Idempotency, Output-Drivenness and the Faithfulness Triangle Inequality: Some Consequences of McCarthy’s (2003) Categoricity Generalization

Abstract

Idempotency requires any phonotactically licit forms to be faithfully realized. Output-drivenness requires any discrepancies between underlying and output forms to be driven exclusively by phonotactics. These formal notions are relevant for phonological theory (they capture counter-feeding and counter-bleeding opacity) and play a crucial role in learnability. Tesar (Output-driven phonology: theory and learning. Cambridge studies in linguistics, 2013) and Magri (J of Linguistics, 2017) provide tight guarantees for OT output-drivenness and idempotency through conditions on the faithfulness constraints. This paper derives analogous faithfulness conditions for HG idempotency and output-drivenness and develops an intuitive interpretation of the various OT and HG faithfulness conditions thus obtained. The intuition is that faithfulness constraints measure the phonological distance between underlying and output forms. They should thus comply with a crucial axiom of the definition of distance, namely that any side of a triangle is shorter than the sum of the other two sides. This intuition leads to a faithfulness triangle inequality which is shown to be equivalent to the faithfulness conditions for idempotency and output-drivenness. These equivalences hold under various assumptions, crucially including McCarthy’s (Phonology 20(1):75–138, 2003b) generalization that (faithfulness) constraints are all categorical.

A Proofs

Throughout this appendix, I consider four strings ${{\mathbf {\mathsf{{a}}}}}$, ${{\mathbf {\mathsf{{b}}}}}$, ${{\mathbf {\mathsf{{c}}}}}$, and ${{\mathbf {\mathsf{{d}}}}}$, whose generic segments are denoted by ${\textsf {a}}$, ${\textsf {b}}$, ${\textsf {c}}$, ${\textsf {d}}$. For readability, I use statements such as “for every/some segment ${\textsf {a}}$” as a shorthand for “for every/some segment ${\textsf {a}}$ of the string ${{\mathbf {\mathsf{{a}}}}}$”, thus leaving the domain of the quantifier implicit.

1.1 A.1 Proof of Proposition 3

• Proposition 3 Assume the candidate set (2) satisfies the transitivity axiom (6) and only contains one-to-one correspondence relations. Consider a faithfulness constraint F which is C-categorical; or I-categorical and O-monotone; or O-categorical and I-monotone. F satisfies the FIC $_{\text {comp}}^{\,\text {OT}}$ if and only if it satisfies the FTI $_{\text {comp}}$. $\square$

Proof

As shown in Sect. 3.3.1, the FTI$_{\text {comp}}$ entails the FIC$_{\text {comp}}^{\text {OT}}$ in the general case. To prove the reverse entailment, consider a faithfulness constraint F which satisfies the FIC$_{\text {comp}}^{\text {OT}}$ repeated in (87) for any two candidates $( {{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{b}}}}}, \rho _{{{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{b}}}}}} )$ and $( {{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{c}}}}}, \rho _{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{c}}}}}} )$ and their composition candidate $({{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{c}}}}}, \rho _{{{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{b}}}}}}\rho _{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{c}}}}}})$, and let me show that F then satisfies the FTI$_{\text {comp}}$ repeated in (88).

For concreteness, the rest of the proof considers the case where F is I-categorical of order $\ell$ and O-monotone, so that it satisfies the I-additivity condition repeated in (89); the cases where F is instead C-categorical or O-categorical and I-monotone are treated analogously.

The I-additivity condition (89) entails that F assigns zero violations to candidates whose underlying string is shorter than $\ell$, as the sum on the right-hand side is empty in this case (there are no subsequences of length $\ell$). The FTI$_{\text {comp}}$ (88) thus trivially holds when its string ${{\mathbf {\mathsf{{a}}}}}$ is shorter than $\ell$, because its left-hand side $F ( {{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{c}}}}}, \rho _{{{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{b}}}}}}\rho _{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{c}}}}}} )$ is equal to zero. From now on, I assume therefore that the string ${{\mathbf {\mathsf{{a}}}}}$ has length at least $\ell$.

