## Abstract

This paper extends a notion of local grammars in formal language theory to autosegmental representations, in order to develop a sufficiently expressive yet computationally restrictive theory of well-formedness in natural language tone patterns. More specifically, it shows how to define a class ASL\(^g\) of stringsets using local grammars over autosegmental representations and a mapping *g* from strings to autosegmental structures. It then defines a particular class ASL\(^{g_T}\) using autosegmental representations specific to tone and compares its expressivity to established formal language grammars that have been successfully applied to other areas of phonology.

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## Notes

- 1.
Tones have been variously been analyzed as being properties of vowels, moras, or syllables, depending on the language. This paper abstracts away from this issue. For discussion, see, e.g., Yip (2002).

- 2.
This use of the terms ‘obligatoriness’ and ‘culminativity’ is due to Hyman (2009).

- 3.
The tone patterns in the Japanese dialects are often referred to as ‘pitch accent’ patterns (see, e.g., Kubozono 2012), but they fit the definition of tone system in that they use pitch to make lexical contrasts, and it has been argued that there is no reason to treat them as phenomenologically distinct from other tone patterns (Hyman 2009). This is at least true for the patterns discussed here, which appear both in Japanese dialects and tone systems elsewhere.

- 4.
The local/piecewise subregular hierarchy is not the only subregular hierarchy; another example is the dot-depth hierarchy of Cohen and Brzozowski (1971, et seq.). However, for brevity this paper will use ‘subregular hierarchy’ to refer specifically to the local/piecewise subregular hierarchy.

- 5.
For a recent proposal to unify these three classes, see Graf (2017)’s

*interval-based strictly piecewise*grammars. - 6.
This pattern is more precisely described by referring to syllable structure and the relative sonority of consonants in the onset. However, adding syllable structure or featural representations does not change the fundamentally local nature of the pattern, which refers to sequences of adjacent segments. Interested readers are referred to Strother-Garcia (2017) on the local nature of syllabification.

- 7.
For discussion of some non-SP, non-TSL constraints in stress patterns see Rogers et al. (2013).

- 8.
For expositional simplicity, we abstract away from the constraint in Kameyama Japanese against one-mora words. This is a SL\(_3\) constraint, as witnessed by \(\left\{ \rtimes {\text {F}}\ltimes ,\rtimes {\text {H}}\ltimes ,\rtimes {\text {L}}\ltimes \right\} \).

- 9.
Likewise, because it is not SP or TSL for a single tier, it is not interval-based strictly piecewise (see Fn. 5).

- 10.
Alternative representations for tone exist (Cassimjee and Kisseberth 1998; Leben 2006; Shih and Inkelas 2014), but a survey of the arguments for using ARs is beyond the purview of this paper. For arguments in favor of ARs, see, for example, Goldsmith (1976), Clements (1977), Archangeli and Pulleyblank (1994) and Hyman (2014).

- 11.
- 12.
Thanks to two anonymous reviewers for highlighting this point.

- 13.
- 14.
Technically, Jardine and Heinz show this only for a binary partition

*P*, but as they note, their reasoning extends straightforwardly to partitions of arbitary size. - 15.
Note this notion of factor is distinct from that in graph theory, which usually refers to a subgraph of a graph containing its entire set of nodes (see, e.g., Diestel 2005).

- 16.
Because for this \(g_T\) multiple association is derived through a merge operation and thus the OCP, it may appear that the OCP is a necessary assumption to capture non-local constraints. However, this is not the case: it is

*multiple association*, and not the OCP, that is responsible for this non-local interaction. If, we were to consider a non-functional*g*that allowed ARs with OCP violations, as in (22), we could still ban OCP violations on a pattern-specific basis by including the following forbidden subgraphs.Thus, any relation between strings and ARs that allow for multiple association can capture such long-distance dependencies.

- 17.
Ideally, the following proofs would be based on an abstract characterization of the \(\mathrm {ASL^{g_T}}\) class, such as SSC for SL sets. As noted in Sect. 7, such a characterization is left for future work.

- 18.
Crucially, this is only defined for graphs in \(g_T(\varSigma ^*)\), not \(\mathrm {GR}(\varGamma )\) in general.

- 19.
For other tone patterns captured by forbidden

*k*-factor grammars over ARs, see Jardine (2017a).

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## Additional information

Many thanks to Jeff Heinz, Jane Chandlee, Thomas Graf, Jim Rogers, the members of the computational linguistics group at the University of Delaware, the members of Thomas Graf’s computational phonology class at Stony Brook University, and the students of the author’s seminar on computation and representation at Rutgers University.

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Jardine, A. The Expressivity of Autosegmental Grammars.
*J of Log Lang and Inf* **28, **9–54 (2019). https://doi.org/10.1007/s10849-018-9270-x

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### Keywords

- Formal language theory
- Graph theory
- Phonology
- Autosegmental representations
- Tone