The Expressivity of Autosegmental Grammars

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.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16

Notes

  1. 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. 2.

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

  3. 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. 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. 5.

    For a recent proposal to unify these three classes, see Graf (2017)’s interval-based strictly piecewise grammars.

  6. 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. 7.

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

  8. 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. 9.

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

  10. 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. 11.

    This interpretation of the NCC, based on that given in Goldsmith (1976), is stronger than that of Coleman and Local (1991), who interpret the NCC as stating that ARs must be planar graphs (and argue that it is not restrictive).

  12. 12.

    Thanks to two anonymous reviewers for highlighting this point.

  13. 13.

    The use of an empty graph with 0 nodes has sometimes been argued against (Harary and Read 1974), but for this purpose (and for, e.g., Engelfriet and Vereijken 1997) it serves as a useful correlate of \(\lambda \).

  14. 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. 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. 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. 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. 18.

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

  19. 19.

    For other tone patterns captured by forbidden k-factor grammars over ARs, see Jardine (2017a).

References

  1. Applegate, R. B. (1972). Ineseño Chumash grammar. Ph.D. thesis, University of California, Berkeley.

  2. Archangeli, D. (1985). Yokuts harmony: Evidence for coplanar representation in nonlinear phonology. Linguistic Inquiry, 16, 335–372.

    Google Scholar 

  3. Archangeli, D., & Pulleyblank, D. (1994). Grounded phonology. Cambridge: MIT Press.

    Google Scholar 

  4. Bird, S., & Ellison, T. M. (1994). One-level phonology: Autosegmental representations and rules as finite automata. Computational Linguistics, 20, 55–90.

    Google Scholar 

  5. Bird, S., & Klein, E. (1990). Phonological events. Journal of Linguistics, 26, 33–56.

    Article  Google Scholar 

  6. Büchi, J. R. (1960). Weak second-order arithmetic and finite automata. Zeitschrift für Mathematische Logik und Grundlagen der Mathmatik, 6, 66–92.

    Article  Google Scholar 

  7. Cassimjee, F., & Kisseberth, C. (1998). Optimal domains theory and Bantu tonology. In C. Kisseberth & L. Hyman (Eds.), Theoretical aspects of Bantu tone (pp. 265–314). Stanford: CSLI.

    Google Scholar 

  8. Chomsky, N. (1956). Three models for the description of language. IRE Transactions on Information Theory, 2, 113–124.

    Article  Google Scholar 

  9. Chomsky, N., & Halle, M. (1965). Some controversial questions in phonological theory. Journal of Linguistics, 1(2), 97–138.

    Article  Google Scholar 

  10. Clements, G. N. (1976). Vowel harmony in nonlinear generative phonology: An autosegmental model. Bloomington: Indiana University Linguistics Club Publications.

    Google Scholar 

  11. Clements, G. N. (1977). Neutral vowels in Hungarian vowel harmony: An autosegmental interpretation. NELS, 7, 49–64.

    Google Scholar 

  12. Clements, G. N. (1985). The geometry of phonological features. Phonology Yearbook, 2, 225–252.

    Article  Google Scholar 

  13. Clements, G . N. (1991). Place of articulation in consonants and vowels: A unified theory. Working Papers of the Cornell Phonetics Laboratory, 5, 77–123.

    Google Scholar 

  14. Cohen, R. S., & Brzozowski, J. A. (1971). Dot-depth of star-free events. Journal of Computer Systems Science, 5, 1–16.

    Article  Google Scholar 

  15. Coleman, J., & Local, J. (1991). The “no crossing constraint” in autosegmental phonology. Linguistics and Philosophy, 14, 295–338.

    Article  Google Scholar 

  16. Diestel, R. (2005). Graph theory. New York: Springer.

    Google Scholar 

  17. Donohue, M. (1997). Tone systems in New Guinea. Linguistic Typology, 1, 347–386.

    Article  Google Scholar 

  18. Engelfriet, J., & Vereijken, J. J. (1997). Context-free graph grammars and concatenation of graphs. Acta Informatica, 34, 773–803.

    Article  Google Scholar 

  19. Ferreira, R. (2013). Efficiently listing combinatorial patterns in graphs. Ph.D. thesis, Università degli Studi di Pisa.

  20. Fu, J., Heinz, J., & Tanner, H. G. (2011). An algebraic characterization of strictly piecewise languages. In M. Ogihara & J. Tarui (Eds.), Theory and applications of models of computation. Lecture notes in computer science (Vol. 6648, pp. 252–263). Berlin: Springer.

    Google Scholar 

  21. García, P., Vidal, E., & Oncina, J. (1990). Learning locally testable languages in the strict sense. In Proceedings of the workshop on algorithmic learning theory (pp. 325–338).