Consider a subsequence ${\textsf {a}}_1 \cdots {\textsf {a}}_\ell$ of ${{\mathbf {\mathsf{{a}}}}}$ of length $\ell$. Let ${{\mathbf {\mathsf{{b}}}}}_{{\textsf {a}}_1\cdots {\textsf {a}}_\ell }$ be the subsequence in ${{\mathbf {\mathsf{{b}}}}}$ which is the surface correspondent of the underlying subsequence ${\textsf {a}}_1 \cdots {\textsf {a}}_\ell$ relative to the correspondence relation $\rho _{{{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{b}}}}}}$ (namely, ${{\mathbf {\mathsf{{b}}}}}_{{\textsf {a}}_1\cdots {\textsf {a}}_\ell }$ is the subsequence of ${{\mathbf {\mathsf{{b}}}}}$ consisting of all and only the segments which are in correspondence with one of the segments ${\textsf {a}}_1,\ldots , {\textsf {a}}_\ell$). The operations of composition and restriction over correspondence relations commute in the sense of the identity (90): the restriction of the composition correspondence relation $\rho _{{{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{b}}}}}} \rho _{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{c}}}}}}$ to the pair of strings $({\textsf {a}}_1\cdots {\textsf {a}}_\ell , {{\mathbf {\mathsf{{c}}}}})$ coincides with the composition of the restrictions of the relations $\rho _{{{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{b}}}}}}$ and $\rho _{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{c}}}}}}$ to the pairs of strings $({\textsf {a}}_1\cdots {\textsf {a}}_\ell , {{\mathbf {\mathsf{{b}}}}}_{{\textsf {a}}_1 \cdots {\textsf {a}}_\ell })$ and $({{\mathbf {\mathsf{{b}}}}}_{{\textsf {a}}_1\cdots {\textsf {a}}_\ell }, {{\mathbf {\mathsf{{c}}}}})$.

The identity (90) says that the candidate (91c) is the composition of the two candidates (91a) and (91b).

The hypothesis that F satisfies the FIC$_{\text {comp}}^{\text {OT}}$ (87) for these two candidates (91a) and (91b) and their composition candidate (91c) becomes:

Since F is I-categorical of order $\ell$ and since the underlying string ${\textsf {a}}_1 \cdots {\textsf {a}}_\ell$ has length $\ell$, the left-hand side of the inequality in the consequent of (92) is equal to either 0 or 1. By reasoning as in Sect. 3.3.2, the FIC$_{\text {comp}}^{\text {OT}}$ (92) thus entails the FTI$_{\text {comp}}$ (93).

The rest of the proof obtains the FTI$_{\text {comp}}$ (88) by summing the inequality (93) over all subsequences ${\textsf {a}}_1\cdots {\textsf {a}}_\ell$ of length $\ell$ of the underlying string ${{\mathbf {\mathsf{{a}}}}}$.

To start, the definition of I-additivity of order $\ell$ applied to the composition candidate $( {{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{c}}}}}, \rho _{{{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{b}}}}}}\rho _{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{c}}}}}} )$ immediately yields the expression (94) for the sum of the terms (93a) over all subsequences ${\textsf {a}}_1\cdots {\textsf {a}}_\ell$ of the underlying string ${{\mathbf {\mathsf{{a}}}}}$.

The sum of the terms (93b) over all subsequences ${\textsf {a}}_1\cdots {\textsf {a}}_\ell$ of the underlying string ${{\mathbf {\mathsf{{a}}}}}$ can be upper bounded as in (95). In step (95a), I have used the hypothesis that F is O-monotone (together with the obvious fact that ${{\mathbf {\mathsf{{b}}}}}_{{\textsf {a}}_1\cdots {\textsf {a}}_\ell }$ is a subsequence of ${{\mathbf {\mathsf{{b}}}}}$). Step (95b) follows from the fact that the restriction of $\rho _{{{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{b}}}}}}$ to the string pair $({\textsf {a}}_1\cdots {\textsf {a}}_\ell , \, {{\mathbf {\mathsf{{b}}}}}_{{\textsf {a}}_1\cdots {\textsf {a}}_\ell })$ is identical to its restriction to the string pair $({\textsf {a}}_1\cdots {\textsf {a}}_\ell , \, {{\mathbf {\mathsf{{b}}}}})$, because ${{\mathbf {\mathsf{{b}}}}}_{{\textsf {a}}_1\cdots {\textsf {a}}_\ell }$ is the subsequence of ${{\mathbf {\mathsf{{b}}}}}$ consisting of those segments which are in correspondence with one of the segments ${\textsf {a}}_1, \dots , {\textsf {a}}_\ell$. Step (95c) follows again from the hypothesis that F is I-additive of order $\ell$.

