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OT grammars, beyond partial orders: ERC sets and antimatroids

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Abstract

Grammars in Optimality Theory can be characterized by sets of Elementary Ranking Conditions (ERCs). Antimatroids are structures that arose initially in the study of lattices. In this paper we prove that antimatroids and consistent ERC sets have the same formal structures. We do so by defining two functions Antimat and RCErc, Antimat being a function from consistent sets of ERCs to antimatroids and RCErc a function from antimatroids to ERC sets. We then show that these functions are inverses of each other and that both maintain the structural properties of ERC sets and antimatroids. This establishes that antimatroids and consistent ERC sets have the same formal structure, allowing linguists to import from the sizable work done on antimatroids any and all results.

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Notes

  1. A lattice is a partially ordered set in which every pair of elements has a meet and a join (a greatest lower bound and least upper bound).

  2. Note that here we define MChain as a function from a total order to a subset of the power set lattice on the constraints, even though our ultimate goal is to define MChain from consistent ERC sets to a subset of the power set lattice. We will in subsequent sections use this total order MChain to define our ultimate ERC set MChain.

  3. As Gaja Jarosz has pointed out (p.c.) this encoding may include other total orders (cf. the three total orders abcd, abdc, and bacd—when embedded in the power set lattice, the total order badc is also encoded). It is only when the orders are ERC-set representable that the encoding is conservative; that is, no other orders are included in their power-set representation. We will show this in latter sections of this paper.

  4. An ERC with only one W and multiple Ls corresponds to a set of simple ERCs. E.g. \(\langle\mathrm{W}, \mathrm{e}, \mathrm{L}, \mathrm{L}\rangle = \{\langle \mathrm{W}, \mathrm{e}, \mathrm{L}, \mathrm{e}\rangle, \langle\mathrm{W},\mathrm{e}, \mathrm{e}, \mathrm{L}\rangle\}\).

  5. This definition differs slightly from the literature (having the property of augmentation) but is equivalent in the systems investigated here.

  6. It should also be obvious that if given any set of ERCs \(E\) then by defining the constraint set to be the ground set \(G\), and MChain(\(E\)) to be \(F\) defines a set system—this immediacy comes from the triviality of the definition of a set system, any subset of the power set, not to be equated with the more structured accessible set system.

  7. There are numerous equivalent definitions of antimatroids. See Dietrich (1987) for a number of variants.

  8. This antimatroid is equivalent to \(\mathit{MChain}(\langle\mathrm{W},\mathrm{e},\mathrm{e}, \mathrm{L}\rangle)\) depicted in Fig.  above.

  9. Recall that the set of constraints marked with a W in ERC \(\alpha\) is denoted \(\mathrm{W}(\alpha)\), the set of constraints marked L is \(\mathrm{L}(\alpha)\), and those marked e, \(\mathrm{e}(\alpha)\).

  10. This antimatroid is equivalent to \(\mathit{MChain}(\langle\mathrm{W},\mathrm{W},\mathrm{L},\mathrm{L}\rangle)\).

  11. Freedom and rootedness do not exhaust the outcomes of trace production—a trace can be neither free nor a rooted circuit—see \(\{a, b, d\}\) with \(\mathit{Antimat}(\mathrm{W}, \mathrm{e}, \mathrm{e}, \mathrm{L}\rangle)\).

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Acknowledgements

Much thanks goes to Alan Prince for regular discussions about this paper, discussions which greatly facilitated its completion. We are also appreciative of comments by Gaja Jarosz and three anonymous reviewers who suggested numerous improvements. The provenance of errors and miscommunications in the paper falls solely at the feet of its two authors.

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  1. Eckerd College, Dept. of Mathematics, 4200 54th Ave South, St. Petersburg, FL, 33711, USA

    Nazarré Merchant

  2. University of Chicago, Dept. of Linguistics, 1010 E 59th Street, Chicago, IL, 60637, USA

    Jason Riggle

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  1. Nazarré Merchant
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Correspondence to Nazarré Merchant.

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Merchant, N., Riggle, J. OT grammars, beyond partial orders: ERC sets and antimatroids. Nat Lang Linguist Theory 34, 241–269 (2016). https://doi.org/10.1007/s11049-015-9297-5

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