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Computational locality in morphological maps

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Abstract

This paper presents a computational investigation of a range of morphological operations. These operations are first represented as morphological maps, or functions that take a stem as input and return an output with the operation applied (e.g., the ing-suffixation map takes the input ‘dɹɪŋk’ and returns ‘dɹɪŋk+ɪŋ’). Given such representations, each operation can be classified in terms of the computational complexity needed to map a given input to its correct output. The set of operations analyzed includes various types of affixation, reduplication, and non-concatenative morphology. The results indicate that many of these operations require less than the power of regular relations (i.e., they are subregular functions), the exception being total reduplication. A comparison of the maps that fall into different complexity classes raises important questions for our overall understanding of the computational nature of phonology, morphology, and the morpho-phonological interface.

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Notes

  1. Though finite-state approximations of total reduplication have been proposed and implemented by several of these authors.

  2. These terms will be explained in the sections to follow.

  3. There are other definitions of regular languages based in other formalisms (e.g., regular expressions, monadic second order logic, etc.), but this paper will use only automata-theoretic characterizations throughout.

  4. This is not a proof that the language is non-regular, just an intuitive explanation. To see how an actual proof can be constructed, readers are referred to Hopcroft et al. (2000).

  5. For a more comprehensive introduction to the finite state formalism and its application to phonology and morphology, readers are directed to Beesley and Karttunen (2003) and Roark and Sproat (2007).

  6. Note that the strings in this example are orthographic instead of phonemic, which is the norm for a system designed to analyze text.

  7. Formally, this means that if \(R_{1}\) and \(R_{2}\) are regular relations and \((x,y) \in R_{1}\) and \((y,z) \in R_{2}\), then there exists another regular relation \(R_{3}\) such that \((x,z) \in R_{3}\).

  8. A reasonable follow-up question would be whether these patterns are in fact syntactic, where we expect to find non-regular phenomena. Culy gives evidence based on the tone pattern of these nouns that suggests it is in fact a morphological phenomenon.

  9. See Langendoen (1981) for an argument against their conclusion.

  10. A finite language is simply a finite set of strings. The grammar for such a language would not have an infinite generative capacity. For this reason finite formal languages (and by extension finite relations) have little to no theoretical interest for natural languages, under the assumption that there is no upper bound on the length of words in a human language (i.e., human languages are infinite).

  11. This is a simplification of the Navajo facts. More generally, [+anterior] sibilants cannot precede [−anterior] sibilants, and vice versa.

  12. Note that strings that do not contain a voiced obstruent in word-final position are simply mapped to themselves. In other words, D is a total function defined for all strings from \(\varSigma ^{*}\), not just those that satisfy the structural description for final devoicing.

  13. Note that since ⋊ is not part of Σ and is therefore guaranteed to only appear once at the start of the string, the λ state and the ⋊ state could also be collapsed with the ⋊ transition being a self-loop. Keeping the two states distinct is motivated by greater transparency in how they represent the pattern in question. See also Chandlee et al. (2015) for reasons why, at least in OSL FSTs, a distinct ⋊ state is necessary.

  14. The fact that the map is ISL does not depend on this reduced alphabet. It would still be ISL, for the same value of k, if the alphabet included the complete segment inventory for a particular language. The FST in that case would just have more states and therefore be less readable.

  15. ISL FSTs are not designated as left or right because when paying attention to the input the same map will result regardless of whether the string is read from the left or the right. For more on this distinction, see Kaplan and Kay (1994), Hulden (2009a), Heinz and Lai (2013).

  16. This is assuming long vowels are represented as VV; if the alphabet instead includes a V: symbol then only the first 5 segments need to be examined.

  17. Thanks to an anonymous reviewer for bringing these cases to my attention.

  18. See Hayes (1982, 1999) for additional examples of processes in this language that depend on the number of syllables in the word.

  19. Riggle (2006) notes that palatal nasal codas are not banned generally in Pima, just in the context of reduplication.

  20. Riggle (2006) also gives examples of forms with complex onsets, in which the second consonant of the onset copies (along with the vowel according to the generalizations already discussed): kɺavo ↦ kɺa+ɺ+vo, ‘nails’. He notes that only a few such forms exist in the language, but the fact that they follow the general pattern suggests that the infixation map should include them. The FST given in the text could easily be modified to handle complex onsets; this would increase the k-value by 1.