Finally, let me bound the sum of the terms (93c) over all subsequences ${\textsf {a}}_1\cdots {\textsf {a}}_\ell$ of the underlying string ${{\mathbf {\mathsf{{a}}}}}$. To this end, I note that the implication (96) holds for any two subsequences ${\textsf {a}}_1\cdots {\textsf {a}}_\ell$ and $\widehat{{\textsf {a}}}_1\cdots \widehat{{\textsf {a}}}_\ell$ and their surface correspondent subsequences ${{\mathbf {\mathsf{{b}}}}}_{{\textsf {a}}_1\cdots {\textsf {a}}_\ell }$ and ${{\mathbf {\mathsf{{b}}}}}_{\widehat{{\textsf {a}}}_1\cdots \widehat{{\textsf {a}}}_\ell }$ of ${{\mathbf {\mathsf{{b}}}}}$.

In fact, assume by contradiction that the antecedent holds but the consequent fails. Since the surface correspondent string ${{\mathbf {\mathsf{{b}}}}}_{{\textsf {a}}_1\cdots {\textsf {a}}_\ell }$ has length at least $\ell$ and since the correspondence relation $\rho _{{{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{b}}}}}}$ cannot break any underlying segment into two or more surface segments (because it is one-to-one), each underlying segment ${\textsf {a}}_i$ of ${\textsf {a}}_1\cdots {\textsf {a}}_\ell$ must have a surface correspondent in ${{\mathbf {\mathsf{{b}}}}}_{{\textsf {a}}_1\cdots {\textsf {a}}_\ell }$. The hypothesis ${\textsf {a}}_1\cdots {\textsf {a}}_\ell \ne \widehat{{\textsf {a}}}_1\cdots \widehat{{\textsf {a}}}_\ell$ means that there exists at least one segment ${\textsf {a}}_i$ which belongs to ${\textsf {a}}_1\cdots {\textsf {a}}_\ell$ but not to $\widehat{{\textsf {a}}}_1\cdots \widehat{{\textsf {a}}}_\ell$. Let ${\textsf {b}}$ be the surface correspondent of ${\textsf {a}}_i$ in ${{\mathbf {\mathsf{{b}}}}}_{{\textsf {a}}_1\cdots {\textsf {a}}_\ell }$. Because of the contradictory assumption that the consequent of (96) fails, ${{\mathbf {\mathsf{{b}}}}}_{{\textsf {a}}_1\cdots {\textsf {a}}_\ell } = {{\mathbf {\mathsf{{b}}}}}_{\widehat{{\textsf {a}}}_1\cdots \widehat{{\textsf {a}}}_\ell }$. This means that ${\textsf {b}}$ also belongs to ${{\mathbf {\mathsf{{b}}}}}_{\widehat{{\textsf {a}}}_1\cdots \widehat{{\textsf {a}}}_\ell }$, namely must correspond to some segment $\widehat{{\textsf {a}}}_j$ of $\widehat{{\textsf {a}}}_1\cdots \widehat{{\textsf {a}}}_\ell$. Since ${\textsf {a}}_i$ does not belong to $\widehat{{\textsf {a}}}_1\cdots \widehat{{\textsf {a}}}_\ell$, then ${\textsf {a}}_i$ and $\widehat{{\textsf {a}}}_j$ must be different. The conclusion that both $({\textsf {a}}_i, {\textsf {b}})$ and $(\widehat{{\textsf {a}}}_j, {\textsf {b}})$ belong to $\rho _{{{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{b}}}}}}$ despite the fact that ${\textsf {a}}_i\ne \widehat{{\textsf {a}}}_j$ contradicts the hypothesis that $\rho _{{{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{b}}}}}}$ does not coalesce any two underlying segments.

