicant and its base is of course a non-surface-apparent distinction and relies on analytical choices made by the linguist. In Ilokano (4), for example, the surface facts are simply that two instances of the three segments /kal/ or /pus/ surface in the reduplicated form – nothing inherently distinguishes one copy from the other. In all theories of reduplication, this surface form can in principle be predicted from either assuming that the phonological element that triggers copying is prefixed or infixed or that the instruction to copy material applies before all base segments or after the first syllable. This ambiguity is illustrated in the depiction in (5) that also introduces the terminology I use throughout the paper. It already implies the phonological account to reduplication assumed here where affixation of two empty moras triggers copying of a heavy syllable. Crucially, both prefixation (5a) or infixation (5b) of these empty moras will derive the same surface structure. As can be seen in (5), I continue to use the traditional term ‘reduplicant’ for the copied string that is realized in the ‘new’ position where the reduplication-triggering morpheme was underlyingly positioned and use the term ‘copied base’ for the counterpart of the copied string in the ‘old’ position.

(5)	Possible structures
	Underlying	a.	/µ µ - kaldíŋ/	b.	/kal < µ µ > díŋ/
	Terminology used:

One promising way to disambiguate base and reduplicant can be the presence of fixed segments; an argumentation used in sections 2.2.2 and 2.2.3 where the distinction of the copied string into reduplicant and base is particularly important.

With this background, we now turn to the empirical evidence for the CWI. The next two subsections give examples for the two language types 2 and 3 in (3) which show thresholds for reduction that are bound to reduplication contexts. More concretely, languages of type 2 are those that show reduction only for elements that are copied 2.2 and languages of type 3 are those that show reduction only for elements that are copied twice 2.3.

2.2 Threshold 1: Copying enables reduction

This subsection gives examples for language type 2 where reduction only affects copied material but is blocked for non-copied material. As we will see, there are three sub-patterns for this language type, each is illustrated with one language below. The most obvious instantiation of the CWI is a language where reduction applies to the whole copied string, i.e. both the reduplicant and the copied base. Such a pattern can be found in Tagalog 2.2.1. There are, however, also languages that show reduction of copied material in only one of the copied instances, i.e. only in the reduplicant or only in the copied base. As will become clear in section 3, this is an epiphenomenon in a model where fission is distribution of underlying activity. In this model, copying makes all copied material more prone to reduction but whether reduction for all or only certain copied elements surfaces is determined by the general phonology of the language. An example where reduction applies only in the reduplicant is Gitksan 2.2.2 and an example where reduction applies only in the copied base is Lushootseed 2.2.3.

2.2.3 Vowel reduction in Lushootseed

The counterpart to a language with reduction only in the reduplicant is one with reduction only in the copied base. An example for this pattern is vowel reduction in Lushootseed (Broselow 1983; Bates et al. 1994; Urbanczyk 2001), a Salishan language. The classic description of the diminutive formation is that it involves prefixation of a monosyllabic /CV/-reduplicant, accompanied by deletion (10a) or reduction of the copied base vowel (10b). Given that stress is initial, this reduction applies to a copied base vowel that becomes unstressed in a reduplication context. Whether the base vowel is deleted or only reduced to /ǝ/ is partially phonologically predictable and depends on the quality of the base vowel (low /a/ has a preference to be deleted) and the sonority of the resulting cluster (Urbanczyk 2001: 111). However, there are some lexical exceptions to these phonological generalization. For the present argumentation, the choice between the two reduction strategies is not particularly important.

In addition, some stems show a fixed vowel /i/ instead of the copied vowel and hence prefix a /Ci/-reduplicant (10c). The choice between the /CV/- and /Ci/-reduplicant is again largely phonologically predictable. Stems that begin with a long vowel, have /ǝ/ as their first vowel, or begin with a consonant cluster show the fixed vowel /i/ (Urbanczyk 2001: §4). In Urbanczyk (2001), the fixed segment /i/ is consequently taken to be a default segment that is realized instead of a copy-vowel to optimize the phonological structure and, for example, avoid copying of a long vowel.

Interestingly for the present argumentation, vowel reduction and deletion do not apply if the fixed vowel /i/ surfaces, excluding, for example, */dí∼dǝk^w/. Vowel reduction thus only affects copied vowels, it is not restricted to, for example, the diminutive context in general.³

1. (10)

1. Diminutive reduplication in Lushootseed (Urbanczyk 2001)

   	base		distributive
   a.	júbil	‘die, starve’	jú∼jǝbil	‘small animal dies’	p.207
   	s-bádil	‘mountain’	s-bábǝdil	‘small mountain’	p.192
   	ʔágwal-ǝb	‘yawn’	ʔá∼ʔǝgwál-ǝb	‘yawn’	p.191
   b.	ʔúsil	‘dive’	ʔú∼ʔsil	‘shallow dive’	p.192
   	kúpi	‘coffee’	kú∼kpi	‘a little coffee’	p.196
   	sáqw’	‘fly’	sá∼sqw’	‘fly just a little bit’	p.201
   c.	dúːkw	‘knife’	dí∼duːkw	‘small knife’	p.195
   	daʔ(a)	‘name, call’	dí∼daʔa-t-ǝb	<name, calling>	p.194
   	tʃ’saj	‘salmon spear’	tʃ’i∼tʃ’sáj	‘toy spear’	p.194

It has to be emphasized that this generalization that reduction is bound to copied unstressed vowels is different from the theoretical analysis in Urbanczyk (2001) where all unstressed /a/ vowels are predicted to be reduced. However, this is not a general process as can be seen in the data in (11) where unstressed /a/’s are preserved outside of reduplication contexts. It is not easy to decide whether reduction is bound to copying or applies to unstressed vowels of a certain quality given that there are exceptions to both generalizations (cf. footnote 3). The present approach takes the correlation between reduction and copying to be the one capturing more patterns and assumes that the reduction is expected for unstressed copied vowels in the default case (10a+b) but not for unstressed vowels (of a certain quality) in general (11).