  21. This type of reduplication has also been called ‘wrong side reduplication’, and its status is controversial. Nelson (2003) argues that all purported cases are epiphenomenal, while Riggle (2003) and more recently Kusmer and Hauser (2016) argue for genuine examples in Creek/Muskogean and Koasati, respectively. The analysis presented in this paper is not an argument for or against the existence of non-local reduplicative copying; it only reveals the computational properties of such a map.

  22. This is the form used when the noun is an intransitive subject or a transitive object (Bogoras 1969).

  23. It is important to understand the difference between the ?:? self-loop in Fig. 26 and those in the ISL FSTs in Figs. 16, 18, 20, and 21. In the ISL FSTs, state ? and its self-loop are an abbreviation for the states and transitions for all other k − 1 sequences not pictured in the figure. These FSTs proceed through these states depending on the input. In Fig. 26, however, the FST remains in state tal and consumes all additional input with the ?:? loop. It must stay in this state to retain the knowledge that the input began with tal.

  24. Again, this does not mean string reversal is required to model the pattern as subsequential. This is just a way to represent the pattern with an FST that reads left-to-right, to be consistent with the other FSTs presented in the paper. In every such case, there is an equivalent FST that reads from the right and also builds the output string starting from the right, such that no string reversal operation is needed.

  25. Engelfriet and Hoogeboom (2001) show that total reduplication can be modeled using graph transductions defined with Monadic Second Order (MSO) logic formulae. This is an interesting result because in terms of formal languages MSO formulae correspond exactly to the regular languages. The total reduplication example, however, proves that the same is not true for maps: MSO formulae can describe both regular and non-regular relations.

  26. This is not a criticism of the works just cited, as they were clearly motivated by different research questions and objectives.

  27. An anonymous reviewer questions this goal, given that computational complexity does not necessarily correspond to the level of processing difficulty (see, e.g., Bach et al. 1986). But there are other areas of interest that the study of computational properties can inform aside from processing, such as evaluating the generative capacity of a particular theory. In addition, computational properties—particularly subregular ones—provide an inroad to understanding how the grammars used in processing are learned in the first place (for arguments in favor of this approach see Heinz 2007, 2009, 2010).

  28. Mafa distinguishes dental and alveolar: the obstruents /t/, /d/, and /nd/ are classified as dental and therefore not subject to palatalization (Ettlinger 2004).

  29. Recall that in a right subsequential FST, the input is read from the right and the resulting output string is reversed. Thus the input /gudza/ would be read as /azdug/, and the resulting output, [j+azdug], would be reversed to [gudza+j].

  30. They also use this operation in their treatment of reduplication, see Beesley and Karttunen (2003) for details.

  31. The ‘tapes’ of an automaton refer to the number of strings being read or written. In all of the FSTs in this paper one tape corresponds to the input and one to the output.

  32. An anonymous reviewer points out that this operation would be regular if the reversal were bounded. But, as with total reduplication, the assumption is that the domain of the function is any possible noun (i.e., it’s unbounded).

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  1. Haverford College, Tri-Co Department of Linguistics, 370 Lancaster Avenue, Haverford, PA, 19041, USA

    Jane Chandlee

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Appendix

Appendix

For ease of exposition the analysis of German circumfixation in Sect. 4.1 treated separately the two generalizations for the distribution of the suffix allomorph -et. This appendix presents the complete 3-ISL FST that models the allomorphy as a single map. The FST is presented in table form for readability. Each row \(q_{1}\) of the table corresponds to a state in the FST, and each column a corresponds to one of the input segments that can be read from that state. The table cells contain pairs (b, \(q_{2}\)) where b is the output produced for input a and \(q_{2}\) is the destination state of the transition from \(q_{1}\) for input a. To further illustrate how this table representation corresponds to the graphical representations used throughout the paper, those transitions represented graphically in Fig. 15 are shaded in the table. The alphabet is Σ= {L, N, T, ?}, where ‘?’ represents all segments in the German inventory except for liquids, nasals, and alveolar stops.

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Chandlee, J. Computational locality in morphological maps. Morphology 27, 599–641 (2017). https://doi.org/10.1007/s11525-017-9316-9

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