Let me now go back to the goal of bounding the sum of the terms (93c) over all subsequences ${\textsf {a}}_1\cdots {\textsf {a}}_\ell$ of the underlying string ${{\mathbf {\mathsf{{a}}}}}$. Since ${\textsf {a}}_1\cdots {\textsf {a}}_\ell$ has length $\ell$ and since the correspondence relation $\rho _{{{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{b}}}}}}$ cannot break any underlying segment (because it is one-to-one), the surface correspondent string ${{\mathbf {\mathsf{{b}}}}}_{{\textsf {a}}_1\cdots {\textsf {a}}_\ell }$ has length $\ell$ or smaller. If ${{\mathbf {\mathsf{{b}}}}}_{{\textsf {a}}_1\cdots {\textsf {a}}_\ell }$ has length smaller than $\ell$, the corresponding term (93c) is null, because F is I-additive of order $\ell$ and thus assigns zero violations to candidates whose underlying string is shorter than $\ell$, as noted at the beginning. The sum can thus be restricted to candidates whose underlying form ${{\mathbf {\mathsf{{b}}}}}_{{\textsf {a}}_1\cdots {\textsf {a}}_\ell }$ has length exactly $\ell$, as in step (97a). Condition (96) says that the mapping from the subsequences ${\textsf {a}}_1\cdots {\textsf {a}}_\ell$ to the corresponding surface subsequences ${{\mathbf {\mathsf{{b}}}}}_{{\textsf {a}}_1\cdots {\textsf {a}}_\ell }$ (of length $\ell$) is an injection, thus guaranteeing step (97b). Step (97c) follows again from the hypothesis that F is I-additive of order $\ell$.

The FTI$_{\text {comp}}$ (88) follows by summing the inequality (93) over all subsequences ${\textsf {a}}_1\cdots {\textsf {a}}_\ell$ of length $\ell$ of the string ${{\mathbf {\mathsf{{a}}}}}$, using the three expressions (94), (95), and (97) for the sums over the three terms (93a), (93b), and (93c). $\square$

1.2 A.2 Proof of Proposition 5

• Proposition 5 Assume that, for any two candidates $({{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{b}}}}}, \rho _{{{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{b}}}}}})$ and $({{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{c}}}}}, \rho _{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{c}}}}}})$, the candidate set also contains a candidate $({{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{c}}}}}, \rho _{{{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{c}}}}}})$ such that the FIC $^{\,\text {HG}}$ repeated in (98) holds for any faithfulness constraint F in the constraint set.

  

  Then, the HG grammar corresponding to any weighting of the constraint set is idempotent, no matter what the markedness constraints look like. $\square$

Proof

Suppose that the HG grammar $G_{\varvec{\theta }}$ corresponding to some weighting $\varvec{\theta }$ fails at the idempotency implication (32) for some candidate $({{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{b}}}}}, \rho _{{{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{b}}}}}})$, as stated in (99): $G_{\varvec{\theta }}$ maps the underlying form ${{\mathbf {\mathsf{{a}}}}}$ to $({{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{b}}}}}, \rho _{{{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{b}}}}}})$, as required by the antecedent of the idempotency implication; but it fails to map the underlying form ${{\mathbf {\mathsf{{b}}}}}$ to the identity candidate $({{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{b}}}}}, \mathbb {I}_{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{b}}}}}})$, as required by the consequent.

Condition (99b) means that the grammar $G_{\varvec{\theta }}$ maps the underlying form ${{\mathbf {\mathsf{{b}}}}}$ to a candidate $({{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{c}}}}}, \rho _{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{c}}}}}})$ different from $({{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{b}}}}}, \mathbb {I}_{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{b}}}}}})$. This means that either the two strings ${{\mathbf {\mathsf{{b}}}}}$ and ${{\mathbf {\mathsf{{c}}}}}$ differ or else ${{\mathbf {\mathsf{{b}}}}}$ and ${{\mathbf {\mathsf{{c}}}}}$ coincide but the two correspondence relations $\rho _{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{c}}}}}}$ and $\mathbb {I}_{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{b}}}}}}$ differ. The latter option is impossible, because the candidate $({{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{b}}}}}, \rho _{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{b}}}}}})$ with $\rho _{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{b}}}}}}\ne \mathbb {I}_{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{b}}}}}}$ is harmonically bounded by the candidate $({{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{b}}}}}, \mathbb {I}_{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{b}}}}}})$: faithfulness constraints cannot prefer the former candidate, by (8); and markedness constraints cannot distinguish between the two candidates, by (7). The two strings ${{\mathbf {\mathsf{{b}}}}}$ and ${{\mathbf {\mathsf{{c}}}}}$ must therefore differ and condition (99) becomes (100).

