2.1 Representational framework

Consider a finite set of segments (for instance, the segments in the IPA table, or some subset thereof), denoted by a, b, c, … . Strings obtained through segment concatenation are denoted by a , b , c , … . The notation a = a ₁⋯a _ℓ says that the string a  is the concatenation of the segments a ₁, …, a _ℓ and thus has length $\ell$ . This paper assumes the representational framework (4). Underlying and surface forms are strings of segments. Phonological candidates establish a correspondence between the segments of these underlying and surface strings.

Correspondence relations will be denoted by thin lines. To illustrate, (5) represents the candidate ( a , b , 𝜌 _{a , b} ) whose underlying string a  is /bnɪk/, whose surface string b  is [blɪk], whose correspondence relation 𝜌 _{a , b} maps underlying to surface segments respecting their position in the strings.

The representational assumption (4) is consistent with additional restrictions on the candidate set (Blaho, Bye & Krämer Reference Blaho, Bye and Krämer2007). This flexibility will be exploited in this paper, which will explore the implications of various restrictions on the correspondence relations that can figure in the candidate set.

2.2 Identity candidates

As anticipated in Section 1, idempotency is about phonotactically licit forms being mapped to themselves. It thus requires the distinction between underlying and surface forms to be blurred. This is achieved through axiom (6). It can be interpreted as a candidacy reflexivity axiom, as it requires each (surface) string to be in correspondence with itself. This axiom will play a crucial role in the definition of idempotency in the next subsection.

By (6), any surface form can be construed as an underlying form (of the corresponding identity candidate). In other words, the set of surface forms is a subset of the set of underlying forms. This is a slightly weaker condition than Moreton’s (Reference Moreton and McCarthy2004b) homogeneity condition, which requires the sets of underlying and surface forms to coincide. Both reflexivity and homogeneity hold when underlying and surface representations are constructed out of the same ‘building blocks’. Moreton claims that ‘most phonological representations are in fact present in both [underlying and surface forms]’, so that reflexivity and homogeneity hold for ‘much of the core business of phonology’. Yet, the reflexivity axiom (6) obviously does not hold in full generality. To illustrate, suppose that the candidate set contains the candidate ( a , b ) = (/mabap/, [ma.ba]). The reflexivity axiom (6) requires the candidate set to also contain the candidate ( b , b ) = (/ma.ba/, [ma.ba]). This contravenes the plausible assumption that syllabification is a property of the surface representations and is absent in the underlying representations. In the case of constraint-based phonology, this difficulty can be circumvented by switching from the identity candidates required by (6) to McCarthy’s (Reference McCarthy2002) fully faithful candidates, as explained in Section 3.6.

2.3 Idempotency

Within the representational framework just defined, a phonological grammar is a map $G$ that takes an underlying form a  and returns a candidate ( a , b , 𝜌 _{a , b} ) whose underlying string is indeed a .Footnote ^[3] A string b  is called phonotactically licit according to a grammar $G$ provided that there exists at least one string a  (with a  possibly identical to b ) such that the grammar $G$ maps the underlying form a  to a candidate ( a , b , 𝜌 _{a , b} ) whose surface string is b . A grammar $G$ is idempotent provided that it maps any phonotactically licit surface form to itself, as formalized by the implication (7) in the following definition. The antecedent of the implication says that the surface form b  is phonotactically licit relative to the grammar $G$ , because it is the surface realization of some underlying form a . The consequent says that b  is then mapped faithfully to itself. The reflexivity axiom (6) ensures the existence of the identity candidate  in the consequent of (7).

To illustrate, suppose that a grammar raises the low vowel /a/ to [e]. The mid vowel [e] is therefore phonotactically licit. In order for that grammar to comply with condition (7) and thus qualify as idempotent, the underlying form /e/ must be mapped faithfully to [e].Footnote ^[4]

2.4 Chain shifts

The candidate set might provide many different relations 𝜌 _{b , b} to put a string b  in correspondence with itself. Some options are illustrated in (8) in the case where b = amba. Axiom (6) requires one of these correspondence relations provided by the candidate set to be the identity relation  illustrated by the left-most candidate in (8). This identity relation  is intuitively the best way to put the string b  in correspondence with itself. A grammar $G$ is well-behaved provided that it abides by this intuition: whenever $G$ maps an underlying string b  to the same surface string b , it does so through the identity correspondence relation  In other words, G ( b ) = ( b , b , 𝜌 _{b , b} ) is impossible when 

Suppose now that a grammar $G$ fails at the idempotency implication (7) for some candidate ( a , b , 𝜌 _{a , b} ), as stated in (9): $G$ maps the underlying form a  to ( a , b , 𝜌 _{a , b} ), as required by the antecedent of the idempotency implication; but $G$ fails to map the underlying form b  to the identity candidate  as required by the consequent.

Condition (9b) means that $G$ maps the underlying form b  to some candidate ( b , c , 𝜌 _{b , c} ) different from  This means that either the two strings b  and c  differ, or b  and c  coincide but the two correspondence relations 𝜌 _{b , c} and  differ. The latter option is impossible when $G$ is well-behaved. The strings b  and c  must thus differ and condition (9) becomes (10).

Condition (10) says that $G$ maps a  to b  and then in turn maps b  to c . Since b ≠ c , this scheme a → b → c is called a chain shift in the phonological literature (see Łubowicz Reference Łubowicz, van Oostendorp, Ewen, Colin and Rice2011 for a comprehensive review). In conclusion, a (well-behaved) grammar $G$ fails at idempotency if and only if it enforces chain shifts.

3.1 Optimality Theory (OT)

A constraint $C$ is a function that takes a candidate ( a , b , 𝜌 _{a , b} ) and returns a number of violations C ( a , b , 𝜌 _{a , b} ) that is large when the candidate scores poorly from the perspective relevant to that constraint. A constraint $C$ prefers a candidate ( a , b , 𝜌 _{a , b} ) to another candidate ( c , d , 𝜌 _{c , d} ) provided that it assigns fewer violations to the former than to the latter, namely C ( a , b , 𝜌 _{a , b} ) < C ( c , d , 𝜌 _{c , d} ). A constraint ranking is an arbitrary linear order $\gg$ over a set of constraints. A constraint ranking $\gg$ prefers a candidate ( a , b , 𝜌 _{a , b} ) to another candidate ( c , d , 𝜌 _{c , d} ) provided that the $\gg$ -highest constraint among those which assign a different number of violations to the two candidates ( a , b , 𝜌 _{a , b} ) and ( c , d , 𝜌 _{c , d} ), prefers the former candidate to the latter. Given a ranking $\gg$ , the corresponding OT grammar $G_{\gg }$ maps an underlying form a  to a candidate ( a , b , 𝜌 _{a , b} ) that is preferred by the ranking $\gg$ to all other candidates ( a , c , 𝜌 _{a , c} ) that share that underlying form a .Footnote ^[5]

3.2 Classical OT

A faithfulness constraint $F$ has the property that it never assigns any violations to any identity candidate  as stated in (11a). A markedness constraint $M$ has the property that it is blind to underlying forms, so that it assigns the same number of violations to any two candidates ( a , c , 𝜌 _{a , c} ) and ( b , c , 𝜌 _{b , c} ) sharing the surface form c  (independently of their underlying forms), as stated in (11b).

Given the candidacy reflexivity axiom (6), no (non-trivial) constraint can be both a faithfulness and a markedness constraint. In fact, suppose by contradiction that were the case for some constraint $C$ . Consider an arbitrary candidate ( a , b , 𝜌 _{a , b} ) in the candidate set. The reflexivity axiom thus ensures that the candidate set also contains the corresponding identity candidate  These two candidates ( a , b , 𝜌 _{a , b} ) and  share the surface form b . Since $C$ is a markedness constraint, $C$ must assign the same number of violations to those two candidates, as stated in (12a). Since $C$ is also a faithfulness constraint, it does not penalize the identity candidate  as stated in (12b).

In conclusion, $C$ does not penalize any candidate, and it is therefore trivial.Footnote ^[6]

Although no constraint can be both a faithfulness and a markedness constraint, it can easily be neither (for instance, comparative markedness constraints are neither; see McCarthy Reference McCarthy2002, Reference McCarthy2003a as well as Section 7 below for additional references). To rule out the latter case, I assume throughout the paper that the constraint set only consists of faithfulness and markedness constraints (this is Moreton (Reference Moreton and McCarthy2004b) conservativity assumption).

Let me call classical the version of OT endowed with the latter restriction (13) on the constraint set.

3.3 A sufficient condition for chain shifts

The classical assumption (13) that each constraint is either a faithfulness constraint (11a) or a markedness constraint (11b) ensures that the identity candidate  harmonically bounds any candidate ( b , b , 𝜌 _{b , b} ) with  To illustrate, the left-most candidate in (8) outperforms the other candidates listed. In fact, faithfulness constraints cannot prefer ( b , b , 𝜌 _{b , b} ) to  by (11a); and markedness constraints do not distinguish between two such candidates, by (11b). In other words, the OT grammar $G_{\gg }$ corresponding to any ranking $\gg$ is well-behaved in the sense of Section 2.4: G _≫( b )≠( b , b , 𝜌 _{b , b} ) whenever 𝜌 _{b , b} ≠I _{b , b} . The characterization of non-idempotency in terms of chain shifts in Section 2.4 thus applies to (classical) OT grammars. To distill the implications of that characterization, let me weaken the ‘if-and-only-if’ statement (10) into the ‘if’ statement (14). In fact, if the grammar $G_{\gg }$ maps the underlying form a  to the candidate ( a , b , 𝜌 _{a , b} ), as stated in (10a), the ranking $\gg$ must in particular prefer the candidate ( a , b , 𝜌 _{a , b} ) to any other loser candidate ( a , c , 𝜌 _{a , b} ), as stated in (14a). Furthermore, if the grammar $G_{\gg }$ maps the underlying form b  to the candidate ( b , c , 𝜌 _{b , c} ), as stated in (10a), the ranking $\gg$ must in particular prefer this candidate ( b , c , 𝜌 _{b , c} ) to the identity candidate  as stated in (14b).