(11)	No reduction of unstressed non-copied vowels (Urbanczyk 2001)
	t’q’ǝdíʔ=ac	‘western hemlock’	p.73
	plílaʔ	‘wild or bitter cherry’	p.73
	ʔu-c’áp’ac	‘insulted someone’	p.86
	líl=ax̌ad	‘far away on the other side’	p.87

The existence of reduction that only applies in the copied base is a debated issue. There are some claims in the literature that such patterns exist (e.g. Shaw & Howe 1999; Struijke 2000) and that the standard model of BR-faithfulness must be extended to capture them. However, some of the data proving the existence of this pattern is in fact ambiguous and allows a reanalysis in terms of reduction only in the reduplicant. An example is the reduplication pattern in Tohono O’odham that either involves reduction in the copied base if the reduplicant is taken to be prefixing but reduction in the reduplicant if the reduplicant is taken to be infixing (Riggle 2006). This ambiguity is of course expected given that the reduplicant and the copied base are identical in a default case of reduplication (cf. the discussion at the end of 2.1).

For Lushootseed, such a reanalysis as reduction in the reduplicant comes at the price of assuming two suppletive allomorphs for the diminutive reduplication. The illustration in (12) summarizes the surface indicators we have for the position of the reduplicant under the alternative interpretation that reduction applies in the reduplicant. In the patterns (10a), the reduplicant must contain the reduced vowel and one of the instances of the copied consonant, resulting in an infixed CV-reduplicant (12a). For the patterns (10c), on the other hand, the reduplicant must contain the fixed vowel and one of the instances of the copied consonant, resulting in either a prefixed CV-reduplicant or an infixed VC-reduplicant (12b).

(12)	Reduction only in the reduplicant: identifying base and reduplicant

At first glance, these facts might still be compatible with assuming a single /CV/-reduplicant for the diminutive in a theory where morphemes can dislocate to optimize the phonological structure (Prince & Smolensky 1993/2002). We could, for example, assume that the /CV/-reduplicant is preferably prefixed but is infixed and realized after the first base vowel if prefixation results in a marked structure. The comparison in (13) shows that such an account is untenable: Both the prefixing and the infixing realizations of the reduplicant result in a form that is equally (un)marked in terms of syllable structure.

1. (13)

1. Alternative morphological structures: reduction only in the reduplicant
2. 

If reduction applies in the reduplicant in Lushootseed, the account hence needs to rely on phonologically conditioned suppletive allomorphy (Paster 2006) between a reduplicative prefix and a reduplicative infix. An account with only a single underlying representation of a prefixing /CV/-reduplicant whose different surface realizations as /CV/ or /Ci/ are predicted by the general phonological optimization is surely to be preferred for reasons of economy and lexical storage. And under this analysis, it is the copied base vowel that is reduced to /ǝ/ or deleted altogether.

Other examples for type 2 languages that show reduction only in the copied base include vowel deletion in Cupeño (Hill 2005) and Klamath (Barker 1964; Kisseberth 1972; Blevins 1993). The pattern is discussed at length within the theory of existential faithfulness (Struijke 2000) which also predicts reduction in the copied base.

2.3 Threshold 2: More copying enables more reduction

This subsection now turns to languages of type 3 (cf. (3)), namely those that restrict reduction to elements that are copied more than once.

Before we turn to concrete examples, some background on ‘multiple copying’ contexts is in order. Since the paper focusses on reduplication, this situation involves multiple reduplication of the same base material. Such a situation arises if two (or more) reduplicative morphemes are present (and usually adjacent) in a word. An example for multiple reduplication from Bikol is given in (14). The imperfective is marked by monosyllabic prefixing reduplication and the plural by bisyllabic prefixing reduplication as can be seen in (14a). Verbs marked for both imperfective and plural now simply concatenate these reduplicants (14b). From the perspective of phonological copying, the string /ha/ in (14b) is copied twice since three /ha/ sequences surface in the multiply reduplicated form.

2.4 Summary: Reduction and copying thresholds

Table (28) repeats the four language types that are in accordance with the CWI. Different examples for languages that exemplify the two threshold types 2 and 3 were discussed in the last two subsections. Language types 1 and 4 are trivially attested and show no interesting reduction threshold with respect to reduplication. An example for a language of type 1 where the presence of reduction is completely independent from any copying is Palauan where all unstressed vowels are predictably reduced to /ǝ/ (Finer 1990). This reduction applies to vowels that are not copied, to those that are copied once, and to those that ore copied twice. And an example for a type 4 language where no reduction applies in any context would be Papapana (Smith 2015; 2016). Single and multiple reduplication contexts predictably copy a prosodically defined portion of the base and no reduction process applies to any copied or non-copied elements.