By assumption, the two candidates $({{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{b}}}}}, \rho _{{{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{b}}}}}})$ and $({{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{c}}}}}, \rho _{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{c}}}}}})$ come with a companion candidate $({{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{c}}}}}, \rho _{{{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{c}}}}}})$. The “if-and-only-if” statement (100) can thus be weakened to the “if” statement (101). In fact, if the grammar $G_{\varvec{\theta }}$ maps the underlying form ${{\mathbf {\mathsf{{a}}}}}$ to the candidate $({{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{b}}}}}, \rho _{{{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{b}}}}}})$ as stated in (100a), the weights $\varvec{\theta }$ prefer this candidate $({{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{b}}}}}, \rho _{{{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{b}}}}}})$ to the candidate $({{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{c}}}}}, \rho _{{{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{c}}}}}})$, as stated in (101a). Furthermore, if the grammar $G_{\varvec{\theta }}$ maps the underlying form ${{\mathbf {\mathsf{{b}}}}}$ to the candidate $({{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{c}}}}}, \rho _{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{c}}}}}})$ as stated in (101b), the weights $\varvec{\theta }$ prefer this candidate $({{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{c}}}}}, \rho _{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{c}}}}}})$ to the identity candidate $({{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{b}}}}}, \mathbb {I}_{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{b}}}}}})$, as stated in (101b).

Condition (101) can be made explicit as in (102) in terms of the number of constraint violations. These sums run over a generic markedness constraint M with weight $\theta _M$ and a generic faithfulness constraint F with weight $\theta _F$. The faithfulness constraints do not appear on the right-hand side of (102b) because $F({{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{b}}}}}, \mathbb {I}_{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{b}}}}}}) = 0$ for every faithfulness constraint F, by (8).

Let the constant $\xi$ be defined as $\xi = \sum _{M} \theta _M M({{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{b}}}}}, \mathbb {I}_{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{b}}}}}}) - \sum _{M} \theta _M M({{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{c}}}}}, \rho _{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{c}}}}}})$. Since markedness constraints are blind to underlying forms by (7), then also $\xi = \sum _{M} \theta _M M({{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{b}}}}}, \rho _{{{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{b}}}}}}) - \sum _{M} \theta _M M({{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{c}}}}}, \rho _{{{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{c}}}}}})$. Condition (102) thus becomes:

In conclusion, idempotency holds for the HG grammar corresponding to any weighting of the constraint set provided the two conditions (103a) and (103b) can never be both satisfied , no matter the choice of the weights $\theta _F$ and the constant $\xi$. In other words, it suffices to assume that, for every two candidates $( {{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{b}}}}}, \rho _{{{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{b}}}}}} )$ and $( {{\mathbf {\mathsf{{b}}}}}, \mathbf{c}, \rho _{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{c}}}}}} )$, there exists some candidate $( {{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{c}}}}}, \rho _{{{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{c}}}}}} )$ such that:

To conclude the proof, I need to show that (104) is equivalent to the FIC$^{\text {HG}}$ (106). To start, let me show that (98) entails (104). In fact, suppose that the antecedent of the implication (104) holds. For every faithfulness constraint F, let $\xi _F$ be defined as in (105a). The antecedent of the implication (104) can thus be restated as in (105b). I can assume without loss of generality that the weights $\theta _F$ are all different from zero. The position (105a) thus entails (105c). Since the implication (98) holds by hypothesis, (105c) entails (105d). The consequent of the implication (104) thus follows from (105b) by taking the weighted average of the inequalities (105d) over all faithfulness constraints.

