Formal Properties of Metrical Structure

Formal Properties of Metrical Structure Marc van Oostendorp Werkverband Grammaticamodellen Tilburg University P.O.Box 90153 5000 LE Tilburg The Netherlands oostendo~kub.nl Abstract This paper offers a provisional mathemat- ical typology of metrical representations. First, a family of algebras corresponding to different versions of grid and bracketed grid theory is introduced. It is subsequently shown in what way bracketed grid theory differs from metrical theories using trees. Finally, we show that there are no sig- nificant differences between the formalism of bracketed grids (for metrical structure) and the representation used in the work of [Kaye, et al., 1985], [1990] for subsyllabic structure. 1 Introduction The most well-known characteristic of Non-linear Phonology is that it shifted its attention from the theory of rules (like in [Chomsky and Halle, 1968]) to the theory of representations. During the last decade phonologists have developed a theory of representa- tions that is sufficiently rich and adequate to describe a wide range of facts from the phonologies of various languages. It is a fairly recent development that these repre- sentations are being studied also from a purely for- mal point of view. There has been done some work on autosegmental structure (for instance [Coleman and Local, 1991; Bird, 1990; Bird and Klein, 1990]) and also some work on metrical trees (like [Coleman, 1990; Coleman, 1992] in unification phonology and [Wheeler, 1981; Moortgat and Morrill, 1991] in cat- egorial logics). As far as I know, apart from the pio- neering work by [Halle and Vergnaud, 1987], hitherto no attention has been paid to the formal aspects of the most popular framework of metrical phonolog~y nowadays, the bracketed grids framework. Yet a lot of questions have to be answered with regard to bracketed grids. First of all, some authors (for instance [Van der Hulst, 1991]) have expressed the intuition that bracketed grids and tree structures (e.g. the [sw] labeled trees of [Hayes, 1981] and re- lated work) are equivalent. In this paper, I study this intuition in some formal detail and show that it is wrong. Secondly, one can wonder what the exact rela- tion is between higher-order metrical structure (foot, word) and subsyllabic structure. In this paper I will show that apart from a fewempirically unimportant details, bracketed grids are equivalent to the kind of subsyllabic structure that is advocated by [Kaye, et al., 1985], [1990] t. 2 The definition of a bracketed grid Below I give a formal definition of the bracketed grid, as it is introduced by HV and subsequently elabo- rated and revised by these authors and others, most notably [Hayes, 1991]. HV have a major part of their book devoted to the formalism themselves, but there are numerous problems with this formalization. I will mention two of them. First, their formalization is not flexible enough to capture all instances of (bracketed) grid theory as it is actually used in the literature of the last few years. They merely give a sketch of the specific im- plementation of bracketed grid theory as it is used in …

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