3.1 A phonological account of reduplication

The reduplication theory advocated here is purely phonological and assumes that reduplication is the consequence of a phonological copy operation that applies to fill otherwise segmentally empty prosodic nodes. This sets this approach apart from theories assuming reduplication-specific elements like the phonologically empty RED morpheme, reduplication-specific correspondence relations (McCarthy & Prince 1995), reduplication-specific phonological operations (Raimy 2000), or reduplication morpheme-specific cophonologies (e.g Inkelas & Zoll 2005; Inkelas 2008).

Reduplication-triggering affixes in this model contain ‘prosodic affixes’ or empty prosodic nodes that lack segmental content. Apart from this representational peculiarity, there is nothing special about them: phonology alone is left to deal with those empty prosodic nodes and no special morphology or phonology applies in reduplication contexts. The influential theoretical proposals in Marantz (1982) and McCarthy & Prince (1986) can be understood as first arguments for such a phonological account of reduplication based on segmentally empty affixes. It was re-introduced into optimality-theoretic phonology in the work of Saba Kirchner (2010) (cf. also Pulleyblank (2009); Bermúdez-Otero (2012); Bye & Svenonius (2012); McCarthy et al. (2012)) that emphasizes, for example, the existence of phonologically predictable allomorphy between reduplication and other non-concatenative strategies as a strong argument for a prosodic affixation account to reduplication (Saba Kirchner 2010; 2013a; b; Zimmermann 2013). Prosodic affixation can thus not only result in reduplication, but it can also result in other non-concatenative exponents that realize a morpheme, most notably vowel lengthening or gemination. The constraints that ensure realization and prosodic integration of a prosodic affix are hence by no means specific to reduplication (e.g. Samek-Lodovici 1992; Davis & Ueda 2002; van Oostendorp 2005; Wolf 2007; Bye & Svenonius 2012; Zimmermann 2017).

The ‘copying’ process that fills the otherwise empty prosodic nodes in reduplicative contexts is standardly taken to be fission: underlying elements are split up into multiple output instances under violation of INTEGRITY (=INT_s) (29a) (Raimy & Idsardi 1997; Spaelti 1997; Struijke 2000; Gafos 2003; Nelson 2003). The basic logic of this account is illustrated in tableau (30) where a prefix consisting only of an empty mora is affixed to a base. Deletion of this mora is impossible due to undominated MAX-µ demanding preservation of every underlying mora (not given in (30)). If the mora is realized, however, it needs to dominate some segments due to (29b) ensuring proper integration of all prosodic nodes. Two obvious possibilities to fill this mora with segmental content are to link it to the following vowel of its base resulting in vowel lengthening (30b), to insert an epenthetic vowel (30c), or to split up the underlying initial vowel into two output instances (30d). It is easily determinable from the indices that mark the IO-correspondence relations that fission applied in this candidate. Both underlying /s₁/ and /i₂/ have two output correspondents with the same index. Given that DEP (29c) and *Vː penalizing long vowels are ranked above INT_s in this language, segment fission becomes optimal. This repair is often termed ‘copying’ in the following but it should be clear that it is simple double realization of input elements. Crucially, the candidates (29a) and (29b) represent strategies that become optimal in other languages to realize an empty mora – strategies which can even alternate in a phonologically predictable way with reduplication (cf. the discussion above).

It has to be noted that the constraint in (29b) only refers to ‘morphologically coloured’ prosodic nodes. The theory of morphological colours assumes that all phonological elements that are part of the underlying representation of a morpheme have a designated colour that represents morphemic affiliation (van Oostendorp 2003; 2006; 2007; 2008; Revithiadou 2007). It hence relies on the often implicit assumption that the phonology can detect whether a phonological element is epenthetic (=has no colour) or not (=has a colour) and whether two phonological elements belong to the same morpheme or not. Only underlying moras induce a violation of (29b) if they are not dominating a vowel. That epenthetic nodes are exempt from those prosodic integration constraints is a natural assumption given that the insertion of epenthetic nodes is only motivated if they are necessary for prosodic integration to begin with.

(30)	Copying as fission: the basic mechanism

In the following analyses, the basic interaction between these constraints is taken for granted and it is not explicitly shown anymore how and why copying emerges as the optimal strategy to fill the prosodic affixes.

Another background assumption adopted here is the containment assumption that literal deletion of input elements is impossible and that elements can only be marked as phonetically uninterpreted (Prince & Smolensky 1993/2002).⁹ In such systems where unrealized elements are still part of the structure, constraints are taken to exist in two versions (Trommer 2011; Trommer & Zimmermann 2014): one referring only to realized material and the other to all phonological material. This distinction is illustrated in (31) with the constraint (29b) demanding that every mora wants to dominate a segment. The phonetic version of this constraint only ‘sees’ realized elements whereas the generalized version sees all the structure, including unrealized material. It has been argued that the latter type of constraints allows a natural account for various challenging morphological lengthening facts (Trommer 2011; Trommer & Zimmermann 2014; Zimmermann 2017). It will become crucial in the accounts in section 4 that an unrealized element can be enough to fill a prosodic node for the generalized constraints since it allows to predict the apparent conundrum that copying applies to fill a prosodic node but parts of this copy are deleted in reduction contexts.

(31)	a.	: Assign * for every morphologically coloured µ that does not dominate a phonetically interpreted vowel.
	b.	µ>V: Assign * for every morphologically coloured µ that does not dominate a vowel.