Let me now show that (104) vice versa entails (98). In fact, suppose that the antecedent of the implication (98) holds, namely that $F({{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{c}}}}}, \rho _{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{c}}}}}})\le \xi$. Let me distinguish two cases, depending on whether $\xi$ is an integer or not. To start, assume that $\xi$ is not an integer. The assumption $F({{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{c}}}}}, \rho _{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{c}}}}}})\le \xi$ (with the loose inequality) is thus equivalent to the assumption $F({{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{c}}}}}, \rho _{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{c}}}}}}) < \xi$ (with the strict inequality), because constraint violations are integers. The antecedent of the implication (104) thus holds with all the weights set equal to zero but for the weight $\theta _F$ corresponding to the faithfulness constraint F considered, which is equal to 1. The consequent of the implication (104) must therefore hold as well, which is in turn identical to the consequent of the implication (98) with this special choice of the weights. If instead the antecedent of (98) holds with $\xi$ equal to an integer, let $\widehat{\xi } = \xi + 1/2$. By reasoning as above, I conclude that the consequent of the implication (98) holds for $\widehat{\xi }$. Since constraint violations are integers, the latter entails in turn that the consequent of the implication (98) holds for $\xi$. $\square$

1.3 A.3 Proof of Proposition 12

• Proposition 12 Assume that, for any two candidates $({{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{d}}}}}, \rho _{{{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{d}}}}}})$ and $({{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{d}}}}}, \rho _{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{d}}}}}})$ such that $({{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{d}}}}}, \rho _{{{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{d}}}}}}) \le _{\text {sim}} ({{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{d}}}}}, \rho _{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{d}}}}}})$, for every candidate $({{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{c}}}}}, \rho _{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{c}}}}}})$ different from $({{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{d}}}}}, \rho _{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{d}}}}}})$, the candidate set also contains a candidate $({{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{c}}}}}, \rho _{{{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{c}}}}}})$ different from $({{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{d}}}}}, \rho _{{{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{d}}}}}})$ such that the FODC $^{\,\text {HG}}$ repeated in (106) holds for any faithfulness constraint F.

  

  Then, the HG grammar corresponding to any weighting of the constraint set is output-driven relative to the similarity order $\le _{\text {sim}}$. $\square$

Proof

The proof is similar to the proof of Proposition 5 in Appendix A.2. Suppose that the HG grammar $G_{\varvec{\theta }}$ corresponding to some weighting $\varvec{\theta }$ fails at the output-drivenness implication (51) for two candidates $({{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{d}}}}}, \rho _{{{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{d}}}}}})$ and $({{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{d}}}}}, \rho _{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{d}}}}}})$, as stated in (107): the grammar $G_{\varvec{\theta }}$ maps the underlying form ${{\mathbf {\mathsf{{a}}}}}$ to the candidate $({{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{b}}}}}, \rho _{{{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{b}}}}}})$ with less internal similarity, as required by the antecedent of (51); but it fails to map the underlying form ${{\mathbf {\mathsf{{b}}}}}$ to the candidate $({{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{d}}}}}, \rho _{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{d}}}}}})$ with more internal similarity, as required by the consequent of (51).

By assumption, the candidate $({{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{c}}}}}, \rho _{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{c}}}}}})$ different from $({{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{d}}}}}, \rho _{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{d}}}}}})$ comes with a companion candidate $({{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{c}}}}}, \rho _{{{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{c}}}}}})$ different from $({{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{d}}}}}, \rho _{{{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{d}}}}}})$. The “if-and-only-if” statement (107) can thus be weakened into the “if” statement (108).

Condition (108) can be made explicit as in (109) in terms of the numbers of constraint violations. The sums run over a generic markedness constraint M with weight $\theta _M$ and a generic faithfulness constraint F with weight $\theta _F$.

Taking advantage of the fact that markedness constraints are blind to underlying forms by (7), condition (109) can be rewritten as in (110) with the position $\xi = \sum _{M} \theta _M M({{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{d}}}}}, \rho _{{{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{d}}}}}}) - \sum _{M} \theta _M M({{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{c}}}}}, \rho _{{{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{c}}}}}}) = \sum _{M} \theta _M M({{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{d}}}}}, \rho _{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{d}}}}}}) - \sum _{M} \theta _M M({{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{c}}}}}, \rho _{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{c}}}}}})$.