Condition (14b) that the ranking $\gg$ prefers ( b , c , 𝜌 _{b , c} ) to the identity candidate  means that the constraint set contains a constraint that prefers ( b , c , 𝜌 _{b , c} ) to  such that all of the constraints that are ranked by $\gg$ above it assign the same number of violations to the two candidates. By (11a), this constraint that prefers ( b , c , 𝜌 _{b , c} ) to  cannot be a faithfulness constraint and must instead be a markedness constraint $M$ . Condition (14b) can thus be explicitated as (15b) and (15c).

Since faithfulness constraints assign no violations to identity candidates by (11a), condition (15c) that any faithfulness constraint ranked above $M$ assigns the same number of violations to ( b , c , 𝜌 _{b , c} ) and ( b , b , I _{b , b} ) means that it assigns no violations to ( b , c , 𝜌 _{b , c} ). Condition (15c) can thus be made explicit as in (16c) and (16d).

The designated markedness constraint $M$ prefers ( b , c , 𝜌 _{b , c} ) to  by (16b). Furthermore, it is blind to the underlying forms, by (11b). Hence, $M$ also prefers ( a , c , 𝜌 _{a , c} ) to ( a , b , 𝜌 _{b , b} ). Assumption (16a) thus requires $M$ to be ranked below some constraint with the opposite preference. The latter constraint cannot be a markedness constraint, because of (16d). It must therefore be a faithfulness constraint. Condition (16a) can thus be made explicit as in (17a).

Condition (17) just derived is necessary for idempotency to fail.

3.4 The faithfulness idempotency condition (FIC)

I am now ready to tackle the central question of this section: which conditions ensure that the OT grammars corresponding to any ranking of a given constraint set are idempotent? The answer to this question is provided by the following Lemma 1. The assumption made by the lemma is twofold. First, it restricts the candidate set: if it contains two candidates ( a , b , 𝜌 _{a , b} ) and ( b , c , 𝜌 _{b , c} ) that share a string b  as the surface and underlying form respectively, it must also contain a candidate ( a , c , 𝜌 _{a , c} ) that puts the underlying string a  of the former candidate in correspondence with the surface string c  of the latter candidate, as in (18).

Second, the assumption of the lemma restricts the constraint set: it requires all the faithfulness constraints to satisfy the implication (19), which is referred to as the faithfulness idempotency condition (FIC). The specific implication (3) in Section 1 is a concrete example of the FIC.

Lemma 1 follows straightforwardly from the discussion in the preceding subsection: the FIC (19) makes the two conditions (17a) and (17c) incompatible and thus prevents idempotency from failing. In fact, (17a) requires the designated markedness constraint $M$ to be outranked by a faithfulness constraint $F$ which assigns fewer violations to ( a , b , 𝜌 _{a , b} ) than to ( a , c , 𝜌 _{a , c} ). This means that the consequent of the FIC (19) fails. The antecedent must therefore fail as well. This means in turn that $F$ assigns some violations to ( b , c, 𝜌 _{b , c} ), contradicting (17c).

3.5 Composition candidates and the FIC $_{comp}$

Lemma 1 makes no assumptions on the nature of the correspondence relation 𝜌 _{a , c} depicted in (18) and in particular on its relationship with the two other correspondence relations 𝜌 _{a , b} and 𝜌 _{b , c} . For instance, 𝜌 _{a , c} could be the empty relation. This would make the FIC (19) trivial when $F$ is an identity faithfulness constraint (because the quantity on the left-hand side of the inequality in the consequent would be equal to zero) but difficult when $F$ is Dep or Max (because the quantity on the left-hand side of the inequality would be large in this case). At the opposite extreme, 𝜌 _{a , c} could be the total relation, which puts any underlying segment in correspondence with any surface segment. This would make the FIC (19) trivial when $F$ is Dep or Max but difficult when $F$ is an identity faithfulness constraint.

A natural assumption is that 𝜌 _{a , c} is the composition 𝜌 _{a , c} =𝜌 _{a , b} 𝜌 _{b , c} of the two correspondence relations 𝜌 _{a , b} and 𝜌 _{b , c} .Footnote ^[7] This means that a segment a of the string a  and a segment c of the string c  are in correspondence through 𝜌 _{a , b} 𝜌 _{b , c} if and only if there exists some ‘mediating’ segment b of the string b  such that a is in correspondence with b through 𝜌 _{a , b} and furthermore b is in correspondence with c through 𝜌 _{b , c} (many examples will be provided in Sections 4–6). The existence of this composition candidate is guaranteed by (20), which can thus be interpreted as a candidacy transitivity axiom, complementing the reflexivity axiom (6).

The original FIC (19) can now be specialized in terms of this composition candidate as the implication (21), which will be referred to as the FIC $_{\text{comp}}$ to highlight the fact that the left-hand side of the inequality in the consequent features the composition candidate. The FIC $_{\text{comp}}$ entails the original FIC and thus provides a sufficient condition for the idempotency of all of the grammars in an OT typology.

The FIC $_{\text{comp}}$ is only a sufficient condition for idempotency, not a necessary-and-sufficient characterization of idempotency.Footnote ^[8] Yet, the FIC $_{\text{comp}}$ is a tight sufficient condition: for any faithfulness constraint that fails at the FIC $_{\text{comp}}$ , it is possible to construct a counterexample where idempotency indeed fails, as will be shown in Section 7.

3.6 Refinements

The definition of idempotency in Section 2.3 crucially relies on the existence of the identity candidate, as guaranteed by the reflexivity axiom (6). Yet, as discussed in Section 2.2, this reflexivity axiom fails when surface representations are richer than underlying representations. For instance, the identity candidate ( b , b ) = (/ma.ba/, [ma.ba]) makes no sense if syllabification is construed as a surface property. Within a constraint-based framework such as OT, this difficulty can be circumvented as follows.Footnote ^[9] Following McCarthy (Reference McCarthy2002: Section 6.2), ( a , b , 𝜌 _{a , b} ) is called a fully faithful candidate (FFC) relative to a constraint set provided that it violates no faithfulness constraints in that constraint set. Identity candidates  are FFCs, because of the definition (11a) of faithfulness constraints. Yet, non-identity candidates can also qualify as FFCs. For instance, the candidate (/maba/, [ma.ba]) is not the identity candidate and yet qualifies as an FFC, under the plausible assumption that syllabification of tautomorphemic sequences is never contrastive and that no faithfulness constraint is therefore sensitive to syllabification. The reasoning presented in this section holds unchanged if idempotency is re-defined as follows: whenever G ( a ) = ( a , b , 𝜌 _{a , b} ), there exists an FFC (𝛽, b , 𝜌 _{𝛽, b} ) such that G (𝛽) = (𝛽, b , 𝜌 _{𝛽, b} ). This definition of idempotency does not require the existence of the identity candidate  and thus dispenses with the problematic reflexivity axiom (6). Instead, it requires the following weaker axiom on the candidate set: for any candidate ( a , b , 𝜌 _{a , b} ) with a surface string b , the candidate set also contains an FFC (𝛽, b , 𝜌 _{𝛽, b} ) with that same surface form  b . This assumption complements McCarthy’s (Reference McCarthy2002) assumption that each underlying form a  admits an FFC ( a , 𝛼, 𝜌 _{a , 𝛼} ) with that underlying form.

4.1 Max

The faithfulness constraint Max assigns to a candidate ( a , b , 𝜌 _{a , b} ) one violation for each deleted underlying segment, namely for each segment of the underlying string a  that has no corresponding segments in the surface string b  according to 𝜌 _{a , b} (McCarthy & Prince Reference McCarthy, Prince, Beckman, Urbanczyk and Walsh Dickey1995, Harris Reference Harris, van Oostendorp, Ewen, Hume and Rice2011, and references therein). To illustrate, Max assigns two violations to the candidate ( a , b , 𝜌 _{a , b} ) in (23), because of its two underlying deleted segments /s/ and /e/.

Let us consider two candidates ( a , b , 𝜌 _{a , b} ) and ( b , c , 𝜌 _{b , c} ) together with their composition candidate ( a , c , 𝜌 _{a , b} 𝜌 _{b , c} ). Does the faithfulness constraint Max satisfy the FIC $_{\text{comp}}$ (21)?

If the antecedent of the implication is false, the implication trivially holds. Thus, let us suppose that the antecedent is true, namely that the candidate ( b , c , 𝜌 _{b , c} ) does not violate Max. For instance, assume that the strings b  and c  consist of two corresponding consonants each, as represented in (24a).

Let us now turn to the inequality in the consequent of the FIC $_{\text{comp}}$ . If the left-hand side of the inequality is zero, the inequality trivially holds. Thus, let us suppose that the left-hand side is larger than zero, namely that the composition candidate ( a , c , 𝜌 _{a , b} 𝜌 _{b , c} ) does violate Max. For instance, assume that the last of the three consonants of the string a  is deleted in c  according to the composition correspondence relation 𝜌 _{a , b} 𝜌 _{b , c} , as represented in (24b). If the consonant /r/ of a  had a correspondent [s] or [t] in b  according to 𝜌 _{a , b} , then it would also have a correspondent in c  according to 𝜌 _{a , b} 𝜌 _{b , c} , because both segments /s/ and /t/ of b  have a correspondent in c  relative to 𝜌 _{b , c} . Thus, the correspondence relation 𝜌 _{a , b} must fail to provide a surface correspondent of /r/ in b , as represented in (24c). This says in turn that the candidate ( a , b , 𝜌 _{a , b} ) which figures in the right-hand side of the FIC $_{\text{comp}}$ inequality violates Max as well, so that the inequality holds in this case.

