Melody learning and long-distance phonotactics in tone

Abstract

This paper presents evidence that tone well-formedness patterns share a property of melody-locality, and shows how patterns with this property can be learned. Essentially, a melody-local pattern is one in which constraints over an autosegmental melody operate independently of constraints over the string of tone-bearing units. This includes a range of local tone patterns, long-distance tone-patterns, and their interactions. These results are obtained from the perspective of formal language theory and grammatical inference, which focus on the structural properties of patterns, but the implications extend to other learning frameworks. In particular, a melody-local learner can induce attested tone patterns that cannot be learned by the tier projection learners that have formed the basis of work on learning long-distance phonology. Thus, melody-local learning is a necessary property for learning tone. It is also shown how melody-local learners are more restrictive than learning directly over autosegmental representations.

Appendix

This appendix collects the formal details of the paper. Standard notation for set theory is used. Let Σ be a fixed, finite alphabet of symbols and $\Sigma^{*}$ be the set of all strings over Σ, including λ, the empty string. For a symbol σ∈Σ, $\sigma^{n}$ denotes the string resulting from n repetitions of σ. Let |w| indicate the length of a string w. For two strings $w,v\in\Sigma^{*}$, let wv denote their concatenation (likewise for $w\in\Sigma^{*}$ and σ∈Σ, wσ denotes their concatenation). A stringset (or formal language) is a subset of $\Sigma^{*}$; this corresponds to the notion of pattern discussed in Sect. 2.1. Let ⋊ and ⋉ represent special boundary symbols not in Σ that represent the beginning and end of words, respectively; thus, ⋊w⋉ is the string w delineated with the boundary strings. (These correspond to the # boundary used in phonology.)

1.1 A.1 Strictly local grammars and k-factors

A string u is a k-factor of w if |u| = k and $w=v_{1}uv_{2}$ for some $v_{1},v_{2}\in\Sigma^{*}$; that is, u is a substring of w of length k. The k-factors of w are given by the following function $\texttt{fac}_{k}$:

$$\texttt{fac}_{k} (w)\mathrel{\stackrel{\mbox{\footnotesize def}}{=}}\textstyle\begin{array}[t]{ll} \{u~|~u\text{ is a }k\text{-factor of } \rtimes w \ltimes \} & \text{ if }| \rtimes w \ltimes |>k \\ \{ \rtimes w \ltimes \} & \text{ otherwise } \\ \end{array} $$

For instance, fac₃(LHLL)={⋊LH,LHL,HLL,LL⋉}. We extend $\texttt{fac}_{k}$ to stringsets in the natural way; i.e. for $L\subseteq\Sigma^{*}$, $\texttt{fac}_{k} (L)=\bigcup_{w\in L} \texttt{fac}_{k} (w)$.

A strictly k-local (SL_k) grammar is a set $S\subseteq \texttt{fac}_{k} (\Sigma^{*})$; that is, a subset of all of the possible k-factors that can appear in strings in S. For example, for Σ = {L,H},

$$\texttt {fac}_{2} (\Sigma^{*})= \{ \rtimes \text{H} , \rtimes \text{L} , \text{H} \text{H} , \text{H} \text{L} , \text{L} \text{H} , \text{L} \text{L} , \text{H} \ltimes , \text{L} \ltimes \}. $$

Then, for example, $S_{\mathrm {alt}} =\{ \rtimes \text{H} , \text{H} \text{H} , \text{L} \text{L} , \text{L} \ltimes \}$ is a SL₂ grammar because ⋊H, HH, LL, and L⋉ are all 2-factors of strings in $\Sigma^{*}$ (for example, they are all in fac₂(HHLL)).

The for a SL_k grammar S, the stringset described by S, written L(S), is thus the set of strings that contain no k-factors in S; that is,

$$L(S)\mathrel{\stackrel{\mbox{\footnotesize def}}{=}} \{w\in\Sigma^{*}~|~ \texttt{fac}_{k} (w)\cap S=\emptyset\} $$

For example,

$$L( S_{\mathrm {alt}} )= \{ \text{L} \text{H} , \text{L} \text{H} \text{L} \text{H} , \text{L} \text{H} \text{L} \text{H} \text{L} \text{H} ,...\}, $$

that is, the set of strings of alternating Hs and Ls, as this is exactly the set of strings that contain none of the 2-factors in $S_{\mathrm {alt}}$.

A stringset L is thus strictly k-local iff L = L(S) for some SL_k grammar S. We say a stringset is strictly local if it is strictly k-local for some k.

