First-order definability as a constraint on phonological structure

Adam Jardine, Rutgers University Computational analyses have resulted in restrictive characterizations of phonology in terms of string patterns, but a yet unanswered question is what role phonological structure plays in this complexity. This paper addresses this question with a hypothesis that phonological structure is first-order (FO-)definable from strings. It is shown how this hypothesis meaningfully restricts the typology with respect to surface correspondence in ways that current analyses do not. Applications to other structure, as well as methods for positing more restrictive hypotheses, are also discussed. Research applying formal language theory to phonotactics has shown that most fall into restrictive sub-classes of the Star-Free class of patterns (Heinz 2009, 2010; Graf 2010; Rogers et al., 2013; McMullin and Hansson, 2016). The Star-Free class is exactly the class of patterns describable with FO constraints (McNaughton and Papert, 1971), where FO is a logical language with variables ranging over positions in a string and whose basic predicates refer to the properties of those positions (including their order). (For example, (∀x)[[+syl](x) → [+voi](x)] is a FO statement requiring that every position x that is syllabic is also voiced.) However, these characterizations are based on strings lacking the kind of structure phonologists use to explain patterns, such as autosegmental representations, prosodic structure, or surface correspondence. We can develop a theory of structure connected to these characterizations using logical definitions which define properties and relations in additional structure in the FO language of the original string (Courcelle, 1994). Such FO-definable structure cannot be used to describe patterns outside of the Star-Free class. FO-definability is thus a restrictive hypothesis of phonological structure. It also has a clear interpretation in terms of abstractness: FO phonological structure must derive from concrete information in the surface string—it cannot represent arbitrary sets or relations. This idea is simply illustrated with the surface correspondence relations that have been proposed to explain consonant harmony (Hansson 2001, 2010; Rose and Walker 2004). The examples in (1) each represent a logically possible correspondence pattern: (1a) in which all sibilants are in correspondence; (1b) in which adjacent pairs of odd and even sibilants are in correspondence.