This reasoning suggests that the FIC $_{\text{comp}}$ (21) holds because the assumption that no segment of b  is deleted in c  (the antecedent of the FIC $_{\text{comp}}$ ) entails that any segment of a  that is deleted in c  (as quantified by the left-hand side of the inequality in the consequent) is also deleted in b  (as quantified by the right-hand side of the inequality). Lemma 2 thus obtained will be refined in Section 5 and proved in Appendix A.1.

4.2 Dep

The faithfulness constraint Dep assigns to a candidate ( a , b , 𝜌 _{a , b} ) one violation for each epenthetic surface segment, namely for each segment of the surface string b  that has no corresponding segments in the underlying string a  according to 𝜌 _{a , b} (McCarthy & Prince Reference McCarthy, Prince, Beckman, Urbanczyk and Walsh Dickey1995, Hall Reference Hall, van Oostendorp, Ewen, Hume and Rice2011, and references therein). To illustrate, Dep assigns two violations to the candidate ( a , b , 𝜌 _{a , b} ) in (25), because of its two epenthetic vowels [ə] and [e] (from Temiar; Itô Reference Itô1989).

Let us consider two candidates ( a , b , 𝜌 _{a , b} ) and ( b , c , 𝜌 _{b , c} ) together with their composition candidate ( a , c , 𝜌 _{a , b} 𝜌 _{b , c} ). Does the faithfulness constraint Dep satisfy the FIC $_{\text{comp}}$ (21)?

We can reason exactly as in the preceding Section 4.1. We assume that the antecedent of the FIC $_{\text{comp}}$ holds, namely that the candidate ( b , c , 𝜌 _{b , c} ) does not violate Dep, as in (26a). Moreover, we assume that the left-hand side of the inequality in the consequent of the FIC $_{\text{comp}}$ is larger than zero, namely that the composition candidate ( a , c , 𝜌 _{a , b} 𝜌 _{b , c} ) does violate Dep, say because of the surface [ə] with no underlying correspondents in (26b).

By definition of the composition correspondence relation 𝜌 _{a , b} 𝜌 _{b , c} , it follows that the vowel [ə] of b  cannot have a correspondent relative to 𝜌 _{a , b} , as represented in (26c). This says in turn that the candidate ( a , b , 𝜌 _{a , b} ) that figures in the right-hand side of the FIC $_{\text{comp}}$ inequality violates Dep as well, so that the inequality holds in this case.

In order to secure the FIC $_{\text{comp}}$ for Dep, some additional care is needed, though: the correspondence relation 𝜌 _{b , c} must be prevented from breaking any underlying segments into two or more surface segments, as shown by the counterexample (27).

The antecedent of the FIC $_{\text{comp}}$ holds, as shown in (27a): the candidate ( b , c , 𝜌 _{b , c} ) does not violate Dep, because every segment of c  has a correspondent, although the two surface vowels [e] and [i] share the underlying correspondent /ə/. The right-hand side of the FIC $_{\text{comp}}$ inequality is equal to 1, as shown in (27c): the candidate ( a , b , 𝜌 _{a , b} ) violates Dep once, because it has a unique epenthetic vowel [ə]. The FIC $_{\text{comp}}$ inequality fails because its left-hand side is instead equal to 2, as shown in (27b): the composition candidate ( a , c , 𝜌 _{a , b} 𝜌 _{b , c} ) violates Dep twice, because both [e] and [i] are epenthetic.

These considerations lead to Lemma 3, which will be refined in Section 5 and proved in Appendix A.2.

Lemmas 2 and 3 highlight a difference between Max and Dep: the former satisfies the FIC $_{\text{comp}}$ without additional assumptions; the latter instead requires the correspondence relation 𝜌 _{b , c} not to break any underlying segments, forbidding scenarios such as (27a). The reason behind this difference can be intuitively appreciated as follows. Dep quantifies over epenthetic surface segments, and the two candidates ( a , c , 𝜌 _{a , b} 𝜌 _{b , c} ) and ( a , b , 𝜌 _{a , b} ) compared by the inequality in the consequent of the FIC $_{\text{comp}}$ have different surface strings b  and c . In order to make these two strings ‘commensurate’, the correspondence relation 𝜌 _{b , c} that links them cannot break underlying segments. Max instead quantifies over deleted underlying segments, and the two candidates ( a , c , 𝜌 _{a , b} 𝜌 _{b , c} ) and ( a , b , 𝜌 _{a , b} ) compared by the FIC $_{\text{comp}}$ inequality share the underlying form a , so that no additional ‘commensurability’ assumptions are needed.

4.3 Ident

A phonological feature $\unicode[STIX]{x1D711}$ takes a segment a and returns a feature value. A feature is called binary if it takes only two values; otherwise, it is called multi-valued. For instance, the feature [nasal] is binary while the feature [place] could be construed as distinguishing between three major places of articulation, making it multi-valued (de Lacy Reference de Lacy2006: Section 2.3.2.1.1). A feature $\unicode[STIX]{x1D711}$ is called total (relative to the candidate set) provided that there is no underlying or surface string that contains a segment for which the feature $\unicode[STIX]{x1D711}$ is undefined. The identity faithfulness constraint Ident $_{\unicode[STIX]{x1D711}}$ corresponding to a total feature $\unicode[STIX]{x1D711}$ assigns to a candidate ( a , b , 𝜌 _{a , b} ) one violation for each pair (a, b) of an underlying segment a and a surface segment b that are put in correspondence by 𝜌 _{a , b} despite the fact that they are assigned different values by the feature $\unicode[STIX]{x1D711}$ (McCarthy & Prince Reference McCarthy, Prince, Beckman, Urbanczyk and Walsh Dickey1995). To illustrate,  assigns two violations to the candidate ( a , b , 𝜌 _{a , b} ) in (28), because of the two corresponding pairs (/n/, [t]) and (/k/, [ŋ]).

Let us consider two candidates ( a , b , 𝜌 _{a , b} ) and ( b , c , 𝜌 _{b , c} ) together with their composition candidate ( a , c , 𝜌 _{a , b} 𝜌 _{b , c} ). Does the identity faithfulness constraint  satisfy the FIC $_{\text{comp}}$ (21)?

We can reason exactly as in the two preceding Sections 4.1 and 4.2. We assume that the antecedent of the FIC $_{\text{comp}}$ holds, namely that the candidate ( b , c , 𝜌 _{b , c} ) does not violate  as in (29a).

Moreover, we assume that the left-hand side of the inequality in the consequent of the FIC $_{\text{comp}}$ is larger than zero, namely that the composition candidate ( a , c , 𝜌 _{a , b} 𝜌 _{b , c} ) does violate  as in (29b). By definition of the composition correspondence relation 𝜌 _{a , b} 𝜌 _{b , c} , 𝜌 _{a , b} must put in correspondence the underlying nasal /ŋ/ of a  with the surface oral [ɡ] of b , as represented in (29c). This says in turn that the candidate ( a , b , 𝜌 _{a , b} ) that figures in the right-hand side of the FIC $_{\text{comp}}$ inequality violates  as well, so that the inequality holds.

Also for  as for Dep, the FIC $_{\text{comp}}$ requires no underlying segment to be broken by the correspondence relation 𝜌 _{b , c} , as shown by the counterexample (30), analogous to (27).

The antecedent of the FIC $_{\text{comp}}$ holds: gemination preserves nasality in the candidate in (30a) (which could correspond, for instance, to the Japanese loan [fu.róɡ.ɡu] of English frog; Kubozono, Ito & Mester Reference Kubozono, Ito and Mester2008). However, the inequality in the consequent of the FIC $_{\text{comp}}$ fails: the composition candidate (30b) violates  twice because of the gemination while the candidate (30c) violates it only once, so that the left-hand side of the inequality exceeds the right-hand side.

These considerations extend from  to the identity faithfulness constraint  corresponding to any feature $\unicode[STIX]{x1D711}$ , independently of whether it is binary or multi-valued, as long as it is total. The case of partial features is indeed more delicate. Assume that the identity faithfulness constraint Ident $_{\unicode[STIX]{x1D711}}$ corresponding to a partial feature $\unicode[STIX]{x1D711}$ assigns to a candidate ( a , b , 𝜌 _{a , b} ) one violation for each pair (a, b) ∈𝜌 _{a , b} of corresponding segments such that the feature $\unicode[STIX]{x1D711}$ is defined for both segments and assigns them a different value. Thus, Ident $_{\unicode[STIX]{x1D711}}$ is not violated when the feature $\unicode[STIX]{x1D711}$ is undefined for at least one of the two segments.Footnote ^[10] To illustrate, suppose that the feature [strident] is only defined for coronals (Hayes Reference Hayes2009). The corresponding constraint  does not satisfy the FIC $_{\text{comp}}$ (21), as shown by the counterexample (31).

The antecedent of the FIC $_{\text{comp}}$ holds: the candidate ( b , c , 𝜌 _{b , c} ) in (31a) does not violate  because the underlying /f/ of b  is not coronal and is thus undefined for stridency. The right-hand side of the FIC $_{\text{comp}}$ inequality is equal to 0: the candidate ( a , b , 𝜌 _{a , b} ) in (31c) does not violate  because again [f] is undefined for stridency. The FIC $_{\text{comp}}$ inequality fails because its left-hand side is equal to 1: the composition candidate ( a , c , 𝜌 _{a , b} 𝜌 _{b , c} ) in (31b) does violate  because of the two corresponding coronal segments /θ/ and [s].

These considerations lead to Lemma 4, which will be refined in Section 5 and proved in Appendix A.3.

5.1 Max $_{R}$

A restriction $R$ pairs a string a  with a subset R ( a ) of its segments. A segment of the string a  satisfies the restriction provided that it belongs to R ( a ). The faithfulness constraint Max $_{R}$ assigns to a candidate ( a , b , 𝜌 _{a , b} ) one violation for each segment of the underlying string a  that satisfies the restriction $R$ and is deleted. Deletion of underlying segments that do not satisfy the restriction is not penalized. To illustrate, consider the restriction $R$ which pairs a string a  with the set R ( a ) of its consonants. The corresponding constraint Max $_{R}$ is the constraint Max-C which militates against consonant deletion, but is not offended by vowel deletion (it thus assigns only one violation to the candidate (23), while unrestricted Max assigns two violations).