  22. Gentner, T. Q., Fenn, K. M., Margoliash, D., & Nusbaum, H. C. (2006). Recursive syntactic pattern learning by songbirds. Nature, 440, 1204–1207.

    Article  Google Scholar 

  23. Goldsmith, J. (1976). Autosegmental phonology. Ph.D. thesis, Massachussets Institute of Technology.

  24. Graf, T. (2017). The power of locality domains in phonology. Phonology, 34, 385–405.

    Article  Google Scholar 

  25. Hagberg, L. (2006). The place of pitch-accent in a typology of phonological prominence. Paper presented at BeST Conference, SIL, Mexico. University of Leiden, Ms.

  26. Hammond, M. (1988). On deriving the well-formedness condition. Linguistic Inquiry, 19(2), 319–325.

    Google Scholar 

  27. Haraguchi, S. (1977). The tone pattern of Japanese: An autosegmental theory of tonology. Tokyo: Kaitakusha.

    Google Scholar 

  28. Harary, F., & Read, R. C. (1974). Is the null graph a pointless concept? In R. A. Bari & F. Harary (Eds.), Graphs and combinatronics: Proceedings of the capital conference on grpah theory and combinatronics at the George Washington University. New York: Springer.

    Google Scholar 

  29. Hayes, B. (1995). Metrical stress theory. Chicago: The University of Chicago Press.

    Google Scholar 

  30. Heinz, J. (2009). On the role of locality in learning stress patterns. Phonology, 26, 303–351.

    Article  Google Scholar 

  31. Heinz, J. (2010a). Learning long-distance phonotactics. Linguistic Inquiry, 41, 623–661.

    Article  Google Scholar 

  32. Heinz, J. (2010b). String extension learning. In Proceedings of the 48th annual meeting of the association for computational linguistics (pp. 897–906). Association for Computational Linguistics.

  33. Heinz, J., & Idsardi, W. (2011). Sentence and word complexity. Science, 333(6040), 295–297.

    Article  Google Scholar 

  34. Heinz, J., Rawal, C., & Tanner, H. G. (2011). Tier-based strictly local constraints for phonology. In Proceedings of the 49th annual meeting of the association for computational linguistics, Portland, Oregon, USA (pp. 58–64). Association for Computational Linguistics.

  35. Heinz, J., & Rogers, J. (2010). Estimating strictly piecewise distributions. In Proceedings of the 48th annual meeting of the ACL. Association for Computational Linguistics.

  36. Heinz, J., & Rogers, J. (2013). Learning subregular classes of languages with factored deterministic automata. In A. Kornai, & M. Kuhlmann (Eds.), Proceedings of the 13th meeting on the mathematics of language (MoL 13), Sofia, Bulgaria (pp. 64–71). Association for Computational Linguistics.

  37. Hirayama, T. (1951). Kyuusyuu hoogen Onchoo no Kenkyuu (Studies on the Tone of the Kyushu Dialects). Tokyo: Gakkai no shinshin-sha.

    Google Scholar 

  38. Hostetler, R., & Hostetler, C. (1975). A tentative description of Tinputz phonology. Workpapers in Papua New Guinea Languages, 13, 44–51.

    Google Scholar 

  39. Hyman, L. (2009). How not to do typology: the case of pitch-accent. Language Sciences, 31, 213–238.

    Article  Google Scholar 

  40. Hyman, L. (2011). Tone: Is it different? In J. A. Goldsmith, J. Riggle, & A. C. L. Yu (Eds.), The Blackwell handbook of phonological theory (pp. 197–238). New York: Wiley-Blackwell.

    Google Scholar 

  41. Hyman, L. (2014). How autosegmental is phonology? The Linguistic Review, 31, 363–400.

    Article  Google Scholar 

  42. Hyman, L., & Katamba, F. X. (2010). Tone, syntax and prosodic domains in Luganda. In L. Downing, A. Rialland, J.-M. Beltzung, S. Manus, C. Patin, & K. Riedel (Eds.), Papers from the workshop on bantu relative clauses, volume 53 of ZAS Papers in Linguistics (pp. 69–98). ZAS Berlin.

  43. Igarashi, Y. (2007). Typology of prosodic phrasing in japanese dialects. Paper presented at the ICPhS 2007 Satellite Meeting on Intonational Phonology. Ms., Japan Society for the Promotion of Science/National Institute for Japanese Language and Linguistics.