3.2 Gradient Symbolic Representations

This paper assumes a specific version of gradient activity for linguistic elements, namely the theory of ‘Gradient Symbolic Representations’ (=GSR; Rosen 2016; Smolensky & Goldrick 2016; Trommer 2018; Zimmermann 2018a; b; 2019b; Kushnir 2019). The assumption that linguistic elements are not categorical but have activity differences is a rather old one. It was, for example, proposed in Rizzi (1986) and Koster (1986) that functional categories in syntax are either weak or strong and Garde (1965) already observed that some lexical accent system are based on scalar grades of accent strength. The present paper adds to the bulk of recent research that investigates the predictions of numerical gradient activity in an OT-system and argues that it can explain exceptionality. More concretely, elements with different activities can be exceptional (non)undergoers or (non)triggers for phonological processes since they violate constraints to different degrees. GSR thus has the potential for being a general representational account of exceptional phonological behaviour: one morpheme can behave exceptionally with respect to the general phonology of a language since its underlying representation contains a phonological element of a different activity. The proposal in this paper extends the GSR assumption in proposing that the phonology contains operations that predictably adjust the activity of phonological elements. As will become clear in 3.3, the new definition of fission as distribution of underlying activity predictably reduces the activity of copied elements.

Gradient activity of elements can also predict gradient or variable behaviour and hence optionality in grammar (e.g. Rosen 2016), something that will become important in the case study of Sikaiana in section 4.2.1.

This paper thus focusses on the specific predictions that arise from the interaction of gradient activity with the grammatical processes and constraints of a concrete language. Its focus is thus different from other work on non-categorical elements in neural networks (an example that focusses on reduplication is Corina (1994) where it is argued that a neural network can induce prosodic categories).

The crucial mechanism for different phonological behaviour depending on underlying activity are gradient constraint violations: deletion of a weakly active element is, for example, not penalized by MAX as much as deletion of a more active element and is consequently ‘cheaper’ for the phonology. This intuition is illustrated in a first toy example in (32). The input contains three segments with the full default activity of 1 and one segment that only has the weak activity of 0.5. These activities are part of the underlying representations for these morphemes and are hence (arbitrary) lexical properties. The tableau illustrates how these activity differences can result in different phonological behaviour: deletion of a fully active segment (32b) results in a full violation of MAX whereas deletion of the weakly active segment (32c) results in a partial violation of MAX.¹⁰ In the following, underlying activity is notated as a circled number below every segment.¹¹

(32)	Toy example: gradient activity = gradient violations

In the original GSR proposals (Rosen 2016; Smolensky & Goldrick 2016), a GEN restriction excludes all output elements that have an activity other than 1 – a weakly active element like /p/ in (32) must therefore be strengthened by adding 0.5 activity in candidates (32a) and (32c) under violation of DEP. Later work in GSR has argued that the ban on activity other than 1 should be a violable constraint and that languages can tolerate weak or over-strong activity for output elements (Zimmermann 2018b; 2019b). In all the following analyses, realized output elements have to have the full activity of 1 but whether this is ensured by a constraint that happens to be inviolable in those languages or by a GEN restriction is in principle irrelevant. In section 5, it is discussed that there might be evidence from reduplication reduction facts for weak output activity but these arguments are left for future research.

Recall from section 3.1 that the present model is also based on the containment assumption that literal deletion of elements is impossible. GSR allows a natural implementation of this principle in stating that non-realization of an element is setting its activity to zero. This formally implements the implication of GSR that output activity can have phonetic effects (cf. section 5 for a discussion).

All addition or deletion of activity in the phonology is notated in boxes below the underlying activity. Non-realization of an element with activity zero is always marked with a gray background to ease readability. Prosodic nodes also remain unrealized if they only dominate unrealized elements.

Tableau (33) illustrates all these assumptions with another toy example. It contains a floating mora and a vowel that underlyingly only contains an activity of 0.5 and that lacks an underlying association to a mora. Non-realization of this vowel in candidates (33a+c) hence induces a gradient 0.5 violation of MAX. Realization of this vowel in (33b), on the other hand, implies strengthening of this weak segment to a fully active segment under a gradient 0.5 violation of DEP. The floating mora does not dominate any segment in (33a) and induces a violation of both µ>V and (31). Both these violations are avoided in (33b) where the mora dominates a realized and fully active vowel. Candidate (33c), however, only avoids one of these violations since the mora dominates a vowel which is itself deprived of all its activity and thus remains non-realized. This candidate induces a violation of but it crucially avoids a violation of the general µ>V: the mora dominates some vowel, even if this vowel is non-realized. The contrast between candidates (33a) and (33c) is a contrast that is only possible under the containment assumption.

The tableau also illustrates another assumption made in all GSR accounts, namely that the harmony evaluation is based on Harmonic Grammar (=HG) where constraints are weighted, not ranked (Legendre et al. 1990; Potts et al. 2010).¹² In HG, the violations each candidate incurs are taken times the respective weight of the constraints and summed up into a so-called harmony score. The candidate with the highest harmony score (given that constraint violations are negative) is the optimal one. The weight for each constraint is given below them and the harmony score for every candidate in the last column.