While we have seen that unrestricted Max satisfies the FIC $_{\text{comp}}$ , its restricted counterpart Max $_{R}$ can fail at the FIC $_{\text{comp}}$ , as shown by the counterexample (32) for Max $_{R}$   $=$  Max-C.

The antecedent of the FIC $_{\text{comp}}$ holds: the candidate ( b , c , 𝜌 _{b , c} ) in (32a) does not violate Max-C, because the deleted segment is a vowel. The right-hand side of the FIC $_{\text{comp}}$ inequality is equal to zero: the candidate ( a , b , 𝜌 _{a , b} ) in (32c) does not violate Max-C, because it involves no deletion. The FIC $_{\text{comp}}$ inequality thus fails, because its left-hand side is instead equal to 1: the composition candidate ( a , c , 𝜌 _{a , b} 𝜌 _{b , c} ) in (32b) does violate Max-C, because it deletes a consonant.

In order for Max $_{R}$ to fail at the FIC $_{\text{comp}}$ in (32), it is crucial that 𝜌 _{a , b} establishes a correspondence between the consonant /s/ and the vowel [e], namely between a segment that satisfies the restriction $R$ and a segment that does not satisfy it. Given a candidate ( a , b , 𝜌 _{a , b} ), the correspondence relation 𝜌 _{a , b} is said to exit from the restriction $R$ if it puts some underlying segment a that satisfies the restriction $R$ in correspondence with some surface segment b that does not satisfy it, as in (33). The top and bottom rectangles represent the sets of segments of a  and b , with the subsets selected by the restriction $R$ highlighted in gray.

The following lemma ensures that Max $_{R}$ satisfies the FIC $_{\text{comp}}$ provided that no correspondence relation in the candidate set exits from the restriction $R$ . To illustrate, the lemma guarantees that Max-C satisfies the FIC $_{\text{comp}}$ provided that no underlying consonant is in correspondence with a surface vowel. The lemma will be further extended in Section 5.3.

A restriction is trivial provided that it pairs every string with the totality of its segments. The case of unrestricted Max discussed in Section 4.1 follows as a special case of Max $_{R}$ with a trivial restriction $R$ : no correspondence relation can exit from $R$ in this case and (33) is thus contradictory.

5.2 Dep $^{S}$

The reasoning in Section 5.1 extends straightforwardly from Max to Dep. Given a restriction $S$ , the corresponding faithfulness constraint Dep $^{S}$ assigns to a candidate ( a , b , 𝜌 _{a , b} ) one violation for each segment of the surface string b  that satisfies the restriction $S$ and is epenthetic.Footnote ^[11] To illustrate, consider the restriction $S$ which pairs a string with the set of its vowels. The corresponding constraint Dep $^{S}$ is the constraint Dep-V which militates against vowel epenthesis, but is not offended by consonant epenthesis. Given a candidate ( a , b , 𝜌 _{a , b} ), the correspondence relation 𝜌 _{a , b} is said to enter into $S$ provided that it puts some underlying segment a that does not satisfy the restriction $S$ in correspondence with some surface segment b that does satisfy it, as in (34). Condition (34) is analogous to (33), only with the roles of underlying and surface segments switched.

In the case of Max $_{R}$ , the no-exiting assumption that (33) is impossible suffices to establish the FIC $_{\text{comp}}$ . In the case of Dep $^{S}$ , the no-entering assumption that (34) is impossible needs to be coupled with the no-breaking condition, as expected based on the discussion in Section 4.2. The following lemma guarantees that it suffices to require the no-breaking condition among the segments that satisfy the restriction $S$ , intuitively because Dep $^{S}$ only cares about those segments. To illustrate, the lemma guarantees that Dep-V satisfies the FIC $_{\text{comp}}$ provided that no surface vowel is in correspondence with an underlying consonant and furthermore no underlying vowel is diphthongized.Footnote ^[12] The lemma will be further extended in Section 5.4.

5.3 Max _R ^S

The doubly restricted constraint Max $_{R}^{S}$ assigns to a candidate ( a , b , 𝜌 _{a , b} ) one violation for each segment of the underlying string a  that satisfies the restriction $R$ (namely, it belongs to R ( a )) and has no correspondent segment in the surface string b  that satisfies the restriction $S$ (namely, it belongs to S ( b )), although it might have surface correspondents that do not satisfy the restriction $S$ . To illustrate, consider the restriction $R$ which pairs a string with the set of its consonants and the restriction $S$ which pairs a string with the set of the segments in its initial syllable. Max $_{R}^{S}$ is Beckman’s (Reference Beckman1999) constraint Max-C- $\unicode[STIX]{x1D70E}_{1}$ , which mandates that every consonant has a correspondent in the initial syllable. The following lemma extends the analysis of the singly restricted Max $_{R}$ to the doubly restricted Max $_{R}^{S}$ . This lemma concludes the analysis of segmental Max constraints. The proof is a straightforward verification, as shown in Appendix A.1.

Condition (35) for Max $_{R}^{S}$ and condition (33) considered in Section 5.1 for Max $_{R}$ differ only in that the former has the additional fourth clause b ∈ S ( b ). Because of this additional clause, the assumption that (35) is impossible required for Max $_{R}^{S}$ to satisfy the FIC $_{\text{comp}}$ is weaker than the assumption that (33) is impossible required for Max $_{R}$ . To illustrate, the lemma says that the doubly restricted Max-C- $\unicode[STIX]{x1D70E}_{1}$ satisfies the FIC $_{\text{comp}}$ provided that no consonant is in correspondence with the vowel of the initial syllable, while the singly restricted Max-C was shown in Section 5.1 to require the stronger assumption that no consonant is in correspondence with any vowel.

The case of Max $_{R}$ follows as a special case of Max $_{R}^{S}$ with a trivial restriction $S$ : the additional clause b ∈ S ( b ) is trivially satisfied in this case and the two conditions (33) and (35) thus coincide. Furthermore, the constraint Max $^{S}$ always satisfies the FIC $_{\text{comp}}$ because it coincides with Max $_{R}^{S}$ with a trivial restriction $R$ , whereby the clause b̸ ∈ R ( b ) is impossible. As yet another interesting special case, suppose that the two restrictions $R$ and $S$ coincide. Condition (35) is then contradictory, because it cannot be the case that b̸ ∈ R ( b ) and b ∈ S ( b ). The constraint Max $_{R}^{S}$ thus satisfies the FIC $_{\text{comp}}$ without additional assumptions when $S=R$ . This observation will be used in Section 6.1 to establish the FIC $_{\text{comp}}$ for featural Max constraints.

5.4 Dep _R ^S

The reasoning in Section 5.3 extends straightforwardly from Max $_{R}^{S}$ to Dep $_{R}^{S}$ . The doubly restricted constraint Dep $_{R}^{S}$ assigns to a candidate ( a , b , 𝜌 _{a , b} ) one violation for each segment of the surface string b  that satisfies the restriction $S$ (namely, it belongs to S ( b )) and has no correspondent segment in the underlying string a  that satisfies the restriction $R$ (namely, it belongs to R ( a )), although it might have underlying correspondents that do not satisfy the restriction $R$ . The following lemma extends the analysis of the singly restricted Dep $^{S}$ to the doubly restricted Dep $_{R}^{S}$ and thus concludes the analysis of segmental Dep constraints. The assumption that (36) is impossible is weaker than the assumption that (34) is impossible, because of the additional clause a ∈ R ( a ). The proof of the lemma is a straightforward verification, as shown in Appendix A.2.

5.5 Ident $_{\unicode[STIX]{x1D711},R}$

The faithfulness constraint Ident $_{\unicode[STIX]{x1D711},R}$ corresponding to a total feature $\unicode[STIX]{x1D711}$ and a restriction $R$ assigns to a candidate ( a , b , 𝜌 _{a , b} ) one violation for each corresponding pair (a, b) ∈𝜌 _{a , b} of segments that differ in the value of the feature $\unicode[STIX]{x1D711}$ such that the underlying segment a satisfies the restriction $R$ (namely, it belongs to R ( a )). To illustrate, consider the restriction $R$ which pairs a string with the set of its nasal segments. The corresponding constraint  is the constraint IdentI  $\rightarrow$  O[+nasal] of Pater (Reference Pater, Kager, van der Hulst and Zonneveld1999), which penalizes de-nasalization (i.e., an underlying nasal segment with an oral surface correspondent), but not nasalization. Lemma 4/A guarantees that Ident $_{\unicode[STIX]{x1D711},R}$ satisfies the FIC $_{\text{comp}}$ provided that the candidate set makes (37) impossible (and furthermore satisfies the usual no-breaking assumption).

Condition (37) coincides with condition (33) used above in the analysis of Max $_{R}$ , apart from the additional clause 𝜑(a) =𝜑 (b) that the two segments a andb are assigned the same value by the feature $\unicode[STIX]{x1D711}$ . The assumption that (37) is impossible thus means that the correspondence relation 𝜌 _{a , b} cannot exit from $R$ without changing the value of the feature $\unicode[STIX]{x1D711}$ . The assumption that (37) is impossible is thus weaker than the assumption that (33) is impossible, which was needed above for Max $_{R}$ . To illustrate, consider again the feature $\unicode[STIX]{x1D711}=[\text{nasal}]$ and the restriction $R$ which pairs a string with the set of its nasals. Condition (37) is contradictory in this case, because the three clauses a ∈ R ( a ), b̸ ∈ R ( b ), and 𝜑(a) =𝜑 (b) cannot hold simultaneously. Pater’s constraint IdentI  $\rightarrow \text{O}[\text{+nasal}]$   $=$  Ident $_{[\text{nasal}],R}$ thus satisfies the FIC $_{\text{comp}}$ (provided there is no breaking).