The learning procedure for the class of strictly k-local stringsets amounts to the function $\mathtt {SLlearn}_{k}$ defined as follows. For a finite set $D\subset\Sigma^{*}$,

$$\mathtt {SLlearn}_{k} (D)\mathrel{\stackrel{\mbox{\footnotesize def}}{=}} \texttt{fac}_{k} (\Sigma^{*})- \texttt{fac}_{k} (D) $$

That is, $\mathtt {SLlearn}_{k}$(D) returns the set of possible k-factors minus the set of k-factors observed in D. This means that $\mathtt {SLlearn}_{k}$ returns a strictly k-local grammar consisting of all of the k-factors not observed in D. It should be noted that this is a ‘batch’ conception of the learner, as opposed to the sequential learner presented in the main text. They are equivalent, however. The sequential version of the learner takes some finite sequence of data points $d_{1}$, $d_{2}$, $d_{3}$, ..., $d_{n}$ and returns, at each data point $d_{i}$, $\mathtt {SLlearn}_{k} (\{d_{1},d_{2},d_{3},...,d_{i}\})$.

The following theorem asserts the correctness of $\mathtt {SLlearn}_{k}$.

Theorem 1

For a target strictlyk-local stringsetLand a sampleDofLsuch that $\texttt{fac}_{k} (D)= \texttt{fac}_{k} (L)$, $\mathtt {SLlearn}_{k} (D)$returns a strictlyk-local grammarSsuch thatL(S)=L.

Proof

We show first that w∈L implies w∈L(S) and then that w∈L(S) implies w∈L. Since $\texttt{fac}_{k} (D)= \texttt{fac}_{k} (L)$, then $\texttt{fac}_{k} (\Sigma^{*})- \texttt{fac}_{k} (D)= \texttt{fac}_{k} (\Sigma^{*})- \texttt{fac}_{k} (L)$. Since $S= \mathtt {SLlearn}_{k} (D){=} \texttt{fac}_{k} (\Sigma^{*})- \texttt{fac}_{k} (D)$, then $S= \texttt{fac}_{k} (\Sigma^{*})- \texttt{fac}_{k} (L)$. Thus for every w∈L, $\texttt{fac}_{k} (w)\cap S=\emptyset$, so w∈L(S).

Because L is a strictly k-local set, there is some strictly k-local grammar $S^{\prime}$ such that $L(S^{\prime})=L$. Note that for any string w that if $\texttt{fac}_{k} (w)\subseteq \texttt{fac}_{k} (L)$, then $\texttt{fac}_{k} (w)\cap S^{\prime}=\emptyset$, and so $\texttt{fac}_{k} (w)\in L$. Because $S= \texttt{fac}_{k} (\Sigma^{*})- \texttt{fac}_{k} (L)$. For w∈L(S), then $\texttt{fac}_{k} (w)\cap( \texttt{fac}_{k} (\Sigma^{*})- \texttt{fac}_{k} (L))=\emptyset$ and so $\texttt{fac}_{k} (w)\subseteq \texttt{fac}_{k} (L)$. Thus w∈L(S) implies that w∈L. □

1.2 A.2 Melody-local grammars and their learning

Having defined strictly local stringsets and their learning, we can now define melody-local stringsets.

First, we define the mldy function recursively as follows. For $w\in\Sigma^{*}$,

$$\textstyle\begin{array}{llll} \mathtt {mldy} (w) & \mathrel{\stackrel{\mbox{\footnotesize def}}{=}}& \lambda& \text{ if }w=\lambda, \\ & & \mathtt {mldy} (v)\sigma& \text{ if }w=v\sigma^{n}\text{, }v\neq u\sigma \text{ for some } u\in\Sigma^{*} \end{array} $$

That is, mldy(w) returns λ if w = λ, otherwise it returns mldy(v)σ, where v is the longest string not ending in σ. For example,

$$\mathtt {mldy} (\text{HHLLLH}) \textstyle\begin{array}[t]{ll} = & \mathtt {mldy} (\text{HHLLL})\text{H} \\ = & \mathtt {mldy} (\text{HH})\text{LH} \\ = & \mathtt {mldy} (\lambda)\text{HLH} \\ = & \lambda\text{HLH}=\text{HLH} \end{array} $$

For a stringset $L\subseteq\Sigma^{*}$ let $\mathtt {mldy} (L)=\{ \mathtt {mldy} (w)~|~w\in L\}$.