(33)	Filling prosodic nodes with realized or unrealized material

A final background assumption for the following accounts is a stratal organization of grammar (Kiparsky 2011; Bermúdez-Otero in preparation) where stems are optimized prior to affix concatenation (Trommer 2011). The important consequence for the following analyses is that the prosodic affixes are added to stems that are already fully prosodified. In addition, all phonological structure at the end of each stratum receives a unified new morphological colour (Trommer 2011). This implements the concept of Bracket Erasure since morpheme differences are neutralized at the end of each stratum. Similarly, the distinction into underlying and epenthetic elements is lost after each stratum. This has important consequences for the present model since it predicts that prosodic nodes that were assigned in an earlier stratum are due to the same constraints µ>V/ as affixed prosodic nodes that are added in the new stratum (cf. especially the discussion of Sikaiana in section 4.2.1).

3.3 Fission as distribution of activity

The new proposal in this paper is that the fission operation creating copies in the phonology is the distribution of underlying activity. This new definition of the GEN operation fission is given in (34).

Fission in this novel sense is illustrated in (35) where two underlying segments /so/ are split into two (35a) and three (35b) output instances of themselves, visible by the corresponding input-output indices. The first intuitive consequence from fission as distribution of activity is that elements that result from fission receive a comparatively small amount of underlying activity. More concretely, all the output instances of /so/ in (35) are weaker than /po/ though all four segments had the same input activity. A second important result is the fact that all elements resulting from fission of the same element have the same amount of (reduced) activity. There is thus no /so/ sequence in (35) that is different from any other /so/ copy, they all have the same activity. Weakening resulting from copying is hence crucially symmetrical. A third important consequence is that more copying weakens elements even more. One copy operation (35a) thus leaves more activity for each segment than two copy operations (35b).

(35)	Copying as distribution of underlying activity

If fission redistributes underlying activity, the evaluation of faithfulness constraints need to be able to distinguish between underlying and epenthetic activity of a segment, even if the former is not necessarily the same as the input activity of the segment in question. More concretely, the first output segment /s/ in (35a) only received 0.5 underlying activity after fission though the input segment it corresponds to has an underlying activity of 1. One way to model this distinction is to extend the assumption of morphological colours to activity. The elements that underwent fission in (35a) thus only have 0.5 underlying or morphologically coloured activity. The faithfulness constraints DEP and MAX penalizing the deletion or addition of activity are now calculated with respect to this (possibly redistributed) underlying or morphologically coloured activity (36). This formalization thus takes the fission intuition to its literal extreme: fissioned or copied elements are indeed the same element distributed among multiple surface instances.

How distribution of elements results in weakening and thus different faithfulness violations is illustrated in (37) where two input segments with activity 1 are split into two respective output instances that get only 0.5 underlying activity each. Realization of every fissioned element consequently induces 0.5 DEP violations. On the other hand, deletion of an element that underwent fission only induces 0.5 violations of MAX since only 0.5 underlying activity remains unrealized. Deletion of a fissioned segment is hence cheaper for MAX than deletion of a fully active element that did not undergo fission. In (37b), this can be seen in the contrast between deleting copied /o/ that is penalized by 0.5 MAX violations and deleting non-copied /s/ that is penalized by 1 MAX violations. Though weakening of copied elements is illustrated with deletion and the gradient violations of MAX here, weakening also affects the violations of IDENT constraints. A weakened element is therefore also more prone to a change of its feature values (cf. section 4.1).

(37)	Fission and gradient DEP/MAX violations

That all copied elements are more prone to reduction (cf. section 2.2) thus follows in this model since they have less underlying activity and induce less faithfulness violations if they undergo any change or remain unrealized. And it automatically falls out that more copying increases the chance of reduction (cf. section 2.3) since more copying implies less underlying activity for each segment. In this model, the four attested language types 1–4 (cf. (3), repeated in (38)) simply differ in the thresholds for reduction set by the weightings of faithfulness constraints.

These different thresholds are illustrated in the toy tableaux (39) and the weighting relations (40). The tableaux compare contexts where there is no copying (39–1), one copy operation (39–2), and two copy operations (39–3). Only two candidates are considered: one candidate without deletion (a.) and one where the penult vowel is deleted (b.) to satisfy some markedness constraint (abbreviated for convenience as DELETEPENULT!). The a.-candidates are preferred by MAX and the b.-candidates by DELETEPENULT! The crucial difference between the contexts are the decreased violations of MAX: it is fully violated in (39–1b), only by 0.5 in (39–2b) and only by 0. in (39–3b). These gradient violations predict exactly the thresholds for deletion in the four language types (38). The weighting relations for language types 1–4 are given in (40). It has to be noted that those weighting relations include ganging effects that are only possible in HG and are impossible under a strict ranking of constraints. More concretely, the weighting relations (40b+c) result in a ranking paradox in OT with ranked constraints. Some exemplifying constraint weights for each language are given in (40) to illustrate that they are indeed possible in HG.

(40)	Reduction thresholds: weighting relations for language types 1–4
	a.	Lg. 1: Always reduction (e.g. Palauan)
		DELETEPENULT! > MAX
		Example weights: DELETEPENULT!=100; MAX=5
	b.	Lg 2: Reduction only if single or multiple reduplication (e.g. Lushootseed)
		MAX > DELETEPENULT! and
		DELETEPENULT! > 0.5×MAX
		Example weights: MAX=100, DELETEPENULT!=99
	c.	Lg 3: Reduction only if multiple reduplication (e.g. Sikaiana)
		0.5×MAX > DELETEPENULT! and
		DELETEPENULT! > 0.×MAX
		Example weights: MAX=100, DELETEPENULT!=49
	d.	Lg 4: No reduction (e.g. Papapana)
		0.×MAX > DELETEPENULT!
		Example weights: MAX=100, DELETEPENULT!=5

The violation profiles in (39) exclude the four other imaginable language types 5–8 in (38) that violate the CWI. This is shown in (41) where the contradictory weighting relations are given that would be necessary to predict such languages. Section 2 established the existence of the four language types in ːast and I’m not aware of any language that resembles the excluded four language types in (41).