5.6 Ident _𝜑 ^S

The faithfulness constraint Ident $_{\unicode[STIX]{x1D711}}^{S}$ is defined analogously, the only difference being that the restriction is applied to surface rather than underlying segments. To illustrate, consider the restriction $S$ which pairs a string with the set of segments that belong to its initial syllable. The corresponding constraint Ident $_{[\text{high}]}^{S}$ is the constraint Ident $_{[\text{high}]}^{\unicode[STIX]{x1D70E}_{1}}$ of Beckman (Reference Beckman1997, Reference Beckman1999), which is violated by a surface vowel in the initial syllable in correspondence with an underlying vowel that differs with respect to the feature [high]. As another example, consider the restriction $S$ which pairs a string with the set of its nasal segments. The corresponding constraint Ident $_{[\text{nasal}]}^{S}$ is the constraint IdentO $\rightarrow \text{I}[\text{+nasal}]$ of Pater (Reference Pater, Kager, van der Hulst and Zonneveld1999), which penalizes nasalization (i.e., an underlying oral segment with a nasal surface correspondent), but not de-nasalization. Lemma 4/B guarantees that Ident $_{\unicode[STIX]{x1D711}}^{S}$ satisfies the FIC $_{\text{comp}}$ provided that condition (38) is impossible. This assumption means that the correspondence relation 𝜌 _{a , b} cannot exit from $R$ without changing the value of the feature $\unicode[STIX]{x1D711}$ . The only difference between Lemmas 4/A and 4/B is that the no-breaking assumption in the latter lemma is restricted to the segments that satisfy the restriction (as underlined). The proof of both lemmas is a straightforward verification, as shown in Appendix A.3.

5.7 Ident _{𝜑, R} ^S

For completeness, let us also consider the faithfulness constraint Ident $_{\unicode[STIX]{x1D711},R}^{S}$ corresponding to a total feature $\unicode[STIX]{x1D711}$ and two restrictions $R,S$ , which assigns to a candidate ( a , b , 𝜌 _{a , b} ) one violation for each pair (a, b) ∈𝜌 _{a , b} of an underlying segment a that satisfies the restriction $R$ (namely, it belongs to R ( a )) and a surface segment b that satisfies the restriction $S$ (namely, it belongs to S ( b )) such that 𝜑(a)≠𝜑(b). To illustrate, consider the constraint *Replace (coronal, labial) proposed in Boersma (Reference Boersma1998): it is violated by an underlying coronal consonant with a labial surface correspondent. It can be reinterpreted as the constraint Ident $_{[\text{place}],R}^{S}$ corresponding to a tri-valued feature [place] where the restrictions $R$ and $S$ pair a string with the set of its coronal segments and the set of its labial segments, respectively. As another example, consider the *Map constraint in (39), proposed by White (Reference White2013) and Hayes & White (Reference Hayes and Whiteto appear) building on Zuraw (Reference Zuraw2007, Reference Zuraw2013): it is violated by an underlying voiceless stop which is in correspondence with a surface voiced fricative. This constraint can be reinterpreted as the constraint Ident $_{[\text{voice}],R}^{S}$ or Ident $_{[\text{cont}],R}^{S}$ where the restrictions $R$ and $S$ pair a string with the set of its voiceless stops and the set of its voiced fricatives, respectively.Footnote ^[13]

No simple conditions on the correspondence relations and the restrictions $R,S$ seem to suffice to ensure that the doubly restricted constraint Ident $_{\unicode[STIX]{x1D711},R}^{S}$ satisfies the FIC $_{\text{comp}}$ . In particular, it does not suffice to simply assume that the two conditions (37) and (38) for Ident $_{\unicode[STIX]{x1D711}}^{S}$ and Ident $_{\unicode[STIX]{x1D711},R}$ are both impossible. Here is a counterexample. Consider a feature $\unicode[STIX]{x1D711}$ that is partial and binary. Consider the corresponding feature $\widehat{\unicode[STIX]{x1D711}}$ that is total and ternary, in the sense that $\widehat{\unicode[STIX]{x1D711}}$ coincides with $\unicode[STIX]{x1D711}$ for any segment that $\unicode[STIX]{x1D711}$ is defined for, while $\widehat{\unicode[STIX]{x1D711}}$ assigns the dummy value ‘0’ to the segments that $\unicode[STIX]{x1D711}$ is undefined for. Consider the identity faithfulness constraint Ident $_{\unicode[STIX]{x1D711}}$ relative to the partial feature $\unicode[STIX]{x1D711}$ , which only penalizes an underlying and a corresponding surface segment when the feature is defined for both and assigns them a different value (Ident $_{\unicode[STIX]{x1D711}}$ does not assign a violation when the feature is defined for exactly one of the two corresponding segments). This constraint Ident $_{\unicode[STIX]{x1D711}}$ is identical to the doubly restricted identity faithfulness constraint Ident $_{\widehat{\unicode[STIX]{x1D711}},R}^{S}$ relative to the total three-valued feature $\widehat{\unicode[STIX]{x1D711}}$ and the restrictions $R=S$ which pair a string with the set of its segments for which the feature $\unicode[STIX]{x1D711}$ is defined (namely the set of segments to which the corresponding total feature $\widehat{\unicode[STIX]{x1D711}}$ assigns values $\pm$ and not the dummy value). Since the identity constraint Ident $_{\unicode[STIX]{x1D711}}$ corresponding to the partial feature $\unicode[STIX]{x1D711}$ has been shown not to satisfy the FIC $_{\text{comp}}$ in Section 4.3, the doubly restricted constraint Ident $_{\widehat{\unicode[STIX]{x1D711}},R}^{S}$ cannot satisfy the FIC $_{\text{comp}}$ either. Yet, conditions (37) and (38) are both contradictory, because the restrictions $R,S$ are defined in terms of the values of the feature $\widehat{\unicode[STIX]{x1D711}}$ .

6.1 Max $_{[\text{+}\unicode[STIX]{x1D711}]}$ , Dep $_{[\text{+}\unicode[STIX]{x1D711}]}$

Let ‘+’ be a designated value of a feature $\unicode[STIX]{x1D711}$ , either partial or total, either binary or multi-valued. The faithfulness constraint Max $_{[\text{+}\unicode[STIX]{x1D711}]}$ assigns to a candidate ( a , b , 𝜌 _{a , b} ) one violation for each segment of the underlying string a  that has the designated value + for the feature $\unicode[STIX]{x1D711}$ but has no correspondent in the surface string b  that shares the value + for the feature $\unicode[STIX]{x1D711}$ (Casali Reference Casali1997, Reference Casali1998; Walker Reference Walker1999; Lombardi Reference Lombardi and Lombardi2001).Footnote ^[14] To illustrate, Max $_{[\text{+voice}]}$ assigns two violations to the candidate (40), because both /b/ and /d/ lose their voicing (through devoicing and deletion, respectively).

The featural constraint Max $_{[\text{+}\unicode[STIX]{x1D711}]}$ coincides with the doubly restricted segmental constraint Max $_{R}^{S}$ when the two restrictions $R$ and $S$ both pair up a string with the set of its segments that have the value + for the feature $\unicode[STIX]{x1D711}$ . Since, in particular, $R=S$ , condition (35) of Lemma 2 is impossible, because its last two clauses b̸ ∈ R ( b ) and b ∈ S ( b ) are contradictory. The following result thus follows as a special case of Lemma 2.

Analogous considerations hold for the constraint Dep $_{[\text{+}\unicode[STIX]{x1D711}]}$ , which assigns to a candidate ( a , b , 𝜌 _{a , b} ) one violation for each segment of the surface string b  that has the designated value + for the feature $\unicode[STIX]{x1D711}$ but has no correspondent in the underlying string a  that shares the value +.

The featural constraints Max $_{[\text{+}\unicode[STIX]{x1D711}]}$ /Dep $_{[\text{+}\unicode[STIX]{x1D711}]}$ differ subtly from the restricted segmental constraints Max $_{R}$ /Dep $^{S}$ , where $R$ and $S$ both pair a string with the set of its segments that have the value ‘+’ for the feature $\unicode[STIX]{x1D711}$ . In fact, Max $_{R}$ /Dep $^{S}$ are violated by an underlying/surface segment that has the value ‘+’ and is deleted/epenthesized, while Max $_{[\text{+}\unicode[STIX]{x1D711}]}$ /Dep $_{[\text{+}\unicode[STIX]{x1D711}]}$ are violated by an underlying/surface segment that has the value ‘+’ and is deleted/epenthesized or put in correspondence with segments with a different value for feature $\unicode[STIX]{x1D711}$ . This subtle difference is computationally substantial: Max $_{[\text{+}\unicode[STIX]{x1D711}]}$ /Dep $_{[\text{+}\unicode[STIX]{x1D711}]}$ satisfy the FIC $_{\text{comp}}$ under no additional assumptions, but Max $_{R}$ /Dep $^{S}$ require the additional no-entering and no-exiting assumptions that (33) and (34) are impossible, as seen in Sections 5.1–5.2. Formally, this difference is due to the fact that Max $_{[\text{+}\unicode[STIX]{x1D711}]}$ /Dep $_{[\text{+}\unicode[STIX]{x1D711}]}$ coincide not with Max $_{R}$ /Dep $^{S}$ but with Max $_{R}^{S}$ /Dep $_{R}^{S}$ with $S=R$ .