  44. Jardine, A. (2014). Logic and the generative power of Autosegmental phonology. In J. Kingston, C. Moore-Cantwell, J. Pater, & R. Staubs (Eds.), Supplemental proceedings of the 2013 meeting on phonology (UMass Amherst), Proceedings of the annual meetings on phonology. LSA.

  45. Jardine, A. (2016a). Computationally, tone is different. Phonology, 33, 247–283.

    Article  Google Scholar 

  46. Jardine, A. (2016b). Locality and non-linear representations in tonal phonology. Ph.D. thesis, University of Delaware.

  47. Jardine, A. (2017a). The local nature of tone-association patterns. Phonology, 34, 385–405.

    Article  Google Scholar 

  48. Jardine, A. (2017b). On the logical complexity of autosegmental representations. In M. Kanazawa, P. de Groote, & M. Sadrzadeh (Eds.), Proceedings of the 15th meeting on the mathematics of language, London, UK (pp. 22–35). Association for Computational Linguistics.

  49. Jardine, A., & Heinz, J. (2015). A concatenation operation to derive autosegmental graphs. In Proceedings of the 14th meeting on the mathematics of language (MoL 2015), Chicago, USA (pp. 139–151). Association for Computational Linguistics.

  50. Jardine, A., & Heinz, J. (2016a). Markedness constraints are negative: An autosegmental constraint definition language. In Proceedings of CLS 51.

  51. Jardine, A., & Heinz, J. (2016b). Learning tier-based strictly 2-local languages. Transactions of the Association for Computational Linguistics, 4, 87–98.

    Article  Google Scholar 

  52. Jardine, A., & McMullin, K. (2017). Efficient learning of tier-based strictly \(k\)-local languages. In F. Drewes, C. Martín-Vide, & B. Truthe (Eds.), Language and automata theory and applications, 11th international conference, Lecture notes in computer science (pp. 64–76). Springer.

  53. Johnson, C. D. (1972). Formal aspects of phonological description. Berlin: Mouton.

    Google Scholar 

  54. Kager, R. (1995). The metrical theory of word stress. In J. A. Goldsmith (Ed.), The handbook of phonological theory, chapter 10 (pp. 367–402). New York: Blackwell.

    Google Scholar 

  55. Kaplan, R., & Kay, M. (1994). Regular models of phonological rule systems. Computational Linguistics, 20, 331–78.

    Google Scholar 

  56. Kay, M. (1987). Nonconcatenative finite-state morphology. In Proceedings, third meeting of the European chapter of the Association for Computational Linguistics (pp. 2–10).

  57. Kenstowicz, M., & Kisseberth, C. (1990). Chizigula tonology: The word and beyond. In S. Inkelas & D. Zec (Eds.), The phonology-syntax connection (pp. 163–194). Chicago: The University of Chicago Press.

    Google Scholar 

  58. Kisseberth, C. W. (1970). On the functional unity of phonological rules. Linguistic Inquiry, 1(3), 291–306.

    Google Scholar 

  59. Kisseberth, C., & Odden, D. (2003). Tone. In D. Nurse & G. Philippson (Eds.), The Bantu languages. New York: Routledge.

    Google Scholar 

  60. Kleene, S. C. (1956). Representation of events in nerve nets and finite automata. In Automata studies (pp. 3–42). Princeton: Princeton University Press.

  61. Kornai, A. (1991). Formal phonology. Ph.D. thesis, Stanford University.

  62. Kornai, A. (1995). Formal phonology. Abingdon: Garland Publication.

    Google Scholar 

  63. Kubozono, H. (2012). Varieties of pitch accent systems in Japanese. Lingua, 122, 1395–1414.

    Article  Google Scholar 

  64. Leben, W. R. (1973). Suprasegmental phonology. Ph.D. thesis, Massachussets Institute of Technology.

  65. Leben, W. R. (2006). Rethinking autosegmental phonology. In J. Mugane (Ed.), Selected proceedings of the 35th annual conference on African linguistics (pp. 1–9). Somerville, MA: Cascadilla Proceedings Project.

    Google Scholar 

  66. Lombardi, L. (1999). Positional faithfulness and voicing assimilation in Optimality Theory. NLLT, 17, 267–302.

    Google Scholar 

  67. McCarthy, J. J. (1979). Formal problems in semitic phonology and morphology. Ph.D. thesis, MIT.

  68. McCarthy, J. J. (1985). Formal problems in semitic phonology and morphology. New York: Garland.

    Google Scholar 

  69. McMullin, K., & Hansson, G. O. (2016). Long-distance phonotactics as tier-based strictly 2-local languages. In Proceedings of AMP 2015.