(41)	Impossible reduction thresholds: weighting relations for language types 5–8
	a.	*Lg. 5: Reduction without and with single reduplication
		DELETEPENULT! > 0.5×MAX and
		0.×MAX > DELETEPENULT!
	b.	*Lg 6: Reduction without and with multiple reduplication
		DELETEPENULT! > MAX and
		0.5×MAX > DELETEPENULT! and
		DELETEPENULT! > 0.×MAX
	c.	*Lg 7: Reduction only if single reduplication
		MAX > DELETEPENULT! and
		DELETEPENULT! > 0.5×MAX and
		0.×MAX > DELETEPENULT!
	d.	*Lg 8: Reduction only without reduplication
		DELETEPENULT! > MAX and
		0.×MAX > DELETEPENULT!

Before we can turn to concrete analyses for the patterns we saw in 2, it is important to point out one final prediction of this system. Apart from being less protected by faithfulness and hence being more prone to reduction, copied segments are also more costly to realize since this induces violations of DEP. Recall from section 3.2 that it is taken for granted for now that output elements have to have the fully activity of 1 – either due to a language-specific high-weighted constraint or due to a GEN restriction. The picture in (42) emerges where the full activity of 1 for one output segment is depicted as a box, filled with either underlying and/or epenthetic activity. The increase of DEP violations induced by realizing copied segments and the decrease of MAX violations induced by non-realizing copied segments predicts that copied elements might remain unrealized simply because they are copied and thus weakened. Such an instance is discussed in the Southern Wakashan case study in 4.2.

4.1 Threshold 1: Copying enables reduction

4.1.2 Vowel reduction in Lushootseed

If a language only shows reduction in either reduplicant or copied base, there thus has to be an independent reason why simultaneous reduction is either blocked by independent constraints or not required for all copied elements. Reduction in the copied base in Lushootseed (cf. (10)) is an example of the latter pattern. It is analysed here as avoidance of unstressed full vowels. Since stress is regularly on the first non-/ǝ/ vowel in the language¹⁴ and reduplication is prefixing, the reduplicant usually ends up to be stressed and the copied base as unstressed. Reduction of unstressed full vowels to /ǝ/ is a common pattern in Salishan languages and one of at least three possible sources for /ǝ/ in most of these languages (cf., for example, Kinkade 1993; Shaw et al. 1999; Blake 2000).

The trigger for vowel reduction is the constraint (46a), penalizing unstressed vowels with a place feature.¹⁵ The constraint is part of a larger family of constraints proposed to account for sonority-driven stress systems (cf., for example, Kenstowicz (1996); Morén (1999); de Lacy (2002), or de Lacy (2006) for discussion of such systems and (de Lacy 2009) and Shih (2016) for the critical discussion that such systems might in fact not exist). The faithfulness constraint preserving place feature specifications for vowels and effectively penalizing vowel reduction is IDENT(VPLACE) (=IdVPl) (46b). The GSR formulation of standard IDENT constraints is – as MAX and DEP – sensitive to the activity of segments. IDENT thus preserves the feature specification of a segment with higher activity more than the one of a segment with weaker activity.

The tableau (47) shows how the Lushootseed vowel reduction is derived. The prefixing reduplication in the diminutive is taken to result from an affixed empty mora. The first important weighting relation is that a non-copied vowel is never reduced, even if it is unstressed. This can be seen in comparing candidates (47a), (47c), and (47d) with candidate (47b). The former realize the final unstressed vowel as a full vowel and induce a violation of *UNSTRV. Reduction of this vowel in (47b) avoids this violation but induces a violation of IdVPl. The harmony score resulting from this constraint violation is worse than the one resulting from a violation of *UNSTRV. Unstressed non-copied vowels are hence not reduced in Lushootseed. Reduction of an unstressed vowel that is copied as in candidate (47c), however, only induces 0.5 violations of IdVPl. This gradient violation is now outweighed by the full violation of *UNSTRV and candidate (47c) emerges as optimal. Importantly, the first copied vowel is equally weakened and its reduction also only penalized by 0.5 violations of IdVPl as can be seen in candidate (47d). However, this vowel is stressed and does not induce a violation of *UNSTRV; its reduction does hence not improve on any constraint. For completeness, the tableau also gives the DEP violations that are identical for all candidates. Four copied and thus weakened segments are strengthened to fully active output segments in each candidate.

In Lushootseed, reduction hence only applies to the copied base since the relevant markedness constraint is simply not violated in the reduplicant. Another straightforward reason for reduction only in the copied base can be a positional faithfulness constraint protecting a prefixed word-initial reduplicant. An example for this is reduction in the copied base in Klamath or Cupeño (cf. section 2.2.3). The same argumentation holds for instances of reduction only in the reduplicant: independent constraints ensure that it is not the whole copied string which is reduced. In Gitksan (cf. 2.2.2, for example, a high-weighted faithfulness constraint preserving material that is already integrated under prosodic structure protects the base to which a reduplicative morpheme is added and confines reduction to the reduplicant.