6.2 Uniformity and Integrity

The faithfulness constraint Uniformity assigns to a candidate ( a , b , 𝜌 _{a , b} ) one violation for each surface coalescence, namely for each segment of the surface string b  that has two or more correspondents in the underlying string a  according to 𝜌 _{a , b} (McCarthy & Prince Reference McCarthy, Prince, Beckman, Urbanczyk and Walsh Dickey1995). To illustrate, Uniformity assigns two violations to the candidate ( a , b , 𝜌 _{a , b} ) in (41), because of its two surface coalescences [b] and [f]. The constraint thus defined is coarse (Wheeler Reference Wheeler2005): it does not distinguish between a coalescence of just two segments (such as [f] below) and a coalescence of more than two segments (such as [b]). This distinction can be captured through the following alternative gradient definition: the faithfulness constraint  assigns $k$ violations for each coalescence of $k\geqslant 2$ underlying segments.

While Dep penalizes surface segments that have no underlying correspondents, Uniformity penalizes surface segments that have too many. The analysis of Dep in Section 4.2 extends to Uniformity, yielding the following lemma 7, whose simple verification is omitted for brevity.

Analogous considerations hold for the faithfulness constraint Integrity, which assigns to a candidate ( a , b , 𝜌 _{a , b} ) one violation for each broken underlying segment, namely for each segment of the underlying string a  that has two or more correspondents in the surface string b  according to 𝜌 _{a , b} (see McCarthy & Prince Reference McCarthy, Prince, Beckman, Urbanczyk and Walsh Dickey1995 and Staroverov Reference Staroverov2014 for discussion). The corresponding gradient constraint Integrity $^{\text{grad}}$ assigns instead $k$ violations for each underlying segment that is broken into $k\geqslant 2$ surface segments. While Max penalizes underlying segments that have no surface correspondents, Integrity penalizes underlying segments that have too many. The analysis of Max in Section 4.1 extends to Integrity, yielding the following Lemma 8.

6.3 Contiguity

The faithfulness constraint I-Contiguity assigns to a candidate ( a , b , 𝜌 _{a , b} ) one violation for each skipped underlying segment, namely for each segment of the underlying string a  that has no correspondents in the surface string b  according to 𝜌 _{a , b} and furthermore is flanked both on the left and on the right by underlying segments that instead do admit surface correspondents.Footnote ^[15] The faithfulness constraint I-Contiguity fails at the FIC $_{\text{comp}}$ , as shown by the counterexample (42). The antecedent of the FIC $_{\text{comp}}$ holds: the candidate ( b , c , 𝜌 _{b , c} ) in (42a) does not violate I-Contiguity because it has no skipped segments (the deleted coda /k/ does not count as skipped because it is string-final). The right-hand side of the FIC $_{\text{comp}}$ inequality is small, namely equal to zero: the candidate ( a , b , 𝜌 _{a , b} ) in (42c) (modeled on metathesis in Rotuman; Carpenter Reference Carpenter, Carpenter, Coetzee and de Lacy2002) does not violate I-Contiguity, because it has no skipped segments (because no underlying segment is deleted). The FIC $_{\text{comp}}$ inequality fails because its left-hand side is large, namely equal to 1: the composition candidate ( a , c , 𝜌 _{a , b} 𝜌 _{b , c} ) in (42b) violates I-Contiguity because /k/ is skipped.

As McCarthy & Prince (Reference McCarthy, Prince, Beckman, Urbanczyk and Walsh Dickey1995) note, I-Contiguity can in most applications be re-defined as penalizing the deletion of internal segments, namely underlying segments that are flanked on both sides by other underlying segments, no matter whether these flankers have correspondents. The counterexample (42) shows that the constraint thus re-defined fails at the FIC $_{\text{comp}}$ as well. This is not surprising, because the constraint thus redefined coincides with Max $_{R}$ , where the restriction $R$ pairs a string with the set of its internal segments. As seen in Section 5.1, Max $_{R}$ fails at the FIC $_{\text{comp}}$ when the correspondence relations can exit from the restriction $R$ . That is precisely the case in the counterexample (42), as the correspondence relation 𝜌 _{a , b} establishes a correspondence between underlying /k/ (which satisfies the restriction $R$ because it is internal to a ) and surface [k] (which does not satisfy the restriction $R$ because it is not internal to b ). Analogous considerations hold for O-Contiguity.

6.4 Adjacency

Carpenter (Reference Carpenter, Carpenter, Coetzee and de Lacy2002) argues that the faithfulness constraint I-Contiguity should be replaced with O-Adjacency. The latter assigns to a candidate ( a , b , 𝜌 _{a , b} ) one violation for any two pairs (a ₁, b ₁), (a ₂, b ₂) of underlying segments a ₁, a ₂ and surface segments b ₁, b ₂ that are in correspondence according to 𝜌 _{a , b} despite the fact that b ₁, b ₂ are adjacent in the surface string b  while a ₁, a ₂ are not adjacent in the underlying string a . The faithfulness constraint I-Adjacency is defined analogously, by looking at adjacency relative to the underlying string. To appreciate the difference between I-Contiguity and O-Adjacency, consider again the counterexample (42) used to show that I-Contiguity fails at the FIC $_{\text{comp}}$ . This counterexample raises no problems for O-Adjacency. The crucial difference is that O-Adjacency assigns one violation to the candidate ( a , b , 𝜌 _{a , b} ) in (42c), because of the two pairs of corresponding segments (/t/, [t]) and (/a/, [a]). Indeed, despite the fact that O-Adjacency and I-Contiguity are shown by Carpenter to do much of the same work, they differ with respect to the FIC $_{\text{comp}}$ : I-Contiguity fails at the FIC $_{\text{comp}}$ , as seen in the preceding subsection; O-Adjacency instead satisfies the FIC $_{\text{comp}}$ , as stated by the following lemma, whose simple verification is omitted for brevity.

6.5 Linearity

The faithfulness constraint Linearity penalizes metathesis. McCarthy (Reference McCarthy2008: 198) defines this constraint as follows:  assigns to a candidate( a , b , 𝜌 _{a , b} ) one violation for each pair of underlying segments a ₁ and a ₂ that admit two swapped surface correspondents, namely there exist two surface segments b ₁ and b ₂ such that a ₁ corresponds through 𝜌 _{a , b} to b ₁,  a ₂ corresponds to b ₂, and yet a ₁ precedes a ₂ while b ₁ follows b ₂. Heinz (Reference Heinz, Alderete, Han and Kochetov2005) offers the following alternative definition:  assigns to a candidate ( a , b , 𝜌 _{a , b} ) one violation for each pair of underlying segments a ₁ and a ₂ that admit no non-swapped surface correspondents, namely there exist no two surface segments b ₁ and b ₂ such that a ₁ corresponds through 𝜌 _{a , b} to b ₁, a ₂ corresponds to b ₂, and both a ₁ precedes a ₂ and b ₁ precedes b ₂.Footnote ^[16] Footnote ^[17] The faithfulness constraint  fails at the FIC $_{\text{comp}}$ when the candidate set allows both coalescence and breaking, as shown by the counterexample (43). The antecedent of the FIC $_{\text{comp}}$ holds: the candidate ( b , c , 𝜌 _{b , c} ) in (43a) does not violate  because it has a unique underlying segment. The right-hand side of the FIC $_{\text{comp}}$ inequality is small, namely equal to zero: the candidate ( a , b , 𝜌 _{a , b} ) in (43c) does not violate  because it has a unique surface segment. The FIC $_{\text{comp}}$ inequality fails because its left-hand side is large, namely equal to 1: the composition candidate ( a , c , 𝜌 _{a , b} 𝜌 _{b , c} ) in (43b) violates  because the two underlying segments /a/ and /i/ are in correspondence with the two surface segments [e] and [i] which have the opposite linear order.Footnote ^[18]

This counterexample does not threaten  Although the composition candidate (43b) violates  because the two underlying segments admit swapped surface segments, it does not violate  because the two underlying segments also admit non-swapped correspondents. These considerations lead to the following lemma, whose proof is a simple verification which is omitted for brevity.

McCarthy & Prince’s (Reference McCarthy, Prince, Beckman, Urbanczyk and Walsh Dickey1995) Contiguity and Carpenter’s (Reference Carpenter, Carpenter, Coetzee and de Lacy2002) Adjacency are closely related constraints meant to serve the same purpose. The same holds for McCarthy’s (Reference McCarthy2008) and Heinz’s (Reference Heinz, Alderete, Han and Kochetov2005) slightly different implementations of Linearity constraints. The discussion in the last two subsections has shown that these small differences in the definition of the constraints can have substantial formal consequences for idempotency.

6.6 Constraint conjunction and disjunction

The OT literature has made use of constraints defined as boolean combinations of other constraints (Crowhurst & Hewitt Reference Crowhurst and Hewitt1997, Wolf Reference Wolf2007). Two boolean operations that have figured prominently are constraint conjunction (Smolensky Reference Smolensky1995, Moreton & Smolensky Reference Moreton, Smolensky, Mikkelsen and Potts2002) and disjunction (Downing Reference Downing1998, Reference Downing2000). Constraint conjunction fails at the FIC $_{\text{comp}}$ , as shown by the counterexample in (44). The conjoined constraint Ident $_{[\text{low}]}$   $\wedge$  Ident $_{[\text{high}]}$ assigns one violation for each pair of corresponding segments that differ for both features [low] and [high]. The antecedent of the FIC $_{\text{comp}}$ holds: the candidate ( b , c , 𝜌 _{b , c} ) in (44a) does not violate the conjoined constraint, because /e/ and [i] only differ for the feature [high]. The right-hand side of the FIC $_{\text{comp}}$ inequality is small, namely equal to zero: the candidate ( a , b , 𝜌 _{a , b} ) in (44c) does not violate the conjoined constraint, because /a/ and [e] only differ for the feature [low]. The FIC $_{\text{comp}}$ inequality fails because its left-hand side is large, namely equal to 1: the composition candidate ( a , c , 𝜌 _{a , b} 𝜌 _{b , c} ) in (44b) violates the conjoined constraint, because /a/ and [i] differ for both features [low] and [high].