In conclusion, output-drivenness holds for the HG grammar corresponding to any constraint weighting provided the two conditions (110a) and (110b) can never be satisfied both, no matter the choice of the weights $\theta _F$ and the constant $\xi$. In other words, it suffices to assume that for every candidate $( {{\mathbf {\mathsf{{b}}}}}, \mathbf{c}, \rho _{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{c}}}}}} )$ different from $({{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{d}}}}}, \rho _{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{d}}}}}})$ there exists a candidate $( {{\mathbf {\mathsf{{a}}}}}, \mathbf{c}, \rho _{{{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{c}}}}}} )$ different from $({{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{d}}}}}, \rho _{{{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{d}}}}}})$ such that:

To conclude the proof, condition (111) can be shown to be equivalent to the FODC$^{\text {HG}}$ (106) by reasoning as at the end of the proof of Proposition 5 to show that condition (104) is equivalent to the FIC$^{\text {HG}}$ (98). $\square$

1.4 A.4 Proof of Proposition 13

• Proposition 13 The two FODC $^{\, OT}$ implications repeated in (112)

  

  are jointly equivalent to the following condition:

  

Proof

Let me show that the two FODC$^{\text {OT}}$ implications (112) jointly entail the implication (113). Thus, assume that the antecedent of the latter implication holds for some $\xi$. I distinguish two cases, depending on whether $\xi$ is (strictly) smaller than 0 or not. Let me start with the former case, stated in (114a).

Since $\xi$ is strictly negative, (114a) entails the strict inequality (114b). The latter in turn coincides with the antecedent of the second FODC$^{\text {OT}}$ implication (112b), which therefore ensures that its consequent holds as well, repeated in (114c). Since $\xi$ is larger than $-1$ and constraint violations are integers, (114c) in turn entails (114d), which is the desired consequent of the implication (113). Note that this reasoning has used only the second FODC$^{\text {OT}}$ implication (112b).

Consider next the complementary case where the antecedent of the implication (113) holds with a nonnegative $\xi$, as stated in (115a).

Since $\xi$ is smaller than $+1$ and constraint violations are integers, (114a) entails (114b). The latter in turns says that the consequent of the first FODC$^{\text {OT}}$ implication (112a) fails. The antecedent must therefore fail as well, as stated in (114c). Since $\xi$ is nonnegative, the latter in turn entails (114d), which is the desired consequent of the implication (113). Note that this reasoning has used only the first FODC$^{\text {OT}}$ implication (112a).

Next, let me show that condition (113) with $0< \xi < +1$ in turn entails the first FODC$^{\text {OT}}$ implication (112a). In fact, suppose that the antecedent of the latter implication holds, namely that $F ( {{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{d}}}}}, \rho _{{{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{d}}}}}} ) < F({{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{c}}}}}, \rho _{{{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{c}}}}}})$. Since $\xi$ is smaller than $+1$ and constraint violations are integers, the latter entails that $F ( {{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{d}}}}}, \rho _{{{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{d}}}}}} ) + \xi < F ({{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{c}}}}}, \rho _{{{\mathbf {\mathsf{{a}}}}}, {{\mathbf {\mathsf{{b}}}}}} \rho _{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{c}}}}}})$. The consequent of the implication (113) thus fails. Its antecedent must therefore fail as well, namely $F ( {{\mathbf {\mathsf{{b}}}}}, \mathbf{c}, \rho _{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{c}}}}}} )\ge F ( {{\mathbf {\mathsf{{b}}}}}, \mathbf{d}, \rho _{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{d}}}}}} ) + \xi$. The latter entails that $F ( {{\mathbf {\mathsf{{b}}}}}, \mathbf{c}, \rho _{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{c}}}}}} ) > F ( {{\mathbf {\mathsf{{b}}}}}, \mathbf{d}, \rho _{{{\mathbf {\mathsf{{b}}}}}, {{\mathbf {\mathsf{{d}}}}}} )$, establishing the consequent of the first FODC$_{\text {comp}}^{\text {OT}}$ implication (112a). An analogous reasoning shows that condition (113) with $-1< \xi < 0$ entails the second FODC$_{\text {comp}}^{\text {OT}}$ implication (112b). $\square$