To summarize, these case studies showed how all copied elements are symmetrically weakened and reduction can be expected for all copied elements. If reduction can only be observed in reduplicant or copied base, this falls out from orthogonal properties of the phonology of a language. The former case was illustrated with Tagalog in 4.1.1 and the latter with Lushootseed in 4.1.2.

4.2 Threshold 2: More copying enables more reduction

In this subsection, we now turn to type 3 languages that only show reduction in multiple reduplication contexts but never in single reduplication contexts or contexts without any reduplication.

4.2.1 Vowel deletion in Sikaiana

The first example for a type 3 language is Sikaiana (cf. 2.3.1) where an optional vowel deletion process (sosopo/ssopo, cf. (15)) becomes obligatory if the vowel is copied twice (sopossopo/*sopososopo, cf. (17)).

The optionality of vowel deletion outside of reduplication and in single reduplication contexts is formally implemented in a model where harmony scores represent probabilities (Maxent; Johnson 2002; Goldwater & Johnson 2003; Wilson 2006). The C₁VC₁/C₁C₁ alternation is taken to result from vowel deletion triggered by the markedness constraint (49). Importantly, this alternation can not only be found in many other Austronesian languages of this area (e.g. Doku/Lengo (Unger 2018), Dobel (Hughes 2000), or Takuu, Totoli, and Mussau (Blust 2007)), it is also attested in a variety of completely unrelated languages (often termed ‘anti-anti-gemination’; Odden 1988; Bakovic 2005). It could alternatively triggered by an OCP constraint that is violated by C₁VC₁ but not by C₁C₁ (Rose 2000).

(49)	*C1VC1: Assign -1 violation for every vowel that is directly preceded and followed by identical consonants C1.

The first weighting relation that predicts the general optionality of C₁VC₁/C₁C₁ (cf. (16)) is given in (50): a violation of MAX_v is taken to result in an equally good/bad harmony score as a violation of *C₁VC₁. This weighting relation implies that stem vowels lack moras in their underlying representation. If a vowel would be dominated by a mora underlyingly, an additional violation of followed if this vowel is deleted and only realization of the marked C₁VC₁ sequence became optimal. Given that moras on short vowels are completely predictable, it is uncontroversial from a Lexicon Optimization perspective that they are absent in underlying representations.

(50)	Weighting relation
	MAXv = *C1VC1	Optional V-deletion in C1VC1

The two reduplicative morphemes are taken to consist of empty syllable and mora nodes. More concretely, the repetitive consists of two empty syllables resulting in bisyllabic reduplication and the plural of one empty mora resulting in monosyllabic reduplication.¹⁶ These empty prosodic nodes must be filled with segmental material due to σ>V (cf. (31)–b) and µ>V respectively. As was shown in 3.1, the containment assumption in this paper predicts the additional constraints and demanding that the vowels filling a prosodic node must be phonetically realized (cf. (31)–a). In the present account, the former constraint type will enforce copying and the second will penalize any vowel deletion within the new copy.

Both σ>V and µ>V enforcing copying to begin with are taken to have a very high weight and are not shown in the following tableaux.¹⁷ The first relevant context is a single reduplication pattern where an empty mora affix is added to a base. The creation of a monosyllabic reduplicant to fill this mora necessarily induces a violation of *C₁VC₁ as in candidate (51a). Vowel deletion in candidate (51b) avoids this violation but induces a 0.5 violation of MAX and a violation of . The weighting relation (52a) ensures that these two strategies result in nearly the same harmony score and optionality between these two candidates emerges. Deletion of a consonant in candidate (51c) is an alternative strategy to avoid a violation of *C₁VC₁ but MAX is taken to have a quite high weight in Sikaiana.

Note that all epenthetic elements are marked with a circle around them in the following tableaux. This distinction is especially important if we recall that σ>V/ and µ>V/ (31) only require that non-epenthetic prosodic nodes dominate segmental material.¹⁸

(51)	Sikaiana: optional syncope for single monosyllabic reduplicant

(52)	Weighting relations
	a.	*C1VC1 + 0.5×DEPv = + 0.5×MAXv	V-deletion or C1VC1
	b.	0.5×MAXc > *C1VC1 + 0.5×DEPc	No C-deletion
	c.	0.5×DEPv + 0.5×MAXc > + 0.5×MAXv + 0.5×DEPc	No C-deletion

A single reduplication context with only the repetitive plural morpheme does not add anything interesting for a C₁V₁C₂V₂-base. Since a bisyllabic string is copied, no additional violation of *C₁VC₁ is induced and no reason for segment deletion arises: faithful bisyllabic copying emerges as optimal. For a base of the pseudoreduplicated shape C₁V₁C₁V₁, however, the present model predicts optional reduction in both reduplicant and copied base. I was not able to find data to confirm this prediction.

The crucial context is now one where both prosodic affixes are combined and we hence expect a monosyllabic and a bisyllabic reduplicant. Crucially, the string /so/ is copied twice to satisfy both µ>V and σ>V. Consequently, the DEP constraints are violated more if all copied segments are realized whereas the MAX constraints are violated less if /s/ or /o/ are deleted. Candidate (53a) fills all prosodic nodes with segmental material without any deletion and hence avoids any violations of and MAX_v. On the other hand, it induces the violation of *C₁VC₁ that was optionally avoided in a single reduplication context (51). The vowel deletion candidate (53b) only induces 0. violations of MAX_v since the unrealized vowel was copied twice. This difference in faithfulness violations now crosses a threshold were optionality becomes impossible: only the deletion candidate (53b) is optimal. Candidate (53c) shows that consonant deletion is again impossible although it also solves the *C₁VC₁ problem. Since MAX_c has such a high weight, even 0. violations of this constraint are enough to exclude such a candidate.