The case of constraint disjunction is different. For concreteness, consider the disjunction Ident $_{\unicode[STIX]{x1D711}}$   $\vee$  Ident $_{\unicode[STIX]{x1D713}}$ of two identity faithfulness constraints Ident $_{\unicode[STIX]{x1D711}}$ and Ident $_{\unicode[STIX]{x1D713}}$ corresponding to two (total) features $\unicode[STIX]{x1D711}$ and $\unicode[STIX]{x1D713}$ . This constraint assigns one violation for each pair of corresponding segments that differ for either feature $\unicode[STIX]{x1D711}$ or $\unicode[STIX]{x1D713}$ (possibly both). Lemma 11 ensures that it satisfies the FIC $_{\text{comp}}$ . The proof is a straightforward verification, which is omitted for brevity. This conclusion easily extends to the disjunction of other (disjoinable) faithfulness constraints: conditions on the FIC $_{\text{comp}}$ -compliance of the constraint disjunction follow by combining conditions on the FIC $_{\text{comp}}$ -compliance of the faithfulness constraints being combined in the disjunction.Footnote ^[19]

The difference between constraint conjunction and disjunction with respect to the FIC $_{\text{comp}}$ can be appreciated as follows. Suppose that the antecedent of the FIC $_{\text{comp}}$ holds for the disjunction Ident $_{[\text{low}]}$   $\vee$  Ident $_{[\text{high}]}$ . This means that the candidate ( b , c , 𝜌 _{b , c} ) does not violate it. This entails in turn that the candidate violates neither Ident $_{[\text{low}]}$ nor Ident $_{[\text{high}]}$ . The FIC $_{\text{comp}}$ for the disjunction thus follows from the FIC $_{\text{comp}}$ previously established for the individual disjuncts. The case of conjunction is different: even if the candidate ( b , c , 𝜌 _{b , c} ) does not violate the conjunction Ident $_{[\text{low}]}$   $\wedge$  Ident $_{[\text{high}]}$ as required by the antecedent of the FIC $_{\text{comp}}$ , it could nonetheless violate one of the two conjuncts Ident $_{[\text{low}]}$ or Ident $_{[\text{high}]}$ . The fact that the conjuncts satisfy the FIC $_{\text{comp}}$ thus provides no guarantees that their conjunction satisfies it as well.

7.1 Only a sufficient condition?

Consider an arbitrary (reflexive and transitive) candidate set, an arbitrary constraint set, and an arbitrary constraint ranking. Section 3 has established that the FIC $_{\text{comp}}$ is a sufficient condition for the idempotency of the corresponding OT grammar. This statement contains three universal quantifications: over candidate sets, over constraint sets, and over rankings. At this level of generality, the FIC $_{\text{comp}}$ is not only a sufficient but also a necessary condition for idempotency, in the following sense. Consider a faithfulness constraint $F$ that does not satisfy the FIC $_{\text{comp}}$ (21). This means that there exist two candidates ( a , b , 𝜌 _{a , b} ) and ( b , c , 𝜌 _{b , c} ) such that $F$ assigns no violations to the candidate ( b , c , 𝜌 _{b , c} ), so that the antecedent of the FIC $_{\text{comp}}$ holds; yet, $F$ assigns more violations to the candidate ( a , c , 𝜌 _{a , b} 𝜌 _{b , c} ) than to the candidate ( a , b , 𝜌 _{a , b} ), so that the consequent fails. Suppose that the constraint set also contains a markedness constraint $M$ that assigns more violations to the surface form b  than to the surface form c .Footnote ^[20] The OT grammar corresponding to the ranking $F\gg M$ displays the chain shift a → b → c and thus fails at idempotency: the string b  is phonotactically licit, because the underlying form a  is mapped to the candidate ( a , b , 𝜌 _{a , b} ), as shown in the left-hand side tableau in (45); yet, the string b  does not surface faithfully, because the underlying form b  is not mapped to the identity candidate ( b , b , I _{b , b} ), as shown in the right-hand side tableau.

The rest of this section shows how various approaches to chain shifts in the classical OT literature fit within the schema (45), where $F$ is one of the faithfulness constraints shown in Sections 4–6 to fail the FIC $_{\text{comp}}$ .

7.2 Chain shifts through constraint conjunction

As noted in Section 6.6, constraint conjunction yields faithfulness constraints that fail the FIC $_{\text{comp}}$ . The use of constraint conjunction to model chain shifts within classical OT has been pioneered by Kirchner (Reference Kirchner1996) and systematized by Moreton & Smolensky (Reference Moreton, Smolensky, Mikkelsen and Potts2002). To illustrate, consider the chain shift in the Lena dialect of Spanish in (46) (data from Hualde Reference Hualde1989 and Gnanadesikan Reference Gnanadesikan1997: Section 5.4.3): because of the high final vowel in the masculine form, the underlying low and mid vowels (as revealed by the feminine form) raise to mid and high vowels, respectively.

The markedness constraint Raise favors higher vowels before a high vowel. The conjunction of the two faithfulness constraints Ident $_{[\text{high}]}$ and Ident $_{[\text{low}]}$ penalizes underlying low vowels mapped to surface high vowels. The analysis (47) is an instance of the scheme (45), with the conjoined constraint playing the role of the non-FIC $_{\text{comp}}$ -complying faithfulness constraint.

Other approaches proposed in the literature are equivalent to the approach based on the conjunction of identity faithfulness constraints (see also the discussion of V-HeightDistance in Kirchner Reference Kirchner1995). For instance, Gnanadesikan (Reference Gnanadesikan1997: chapter 3) accounts for the chain shift p→b→m (in post-nasal position) through the faithfulness constraint Ident-Adj, which is violated by a voiceless obstruent and a corresponding sonorant because they are separated by a distance larger than 2 on the inherent voicing scale. This constraint is thus equivalent to the conjunction Ident $_{[\text{voice}]}$   $\wedge$  Ident $_{[\text{son}]}$ . Analogously, Dinnsen & Barlow (Reference Dinnsen and Barlow1998) account for the chain shift s→𝜃→f through the faithfulness constraint DistFaith, which is violated when the underlying and surface forms differ by more than 1 on the scale f = 1, 𝜃 = 2, and s = 3 and is thus equivalent to the conjunction Ident $_{[\text{coronal}]}$   $\wedge$  Ident $_{[\text{strident}]}$ .

Yet, the approach based on constraint conjunction is more general than the latter approaches based on ‘scales’, as the former but not the latter extends to chain shifts that involve deletion (Moreton & Smolensky Reference Moreton, Smolensky, Mikkelsen and Potts2002). To illustrate, consider the Sea Dayak chain shift in (48) (data from Kenstowicz & Kisseberth Reference Kenstowicz and Kisseberth1979).

The analysis based on constraint conjunction extends as in (47) (based on Łubowicz Reference Łubowicz, van Oostendorp, Ewen, Colin and Rice2011), which is another instance of the scheme (45).

7.3 Chain shifts through constraint restrictions

As noted in Section 5, constraint restriction yields faithfulness constraints that fail the FIC $_{\text{comp}}$ when the correspondence relations are allowed to ‘cross’ the restriction. This observation systematizes various approaches to chain shifts proposed in the classical OT literature. For instance, Orgun (Reference Orgun1995) considers the chain shift in (50) from Bedouin Hijazi Arabic: /a/ is raised to [i] but /i/ is deleted (both processes are restricted to short vowels in non-final open syllables; the data come from McCarthy Reference McCarthy1993).

Orgun’s analysis is summarized in (51), plus a markedness constraint [*a] which is omitted here. It relies on his constraint Correspond(/a/), which mandates that ‘every input /a/ has an output correspondent’. This constraint can be re-interpreted as Max $_{R}$ , where the restriction $R$ pairs a string with the set of its a’s. As shown in Section 5.1, Max $_{R}$ fails at the FIC $_{\text{comp}}$ when correspondence relations are allowed to exit from the restriction $R$ , namely to put in correspondence an underlying segment that satisfies the restriction with a surface segment that does not, as described in (33). That is precisely the case in (51), as the underlying /a/ (which satisfies the restriction $R$ ) corresponds to the surface [i] (which does not satisfy the restriction). Orgun’s analysis (51) is thus an instance of the scheme (45), with the restricted constraint Max $_{R}$ playing the role of the non-FIC $_{\text{comp}}$ -complying faithfulness constraint.

As another example, Jesney (Reference Jesney2005, Reference Jesney2007) considers the classical child chain shift in (52): coronal stridents are realized as coronal stops across the board, but coronal stops are velarized when followed by a lateral (data from Amahl age 2;2-2;11, as described in Smith Reference Smith1973).

Jesney’s analysis is summarized in (53). It relies on her ‘specific’ faithfulness constraint IdentCoronal/[+strident], which mandates that input stridents preserve their coronality. This constraint can be re-interpreted as Ident $_{\unicode[STIX]{x1D711},R}$ , where $\unicode[STIX]{x1D711}$ is the feature [coronal] and the restriction $R$ pairs a string with the set of its stridents. As shown in Section 5.5, Ident $_{\unicode[STIX]{x1D711},R}$ fails at the FIC $_{\text{comp}}$ when correspondence relations are allowed to exit from the restriction $R$ without changing the value of the feature $\unicode[STIX]{x1D711}$ , namely to put in correspondence an underlying segment that satisfies the restriction with a surface segment that does not and yet has the same value for the feature $\unicode[STIX]{x1D711}$ , as described in (37). That is precisely the case in (53), as the underlying /s/ (which satisfies the restriction $R$ ) corresponds to the surface [t] (which does not satisfy the restriction) and yet they are both coronals. Jesney’s analysis (53) is thus an instance of the scheme (45), with the restricted constraint Ident $_{[\text{cor}],R}$ playing the role of the non-FIC $_{\text{comp}}$ -complying faithfulness constraint.Footnote ^[21]

7.4 Chain shifts through breaking

Let me close this section by discussing a fictional example. Kubozono et al. (Reference Kubozono, Ito and Mester2008) report that English frog is imported as [fu.róg.gu] into Japanese: the velar stop geminates (despite being voiced) because of a requirement on the placement of stress, captured here through a place-holder constraint Stress. Assume an analysis of consonant gemination in terms of breaking of a single underlying consonant into two surface copies, as indicated by the correspondence relations in (54). Section 4.3 has shown that plain identity faithfulness constraints fail at the FIC $_{\text{comp}}$ when the correspondence relations are allowed to break underlying segments. This fact could be used to derive a fictional chain shift such as  through the analysis (54), which is an instance of the scheme (45) with the identity constraint playing the role of the non-FIC $_{\text{comp}}$ faithfulness constraint.