A melody strictly k-local grammar M is thus, like a strictly k-local grammar, a subset of the possible k factors of Σ. That is, $M\subseteq \texttt{fac}_{k} (\Sigma^{*})$. The difference is that we interpret a melody strictly k-local grammar using the mldy function. The stringset described by M is as follows:

$$L(M)\mathrel{\stackrel{\mbox{\footnotesize def}}{=}} \{w\in\Sigma^{*}~|~ \texttt{fac}_{k} ( \mathtt {mldy} (w))\cap M=\emptyset\} $$

Thus, for example, if k = 3 and M = {HLH}, then HHLLLH∉L(M), because mldy(HHLLLH)=HLH and $\texttt{fac}_{k} (\text{HLH})\cap M=\{\text{HLH}\}$. However, HLLLL∈L(M), because mldy(HLLLL)=HL and $\texttt{fac}_{k} (\text{HL})\cap M=\emptyset$.

We can then define a k,j-melody-local grammar G as a tuple G(S,M) where S is a strictly k-local grammar and M is a melody strictly j-local grammar. The stringset described by G is thus

$$L(G)\mathrel{\stackrel{\mbox{\footnotesize def}}{=}}L(S)\cap L(M), $$

that is, the set of strings that satisfy both S and M. We say a stringset is melody-local if it is k,j-melody-local for some k and j.

Learning melody-local stringsets is a straightforward extension of learning strictly local stringsets. If we fix k, we can define a learning function that takes an input D and outputs the following result:

$$\mathtt {MLlearn}_{k,j} (D)\mathrel{\stackrel{\mbox{\footnotesize def}}{=}}\big( \mathtt {SLlearn}_{k} (D), \mathtt {SLlearn}_{j} ( \mathtt {mldy} (D))\big) $$

That is, $\mathtt {MLlearn}_{k,j} (D)$ returns a tuple, the first of which is obtained by running a strictly k-local learning on D, the second of which is a melody strictly j-local grammar obtained by running strictly j-local learning on mldy(D). The following theorem asserts the correctness of $\mathtt {MLlearn}_{k,j}$.

Theorem 2

For a targetk,j-melody-local stringsetLand a sampleDofLsuch that $\texttt{fac}_{k} (D)= \texttt{fac}_{k} (L)$and $\texttt {fac}_{j} ( \mathtt {mldy} (D))= \texttt {fac}_{j} ( \mathtt {mldy} (L))$, $\mathtt {MLlearn}_{k,j} (D)$returns ak,j-melody-local grammarGsuch thatL(G)=L.

Proof

Almost immediate from Theorem 1. If L is k-melody-local, then there is some k-melody-local grammar $G^{\prime}=(S^{\prime},M^{\prime})$ such that $L(G^{\prime})=L$. Let G = (S,M). Because $\texttt{fac}_{k} (D)= \texttt{fac}_{k} (L)$ and $\texttt {fac}_{j} ( \mathtt {mldy} (D))= \texttt {fac}_{j} ( \mathtt {mldy} (L))$, from Theorem 1 we know that $L(S)=L(S^{\prime})$ and $L(M)=L(M^{\prime})$. Thus $L(G)=L(G^{\prime})=L$. □

1.3 A.3 Abstract characterization

We can posit an abstract characterization for melody-local patterns independent of a particular grammar formalism to describe them. This allows us to prove whether or not a pattern is melody-local. We base this off of the abstract characterization of strictly local stringsets. Strictly local stringsets can be characterized by the property of suffix substitution closure (Rogers and Pullum 2011; Rogers et al. 2013), which can be used to prove that a pattern is not strictly local.

Theorem 3

(Suffix substitution closure, Rogers and Pullum 2011)

A stringsetLis SL_kiff for any stringxof lengthk − 1 and any strings $u_{1}$, $u_{2}$, $w_{1}$, and $w_{2}$,

$$\text{if }u_{1}xu_{2}\in L \textit{ and }w_{1}xw_{2}\in L\textit{, then }u_{1}xw_{2} \in L $$

This means that, for any $u_{1}xu_{2}\in L$, and for any $w_{1}xw_{2}\in L$, then, as long as x is of length k − 1, then we can freely replace $u_{2}$ with $w_{2}$ and be guaranteed to produce another string in L. For example, for the stringset $L_{\mathrm {KJ}}$ (penultimate or final H tone) from the main text, we can set x to be HL (because k = 3, x must be of length 2), and $u_{1}$, $u_{2}$, $w_{1}$, and $w_{2}$ as in (71).

Thus, $u_{1}xu_{2}$ is LLLLLH, which is a member of $L_{\mathrm {KJ}}$, and $w_{1}xw_{2}$ is LLHL, which is also a member of $L_{\mathrm {KJ}}$. If we substitute $u_{2}$ for $w_{2}$ in the former, then we obtain a new string $u_{1}xw_{2}=\text{LLLLLHL}$, which is also in $L_{\mathrm {KJ}}$. We can do this for any x of length 2. Another example is given below in (72) for x = LL.