(53)	Sikaiana: obligatory syncope in multiple reduplication contexts

(54)	Weighting relations
	a.	*C1VC1 + 0.×DEPv > 0.×MAXv +	Only V-deletion
	b.	0.×MAXc + 0.×DEPv > 0.×MAXv +	No C-deletion

A final candidate that one has to consider for the context (53) is one that simply integrates the affixed empty mora under one of the affixed empty syllables (=‘prosodic integration’ in the terminology of Zimmermann (to appear)). This structure avoids two INT_s violations since the string /sopo/ needs to be copied only once to fill all prosodic structure. It is taken to fall out from a high-weighted faithfulness constraint against adding new association lines between morphologically coloured prosodic nodes (Zimmermann to appear) that this structure is sub-optimal in Sikaiana. Though morphologically coloured nodes can be integrated under epenthetic prosodic nodes for free, it comes at a cost to integrate them under other morphologically coloured nodes.

The optionality of vowel deletion in Sikaiana hence vanishes as soon as vowels are copied ‘too often’. Distribution of a vowel into three output instances leaves only 0. input activity for each which makes them too easy to delete and too costly to realize. However, only mora affixes are ‘satisfied’ with phonetically unrealized vowels and deletion only ever applies within the monosyllabic reduplicant.

4.2.2 Segment deletion in Southern Wakashan

The account for deletion in multiple reduplication contexts in Southern Wakashan (cf. (22) and (23)) is interestingly different. Whereas we have an independent markedness reason to shorten a reduplicant in Sikaiana, there is no good phonotactic reason in Southern Wakashan why one reduplicant should completely be deleted if another one is adjacent. Southern Wakashan is rather an example for a pattern where copied elements are avoided simply because there are copied too often and realization of weakened elements is too costly. Recall from the discussion of (42) that the present model predicts such a situation since more DEP violations are induced by realizing and fewer MAX violations by deleting a copied and thus weakened segment.

The first obvious question for this account is why only one of the reduplicants is deleted in a multiple reduplication context. If double copying weakens all the copied elements so much that they are rather deleted than strengthened to fully active output elements, we would expect that no reduplicant and no copied base elements surface in a multiple reduplication context. For the Ditidaht pattern, it is argued that the surfacing of one of the reduplicants follows since only morphologically coloured syllables tolerate it to be filled by deleted vowels, morphologically coloured moras have to be filled by realized vowels.¹⁹ Theoretically, this is implemented by the different weights of and that predict that only a violation of the latter is tolerated in Ditidaht. Tableau (55) shows this for a multiple reduplication context where both an empty syllable and two empty moras are added. Note that this tableau is a simplification and only gives the relevant reduplication-triggering portions of the added affix. Filling both these underlyingly empty prosodic nodes with segmental material²⁰ as in candidate (55a) induces quite a number of DEP violations since 6 segments with only 0. activity need to be strengthened to fully active output elements. The optimal candidate (56b) is one that deletes the consonant and vowel that were copied to fill the empty syllable affix. This induces 0. MAX violations in addition to a violation which still results in a better harmony score than the 0. violations of DEP that this candidate avoids in comparison to (56a). Deletion of the material under the affixed mora nodes in candidate (56c), on the other hand, induces violations of since two morphologically coloured moras only dominate a phonetically invisible vowel. Since the weight of is higher than the weight of , this candidate is sub-optimal. For the same reason, the deletion in both reduplicants (56d) and simultaneous deletion in reduplicant and copied base (56e) is excluded. The latter candidate also induces a violation since the mora of the copied base was already present in the input. The general µ>V and σ>V that trigger reduplication to begin with (and which are satisfied with dominating deleted segments) are not given to ease readability of the tableau. Recall again that prosodic nodes that dominate only unrealized elements with an activity of zero remain unrealized themselves and allow line crossing (cf. candidate (55e)).

1. (55)

1. Ditidaht: deletion in multiple reduplication contexts
2. 

(56)	Weighting relations
	a.	1.×DEP > 0.×MAX +	Deletion within a σ-affix
	b.	2×µ>V >	No deletion for a µ-affix
	c.	0.×MAX + 2×µ>V > 1.×DEP	No deletion within a µ-affix
	d.	0.×MAX + + > 1.×DEP	No deletion within the copied base

The weighting relation in (56a) shows that deletion of segments that are copied to fill an affixed syllable node are possible in Southern Wakashan if these segments are split into three output instances. For a complete account, the weighting relation in (57) is also crucial that predicts that no deletion is possible in single reduplication contexts when segments are only split into two output instances. In such a context, the segments still receive 0.5 underlying activity and the violations of MAX and are not out-weighed by the 0.5×DEP violations.

(57)	Weighting relations
	0.5×MAX + > 0.5×DEP	No deletion if single reduplication

To summarize, these final two case studies showed concrete accounts of languages of type 3 that only show reduction in multiple reduplication contexts. The two case studies were interestingly different since the reduction has an independent markedness reason (=*C₁VC₁) in Sikaiana whereas the copied segments are reduced simply because they are copied in Southern Wakashan. It was shown that both patterns straightforwardly fall out under the assumption of fission as distribution of underlying activity and standard markedness and faithfulness constraints within a HG account.