This example is fictional because I have not been able to find a realistic case of chain shift that involves an underlying segment broken into two surface segments.

7.5 Summary

Section 3 has shown that chain shifts require the FIC $_{\text{comp}}$ to fail. Based on Sections 4–6, there are three major ways for the FIC $_{\text{comp}}$ to fail. One option is to use a faithfulness constraint that flouts the FIC $_{\text{comp}}$ , such as a faithfulness constraint obtained through constraint conjunction. A second option is to use the restriction of a faithfulness constraint that would otherwise comply with the FIC $_{\text{comp}}$ . The third option is to let the correspondence relations break underlying segments. The former two options have been exploited in the literature on chain shifts.

8.1 The FIC $_{\text{comp}}$ and the metrical nature of the faithfulness constraints

Theorem 1 places no restrictions on the markedness constraints and only looks at the faithfulness constraints. This result thus motivates a deeper look into the formal underpinning of the Correspondence Theory of faithfulness. Intuitively, faithfulness constraints measure the ‘distance’ between underlying and surface forms along various phonologically relevant dimensions. It thus makes sense to investigate whether faithfulness constraints satisfy formal properties of distances (or metrics). One such important property is the triangle inequality (58): it captures the intuition that the distance between any two points a  and c  is shorter than the distance between a  and b  plus the distance between b  and c , for any choice of the intermediate point b  (Rudin Reference Rudin1953).

A faithfulness constraint $F$ is thus said to satisfy the faithfulness triangle inequality provided that condition (59) holds for any two candidates ( a , b , 𝜌 _{a , b} ) and ( b , c , 𝜌 _{b , c} ) that share the string b  as the underlying and the surface form, respectively.

Magri (Reference Magrito appear) establishes an equivalence result between the triangle inequality (59) and the condition FIC $_{\text{comp}}$ , which was established in this paper to be sufficient for idempotency. This equivalence thus provides an intuitive metric interpretation of the apparently very technical FIC $_{\text{comp}}$ . This equivalence requires two assumptions: first, that the correspondence relations in the candidate set are one-to-one (no breaking nor coalescence); second, that the faithfulness constraints are categorical, in the sense of McCarthy (Reference McCarthy2003b). Building on Lemmas 2–11 established in this paper, this equivalence between the FIC $_{\text{comp}}$ and the triangle inequality yields a straightforward characterization of the faithfulness constraints that satisfy the triangle inequality and thus admit a metric interpretation.

8.2 Idempotency in Harmonic Grammar

This paper has focused on idempotency within the OT implementation of constraint-based phonology. Magri (Reference Magrito appear) extends the theory of idempotency to Harmonic Grammar (HG; Legendre et al. Reference Legendre, Miyata, Smolensky, Gernsbacher and Derry1990a, Reference Legendre, Miyata, Smolensky, Gernsbacher and Derryb; Smolensky & Legendre Reference Smolensky and Legendre2006). Also in HG, idempotency is guaranteed by a condition on the faithfulness constraints, which can be referred to as the FIC $_{\text{comp}}^{\text{HG}}$ , to distinguish it from the OT condition obtained in this paper, which I will now refer to as the FIC $_{\text{comp}}^{\text{OT}}$ . In the general case, the FIC $_{\text{comp}}^{\text{HG}}$ asymmetrically entails the FIC $_{\text{comp}}^{\text{OT}}$ . This makes sense: HG typologies properly contain OT typologiesFootnote ^[22] , so that a stronger condition is needed in order to discipline a larger typology of grammars to all comply with idempotency. The asymmetric relationship between the FIC $_{\text{comp}}^{\text{HG}}$ and the FIC $_{\text{comp}}^{\text{OT}}$ is revealed, for instance, by the following fact: while the no-breaking assumption on correspondence relations suffices to ensure that basic faithfulness constraints (such as Dep and Ident) satisfy the FIC $_{\text{comp}}^{\text{OT}}$ , the FIC $_{\text{comp}}^{\text{HG}}$ also requires no coalescence, effectively restricting all correspondence relations in the candidate set to be one-to-one. Yet, when correspondence relations are one-to-one, the FIC $_{\text{comp}}^{\text{HG}}$ and the FIC $_{\text{comp}}^{\text{OT}}$ can be shown to be equivalent for faithfulness constraints that are categorical in the sense of McCarthy (Reference McCarthy2003b), because both conditions are equivalent to the triangle inequality. McCarthy’s categoricity conjecture thus entails that, although HG idempotency requires additional conditions on the correspondence relations relative to OT idempotency (the former requires neither breaking nor coalescence; the latter only requires no breaking), it effectively places no additional restrictions on the faithfulness constraints.

8.3 Idempotency and output-drivenness

Tesar (Reference Tesar2013) investigates another structural condition on phonological grammars, which he calls output-drivenness. It formalizes the intuition that any discrepancy between an underlying and a surface (or output) form is driven exclusively by the goal of making the surface form fit the phonotactics. Tesar shows that output-drivenness holds provided that the faithfulness constraints satisfy two implications which generalize the FIC and are called the faithfulness output-drivenness condition (FODC). He then investigates which faithfulness constraints satisfy the FODC in the special case where all correspondence relations are one-to-one. The answer to this question is non-trivial. For instance, two pages of Tesar’s book suffice to establish the FODC as a sufficient condition for OT output-drivenness, while the entire chapter 3 is devoted to verifying the FODC for just the three constraints Max, Dep, and Ident.

The results established in this paper on the FIC afford a substantial simplification of Tesar’s theory. In fact, Magri (Reference Magrito appear) looks at the relationship between idempotency and output-drivenness and between the two corresponding sufficient conditions, the FIC and the FODC. In the general case, the FODC is stronger than the FIC, matching the fact that output-drivenness is a stronger condition than idempotency, as shown by derived environment effects or saltations (Łubowicz Reference Łubowicz2002, White Reference White2013), which are idempotent but not output-driven. Yet, the FODC and the FIC are shown to be equivalent when the correspondence relations are all one-to-one (as assumed by Tesar) and the faithfulness constraints are all categorical (as conjectured in McCarthy Reference McCarthy2003b), because both conditions are equivalent to the triangle inequality mentioned in Section 8.1. Crucially, all of the faithfulness constraints analyzed in this paper are categorical. The equivalence result between the FIC and the FODC thus allows the results obtained here concerning the faithfulness constraints that satisfy the FIC to be translated straightforwardly into results concerning the faithfulness constraints that satisfy the FODC. A measure of the improvement obtained is provided by the fact that a large array of faithfulness constraints (beyond the three faithfulness constraints considered by Tesar) are shown in a snap to satisfy the FODC.

8.4 Benign chain shifts

Magri (Reference Magri2015) explores the implications of the theory of idempotency developed in this paper for OT learnability. The literature on the early acquisition of phonotactics usually assumes that the child posits a fully faithful underlying form for each phonotactically licit training surface form (Gnanadesikan Reference Gnanadesikan, Kager, Pater and Zonneveld2004, Hayes Reference Hayes, Kager, Pater and Zonneveld2004, Prince & Tesar Reference Prince, Tesar, Kager, Pater and Zonneveld2004). Is this assumption of faithful underlying forms computationally sound? Suppose that the target grammar is not idempotent because it displays the chain shift (60a). The form b  is phonotactically licit (because it is the surface realization of a ) and yet it is not faithfully mapped to itself. The assumption that every instance of b  in the training set is the surface realization of a faithful underlying form could thus mislead the learner into positing a set of mappings that is not consistent with any grammar in the typology.

Yet, the chain shift (60a) raises no issues for the learner’s assumption of fully faithful underlying forms as long as the chain shift happens to be benign, in the sense that the typology entertained by the learner happens to contain the companion grammar (60b), which is idempotent (there is no chain shift) and yet makes the same phonotactic distinctions ( a  is illicit while b  and c  are licit for both grammars). Under which conditions are chain shifts benign?

The theory of idempotency developed in this paper and summarized above as Theorem 1 says that basic faithfulness constraints all satisfy the FIC $_{\text{comp}}$ , thus explaining the well-known difficulty in modeling chain shifts within (classical) OT. In order to flout the FIC $_{\text{comp}}$ and thus be able to derive chain shifts, we need to look at derived faithfulness constraints. As seen in Section 7, there are two main strategies to construct these derived constraints: through the restriction of some basic and thus FIC $_{\text{comp}}$ -complying faithfulness constraint (such as Max $_{R}$ ) or through the conjunction of two basic and thus individually FIC $_{\text{comp}}$ -complying faithfulness constraints (such as the conjunction Ident $_{[\text{high}]}$   $\wedge$  Ident $_{[\text{low}]}$ ). Magri (Reference Magri2015) then formulates the conjecture that, whenever a chain shift (60a) is obtained through a ranking where one of these derived constraints occupies a prominent position, the ranking with the derived faithfulness constraint replaced by the corresponding basic one (say, Max $_{R}$ replaced by Max, or Ident $_{[\text{high}]}$   $\wedge$  Ident $_{[\text{low}]}$ replaced by either Ident $_{[\text{high}]}$ or Ident $_{[\text{low}]}$ ) yields the idempotent and phonotactically equivalent companion grammar (60b). The exploration of this conjecture would provide a solid foundation for a variety of models of the acquisition of phonotactics that share the assumption of completely faithful underlying forms.