  70. McNaugthon, R. (1974). Algebraic decision procedures for local testability. Mathematical Systems Theory, 8, 60–76.

    Article  Google Scholar 

  71. McNaughton, R., & Papert, S. (1971). Counter-free automata. Cambridge: MIT Press.

    Google Scholar 

  72. Nevins, A. (2010). Locality in vowel harmony. Number 55 in linguistic inquiry monographs. Cambridge: MIT Press.

    Google Scholar 

  73. Odden, D. (1986). On the role of the obligatory contour principle in phonological theory. Language, 62(2), 353–383.

    Article  Google Scholar 

  74. Pulleyblank, D. (1986). Tone in lexical phonology. Dordrecht: D. Reidel.

    Google Scholar 

  75. Rogers, J. (1997). Strict LT2: Regular \(:\): Local : Recognizable. In C. Retoré (Ed.), Logical aspects of computational linguistics: First international conference, LACL ’96 Nancy, France, September 23–25, 1996 selected papers (pp. 366–385). Berlin: Springer.

  76. Rogers, J., Heinz, J., Bailey, G., Edlefsen, M., Visscher, M., Wellcome, D., et al. (2010). On languages piecewise testable in the strict sense. In C. Ebert, G. Jäger, & J. Michaelis (Eds.), The mathematics of language, volume 6149 of Lecture notes in artifical intelligence (pp. 255–265). Berlin: Springer.

    Google Scholar 

  77. Rogers, J., Heinz, J., Fero, M., Hurst, J., Lambert, D., & Wibel, S. (2013). Cognitive and sub-regular complexity. In Formal grammar, volume 8036 of Lecture notes in computer science (pp. 90–108). Springer.

  78. Rogers, J., & Pullum, G. (2011). Aural pattern recognition experiments and the subregular hierarchy. Journal of Logic, Language and Information, 20, 329–342.

    Article  Google Scholar 

  79. Sagey, E. (1988). On the ill-formedness of crossing association lines. Linguistic Inquiry, 19, 109–118.

    Google Scholar 

  80. Schūtzenberger, M. (1965). On finite monoids having only trivial subgroups. Information and Control, 8, 190–194.

    Article  Google Scholar 

  81. Shih, S., & Inkelas, S. (2014). A subsegmental correspondence approach to contour tone (dis)harmony patterns. In J. Kingston, C. Moore-Cantwell, J. Pater, & R. Staubs (Eds.), Proceedings of the 2013 meeting on phonology (UMass Amherst), Proceedings of the annual meetings on phonology. LSA.

  82. Simon, I. (1975). Piecewise testable events. In H. Brakhage (Ed.), Automata theory and formal languages 2nd GI conference Kaiserslautern, May 20–23, 1975. Lecture notes in computer science (Vol. 33, pp. 214–222). Berlin: Springer.

  83. Steriade, D. (1987). Locality conditions and feature geometry. North Eastern Linguistic Society, 18, 595–618.

    Google Scholar 

  84. Strother-Garcia, K. (2017). Imdlawn Tashlhiyt Berber syllabification is quantifier-free. In Proceedings of the first annual meeting of the Society for Computation in Linguistics (Vol. 1, pp. 145–153).

  85. Thomas, W. (1982). Classifying regular events in symbolic logic. Journal of Computer and Systems Sciences, 25, 360–376.

    Article  Google Scholar 

  86. Uwano, Z. (1989). Nihongo no akusento [Japanese accent]. In M. Sugito (Ed.), Koza Nihongo to Nihongo Kyoiku 2: Nihongo no Onsei On’in [Lectures, Japanese and Japanese Education 2: Japanese Phonetics and Phonology] (pp. 178–205). Tokyo: Meiji Shoin.

    Google Scholar 

  87. van der Hulst, H., & Smith, N. (1986). On neutral vowels. In K. Bogers, H. van der Hulst, & M. Mous (Eds.), The phonological representation of suprasegmentals (pp. 233–281). Dordrecht: Foris Publications.

    Google Scholar 

  88. Walker, R. (2011). Vowel patterns in language. Cambridge: Cambridge University Press.

    Google Scholar 

  89. Wiebe, B. (1992). Modelling autosegmental phonology with multi-tape finite state transducers. Master’s thesis, Simon Fraser University.

  90. Williams, E. S. (1976). Underlying tone in Margi and Igbo. Linguistic Inquiry, 7(3), 463–484.

    Google Scholar 

  91. Yip, M. (2002). Tone. Cambridge: Cambridge University Press.

    Google Scholar 

  92. Yli-Jyrä, A. (2013). On finite-state tonology with autosegmental representations. In Proceedings of the 11th international conference on finite state methods and natural language processing (pp. 90–98). Association for Computational Linguistics.

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Adam Jardine.

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.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

Keywords

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