Abstract
This paper presents evidence that tone wellformedness patterns share a property of melodylocality, and shows how patterns with this property can be learned. Essentially, a melodylocal pattern is one in which constraints over an autosegmental melody operate independently of constraints over the string of tonebearing units. This includes a range of local tone patterns, longdistance tonepatterns, 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 melodylocal learner can induce attested tone patterns that cannot be learned by the tier projection learners that have formed the basis of work on learning longdistance phonology. Thus, melodylocal learning is a necessary property for learning tone. It is also shown how melodylocal learners are more restrictive than learning directly over autosegmental representations.
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Notes
 1.
Pierrehumbert and Beckman (1988) propose an underspecificationbased model for (Tokyo) Japanese. A partial stringbased representation of this model would instead represent the Ltoned TBUs as ∅ TBUs, indicating that they are not linked to any tone, and H, which indicates the TBU linked to the H tone. How we might fully capture the spirit of underspecification models via string representations is briefly discussed in Sect. 6.3.
 2.
Hyman and Katamba’s 2010 account of Luganda is derivational, explicitly distinguishing between lexical and postlexical levels of the phonology. This paper follows this assumption that the phonology can be organized into distinct subphonologies, although it abstracts away from the details of how these are organized. In a constraintbased framework, this would require a stratally organized grammar (Kiparsky 2000; BermúdezOtero 2017).
 3.
Recall our assumption that generalizations hold for strings of arbitrary length. Thus, the property of locality is about the phonological pattern itself, independent of generalizations about word length or of constraints on performance or processing.
 4.
 5.
This function is based on Jardine and Heinz’s 2015 concatenation operation for generating OCPobeying ARs from strings. For now, we gloss over the treatment of contours, which can be straightforwardly dealt with but are not necessary to capture the tone patterns from Sect. 3. It will be shown in Sect. 6.2 how to adapt our melody function to incorporate contours.
 6.
 7.
In wordfinal position, this can technically be realized as a rising or falling tone; contours are abstracted away from here to focus on the longdistance nature of the pattern. For more on contours, see Sect. 6.2.
 8.
The second tone in (32a) and (32b) are ‘melody high’ tones assigned to a particular mora by tense, aspect, and mood morphology. As Bickmore and Kula (2013) explain, these tones behave identically to underlying tones with respect to the main spreading generalizations.
 9.
The boundaries here are not strictly word boundaries; to make this explicit, one could replace these with appropriate boundaries that demarcate the stem.
 10.
There are cases, e.g. in Cilungu (Bickmore 2007:16), in which a morpheme introduces two H tones, which associate to the beginning and end of the word. However, as both tones are introduced by a morphological process, this is best characterized as a kind of circumfixation and not phonological agreement. The applicability of melodylocal learning to morphological processes is an interesting question for future work.
 11.
For brevity, this grammar abstracts away from the complete set of constraints that obtain bounded spreading.
 12.
A full analysis would also require AR versions of the local spreading constraints in (42), to eliminate forms of the shape *HLL^{n}. However, this is still describable with AR subgraph grammars; see Jardine (2017).
 13.
 14.
Thanks to an anonymous reviewer for pointing this out.
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Acknowledgements
The author would like to thank Jane Chandlee, Jeffrey Heinz, Arto Anttila, and three anonymous reviewers for their thoughtful comments.
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Appendix
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.)
A.1 Strictly local grammars and kfactors
A string u is a kfactor 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 kfactors of w are given by the following function \(\texttt{fac}_{k}\):
For instance, fac_{3}(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 klocal (SL_{k}) grammar is a set \(S\subseteq \texttt{fac}_{k} (\Sigma^{*})\); that is, a subset of all of the possible kfactors that can appear in strings in S. For example, for Σ = {L,H},
Then, for example, \(S_{\mathrm {alt}} =\{ \rtimes \text{H} , \text{H} \text{H} , \text{L} \text{L} , \text{L} \ltimes \}\) is a SL_{2} grammar because ⋊H, HH, LL, and L⋉ are all 2factors of strings in \(\Sigma^{*}\) (for example, they are all in fac_{2}(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 kfactors in S; that is,
For example,
that is, the set of strings of alternating Hs and Ls, as this is exactly the set of strings that contain none of the 2factors in \(S_{\mathrm {alt}}\).