To show that a stringset is not strictly local, we show that suffix substitution closure fails for some x no matter the size of k. Recall the stringset $L_{\mathrm {Ch}}$ (at least one H) from the main text.

If, as in $L_{\mathrm {KJ}}$, we set k = 3 and choose the string LL, then $L_{\mathrm {Ch}}$ fails suffix substitution closure for x = LL and $u_{1}$, $u_{2}$, $w_{1}$, $w_{2}$ chosen as shown in (74).

Because $u_{1}xw_{2}=\text{LLLL}$ is not a member of $L_{\mathrm {Ch}}$, $L_{\mathrm {Ch}}$ is not strictly 3-local. Furthermore, there is no k for which $L_{\mathrm {Ch}}$ is strictly k-local, because we can simply replace x with $\text{L} ^{k-1}$ (k − 1 repetitions of L).

This shows that, no matter what k − 1, suffix substitution in this case will produce a string LL$^{k-1}$L, which is not a member of $L_{\mathrm {Ch}}$. Thus, $L_{\mathrm {Ch}}$ fails suffix substitution closure for any k. This is a formal version of the intuitive ‘scanning’ proof given in Sect. 2.3, (13).

From the suffix substitution closure characterization of strictly local stringsets, we can posit melody-dependent suffix substitution closure as the abstract characterization of melody-local stringsets.

Theorem 4

(Melody-dependent suffix substitution closure (MSSC))

A stringsetLis melody-local iff, for somekand somej,

• 1. mldy(L) is strictlyj-local and

• 2. for any strings $w_{1},w_{2},u_{1},u_{2}$and for any stringx, |x| = k − 1,

  $$w_{1}xw_{2}\in L\text{ and }u_{1}xu_{2}\in L\textit{ and } \mathtt {mldy} (w_{1}xu_{2}) \in \mathtt {mldy} (L)\textit{ implies }w_{1}xu_{2}\in L $$

Proof

Recall that a stringset is melody-local iff it is describable by some melody-local grammar G = (S,M). Thm. 4a follows directly from the definition of L(M). Thm. 4b follows from suffix substitution closure for L(S) plus the additional requirement that L(G)=L(S)∩L(M). □

Melody-dependent suffix substitution closure adds two conditions on suffix substitution closure. First, Thm. 4a states that mldy(L) (the stringset consisting of the melodies of all strings in L) must be strictly j-local. Second, Thm. 4b adds to the antecedent of the suffix substitution closure implication that $\mathtt {mldy} (w_{1}xu_{2})$ must be in mldy(L). As an example, take $L_{\mathrm {Ch}}$. First, note that $\mathtt {mldy} ( L_{\mathrm {Ch}} )$ (given below in (76)), is strictly 3-local, as witnessed by the melody strictly j-local grammar $M_{\mathrm {Ch}} =\{ \rtimes \text{L} \ltimes \}$ (i.e., it does not contain the string L).

It is also then true that $L_{\mathrm {Ch}}$ satisfies Thm. 4 for k = j = 3. While $L_{\mathrm {Ch}}$ fails the implication in (74) for suffix substitution closure, this implication holds for melody-dependent suffix substitution closure, because mldy(LLLL)=L is not a member of $\mathtt {mldy} ( L_{\mathrm {Ch}} )$, and so it does not matter that $\text{LLLL}\not\in L_{\mathrm {Ch}}$.

It is thus the case that $L_{\mathrm {Ch}}$ satisfies melody-dependent suffix substitution closure.

To give an example that does not, recall the ‘no consecutive spreading Hs’ pattern discussed in Sect. 5. More explicitly, this is the set $L_{\mathrm {No2H}}$ as follows.

That is, $L_{\mathrm {No2H}}$ is exactly the set not containing any strings like *HHLLHH, or *HHHLHH, or *HHLLLHH, where H spans separated by exactly one L-span are both of more than one TBU.

There are no constraints on the melody in this pattern; thus $\mathtt {mldy} ( L_{\mathrm {No2H}} )$ is the full set of alternating strings of Hs and Ls.

We can show that this fails melody-dependent substitution closure using example strings based on the ARs in (63) from the main text.

In this case, $u_{1}xw_{2}=\text{HHHHL$^{k-1}$HHHH}$, in which two consecutive H spans have spread more than two TBUs (as in (63) in the main text). This satisfies the melody constraint (because, e.g., HHHHLH$\in L_{\mathrm {No2H}}$ and so HLH$\in \mathtt {mldy} ( L_{\mathrm {No2H}} )$), but it is not in $L_{\mathrm {No2H}}$, so it fails the implication, for any k. Thus, ‘the no consecutive spreading Hs’ pattern $L_{\mathrm {No2H}}$ is not melody-local.