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Subregular linguistics: bridging theoretical linguistics and formal grammar

  • Thomas Graf EMAIL logo
From the journal Theoretical Linguistics


Subregular linguistics is a fairly new approach that seeks a deeper understanding of language by combining the rigor of formal grammar with the empirical sophistication of theoretical linguistics. The approach started in phonology but has since branched out to morphology and even syntax, unearthing unexpected parallels between these three domains of language. In this paper, I argue based on these results that subregular linguistics has a lot to offer to both fields. Subregular linguistics may be the ideal conduit for knowledge transfer between these two communities.

Corresponding author: Thomas Graf, Department of Linguistics, Stony Brook University, Stony Brook, NY, USA, E-mail:

Award Identifier / Grant number: BCS-1845344


I thank the editors for their very detailed comments and their tremendous patience during the revision process.

  1. Research funding: The work reported in this paper was supported by the National Science Foundation under Grant No. BCS-1845344.

Appendix: String languages in the (refined) Chomsky hierarchy

For the interested reader, I include a brief overview of the classes listed in (1) here. The presentation is deliberately informal and emphasizes general intuitions rather than precise definitions.

REG (regular). In mathematical terms, a string language is regular iff it can be recognized by a finite-state automaton. There are many additional characterizations, a.o. definability in monadic second-order logic with successor, having a Myhill-Nerode relation of finite index, or being a projection of a strictly 2-local string language. Each one of these characterizations has unique advantages—automata provide a way of distinguishing well-formed from ill-formed strings, the Myhill-Nerode characterization makes it easy to show that a given language is not regular, monadic second-order logic offers a very succinct, constraint-based description, and so on. Thanks to the large number of equivalent perspectives, there are many different intuitions for what it means to be regular, but the following is perhaps the most accessible: each string in a regular string language can be correctly built from left-to-right while only memorizing a finitely bounded amount of information about the string built so far. For example, the regular language (aa)* only contains strings over a whose length is even, e.g. aa or aaaa, but not aaa. While building such a string, one does not need to store the exact number of as already built, it suffices to keep track of whether the length of the string built so far is odd or even, and that is a finite amount of information that does not scale with the actual length of the string.

CFL (context-free). A string language is context-free iff it can be generated by a context-free grammar (the mathematical counterpart to the familiar phrase structure grammars). Context-free string languages can exhibit an unbounded number of nested dependencies. The palindrome language, for instance, contains strings that read the same from left to right as from right to left, including aa, abba, aaaa, ababa, abbba, abbcbba, and so on. The string abbab, on the other hand, is not part of the palindrome language because the first symbol does not match the last one (nor does the second symbol match the last but one). Each string of the palindrome language consists of multiple nested dependencies: in a string of length n, the i-th symbol must match the (n − i + 1)-th symbol. Hence in abbba, which has length 5, we have a dependency between the two bs in positions 2 and 5 − 2 + 1 = 4, and this dependency in turn is nested inside another dependency between the two as in positions 1 and 5 − 1 + 1 = 5.

Again there are many equivalent characterizations of the context-free languages, but on an intuitive level the central idea behind context-freeness is the ability to take an assembled object, split it in two pieces, and then wrap those pieces around some other object of bounded size. Hence abbba can be understood as taking aa and breaking it in two pieces a and a, which are then wrapped around bb to yield abba. Then abba is again split into ab and ba and wrapped around b to yield abbba. This view of context-freeness is useful because it can be generalized to yield the mildly context-sensitive classes TAL and MCFL.

TAL (tree-adjoining languages). TALs were originally defined as the string languages that can be generated by Tree Adjoining Grammars, but once again many equivalent definitions have been found, for instance in terms of embedded push-down automata (Vijay-Shanker 1987). One may think of TALs as a generalization of CFLs where a string can be broken into three pieces instead of just two. This makes it possible to generate not only unbounded nested dependencies, but also unbounded crossing dependencies. As a simple example, consider a language where the only available symbols are a, b, and c, and every string exhibits a limited kind of unbounded reduplication: if a string contains a c, then the part before the first c in the string must be the output of reduplication. This means that a well-formed string containing c must be of the form uucv, where u and v are arbitrary strings over a, b, and c. Hence abbabbcb would be well-formed, but abaacb would be ill-formed because abaa does not consist of two identical halves. This kind of reduplication can establish unbounded crossing dependencies because we have two copies, abb and abb, and the i-th symbol of the first copy must match the i-th symbol of the second copy. The bigger the copies, the larger the number of crossing dependencies between them.

A well-formed string like abbabbcb is easy to build as long as we have the ability to break assembled strings into three separate pieces: We start with aacb and split it into the three pieces a, a, and cb. We then take those three pieces and wrap them around b and b to yield ababcb. This is once more broken up into three pieces—ab, ab, and cb—which are again wrapped around b and b to yield the desired abbabbcb.

MCFL (multiple context-free languages). The class MCFL generalizes TAL in two ways. First, there is no longer a universal upper bound on the number of pieces one may have to juggle when building a string. For one language, one may need the ability to split strings into four pieces, for another language the number might be twelve, or a million. Second, pieces can be shuffled around before reassembly, so that the second piece may end up following the fourth piece. This means that MCFLs may exhibit much more complicated crossing dependencies than TALs. However, MCFLs are still fairly similar to TALs, which is why both are considered to be part of the mildly context-sensitive region.

PMCFL (parallel multiple context-free languages). PMCFLs can be regarded as MCFLs with recursive copying. Consider, for instance, the English schm-X construction, where a single noun like rules may be partially copied to yield rules schmules. If this operation could apply recursively, then one could feed it its own output rules schmules and obtain [rules schmules] [schmules schmules], and from that one could build [rules schmules schmules schmules] [schmules schmules schmules schmules]. Now if one considers only the set of strings that this operation can build from rules, one gets the PMCFL rules schmules2n−1 (n ≥ 0), i.e. rules followed by 201=11=0 instances of schmules, or 211=21=1 instance, or 221=41=3 instances, or 231=81=7, or 241=161=15, and so on. Notice how the length of strings grows exponentially, which is a common property of PMCFLs. An MCFL must have an upper bound k such that if one looks at some string s of length n, the shortest string that is longer than s has at most length n+k. PMCFLs may lack this constant growth property.

CSL (context-sensitive) and RE (recursively enumerable). CSL and RE are such powerful classes that it is hard to give them intuitive characterizations. RE is the class of all computable string languages. In linguistic terms, one may think of RE as the class of string languages that can be generated by unrestricted SPE-style rewrite rules of the form αβϕ_ψ (as discussed in Section 3.1, this does not mean that SPE as used by linguists generates RE languages). Here α, β, ϕ and ψ are arbitrary finite sequences of symbols. Essentially, then, anything can be rewritten as anything, conditioned by arbitrary finite contexts. CSL is the special case of RE where β never contains fewer symbols than α, so the output of a rewrite rule cannot be shorter than its input. Both classes allow for dependencies very much unlike what we find in natural language, e.g. that the length of a well-formed string must be a prime number.


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Received: 2020-10-02
Accepted: 2022-01-28
Published Online: 2022-12-01
Published in Print: 2022-10-26

© 2022 Walter de Gruyter GmbH, Berlin/Boston

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