A stringset L is thus strictly klocal iff L = L(S) for some SL_{k} grammar S. We say a stringset is strictly local if it is strictly klocal for some k.
The learning procedure for the class of strictly klocal stringsets amounts to the function \(\mathtt {SLlearn}_{k}\) defined as follows. For a finite set \(D\subset\Sigma^{*}\),
That is, \(\mathtt {SLlearn}_{k}\)(D) returns the set of possible kfactors minus the set of kfactors observed in D. This means that \(\mathtt {SLlearn}_{k}\) returns a strictly klocal grammar consisting of all of the kfactors 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 strictlyklocal stringsetLand a sampleDofLsuch that \(\texttt{fac}_{k} (D)= \texttt{fac}_{k} (L)\), \(\mathtt {SLlearn}_{k} (D)\)returns a strictlyklocal 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 klocal set, there is some strictly klocal 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. □
A.2 Melodylocal grammars and their learning
Having defined strictly local stringsets and their learning, we can now define melodylocal stringsets.
First, we define the mldy function recursively as follows. For \(w\in\Sigma^{*}\),
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,
For a stringset \(L\subseteq\Sigma^{*}\) let \(\mathtt {mldy} (L)=\{ \mathtt {mldy} (w)~~w\in L\}\).
A melody strictly klocal grammar M is thus, like a strictly klocal 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 klocal grammar using the mldy function. The stringset described by M is as follows:
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,jmelodylocal grammar G as a tuple G(S,M) where S is a strictly klocal grammar and M is a melody strictly jlocal grammar. The stringset described by G is thus
that is, the set of strings that satisfy both S and M. We say a stringset is melodylocal if it is k,jmelodylocal for some k and j.
Learning melodylocal 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:
That is, \(\mathtt {MLlearn}_{k,j} (D)\) returns a tuple, the first of which is obtained by running a strictly klocal learning on D, the second of which is a melody strictly jlocal grammar obtained by running strictly jlocal learning on mldy(D). The following theorem asserts the correctness of \(\mathtt {MLlearn}_{k,j}\).
Theorem 2
For a targetk,jmelodylocal 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,jmelodylocal grammarGsuch thatL(G)=L.
Proof
Almost immediate from Theorem 1. If L is kmelodylocal, then there is some kmelodylocal 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 \(L(G)=L(G^{\prime})=L\). □
A.3 Abstract characterization
We can posit an abstract characterization for melodylocal patterns independent of a particular grammar formalism to describe them. This allows us to prove whether or not a pattern is melodylocal. 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_{k}iff for any stringxof lengthk − 1 and any strings \(u_{1}\), \(u_{2}\), \(w_{1}\), and \(w_{2}\),
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 3local. Furthermore, there is no k for which \(L_{\mathrm {Ch}}\) is strictly klocal, because we can simply replace x with \(\text{L} ^{k1}\) (k − 1 repetitions of L).
This shows that, no matter what k − 1, suffix substitution in this case will produce a string LL\(^{k1}\)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 melodydependent suffix substitution closure as the abstract characterization of melodylocal stringsets.
Theorem 4
(Melodydependent suffix substitution closure (MSSC))
A stringsetLis melodylocal iff, for somekand somej,

1.
mldy(L) is strictlyjlocal 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 melodylocal iff it is describable by some melodylocal 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). □
Melodydependent 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 jlocal. 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 3local, as witnessed by the melody strictly jlocal 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 melodydependent 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 melodydependent 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 Lspan 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 melodydependent substitution closure using example strings based on the ARs in (63) from the main text.
In this case, \(u_{1}xw_{2}=\text{HHHHL$^{k1}$HHHH}\), 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 melodylocal.
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Jardine, A. Melody learning and longdistance phonotactics in tone. Nat Lang Linguist Theory 38, 1145–1195 (2020). https://doi.org/10.1007/s1104902009466y
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Keywords
 Tone
 Learnability
 Computational phonology
 Representation