eval - LINGUIST 726: Mathematical Methods in Linguistics...

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LINGUIST 726: Mathematical Methods in Linguistics September 30, 2004 Formalizing Optimality Theory: Some Preliminaries 1 Rajesh Bhatt 1 The Basics of Optimality Theory 1.1 What an OT Grammar Consists Of A set of inputs A function GEN that given an input generates a set of candidates A set of constraints A constraint hierarchy A function EVAL that given a candidate set and a constraint hierarchy re- turns the set of optimal candidates. 1.2 Strictness of Domination An OT grammar is a special case of the computational problem known as con- straint satisfaction/optimization. Input: a set of constraints ( ) and a set of weights associated with the constraints ( ). a set of candidates ( ) Goal: To determine which candidate satisfies the constraints optimally 1 Thanks to Andries Coetzee and Joe Pater for answering my questions. The good ideas within are taken from Chs. 1 and 2 of Coetzee (2004); the inadequacies are mine.
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We can model how well a particular candidate satisfies the constraints by the following function : (1) (to be consistent with how things are done in OT, I assume that if , then satisfies to a greater extent than does.) The optimization problem now reduced to determining the (or the subset of ) such that . We could think of the ordering of the weights as giving us something like the ranking in OT. Consider the following constraints with the suggested weights: i. *A : must not contain an a , weight 5 ii. *B : must not contain a b , weight 1 Case 1: Let a, b . Then ( a )=5and ( b )=1 Hence b is the optimal candidate. Case 2: Let a, bbbbbb . Then ( a )=5and ( bbbbbb )=6 Hence a is the optimal candidate. What we see here is that a ‘lower ranked’ constraint can overturn the judgement of a ‘higher ranked’ constraint. This is quite different from what happens in OT. In OT, a lower ranked constraint can never overturn the judgement of a higher ranked constraint. This property of an OT grammar is referred to as strictness of domination (cf. McCarthy (2002) and references therein), and restricts unbounded constraint interaction - what we could call ‘bargaining’ between groups of con- straints. In an OT grammar which ranked *A above *B , a would not be optimal in either grammar. 2
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We can thus think of OT as a constraint optimization system where if is ranked above , then , the weight associated with is incommensurably larger than , the weight associated with . 1.3 What we want from an OT Grammar Any OT grammar must be able to give us the set of most optimal candidates given a set of candidates and a constraint hierarchy. In addition, there are certain other decisions one can make concerning what we get from an OT grammar. 1.3.1 Bipartition or a Ranking Given a set of candidates , an OT grammar could return us: (i) a bipartition of into a set of winners and a set of losers: Winners Losers The winners are all equally optimal and more optimal than the losers. No infor- mation is represented concerning further ranking among the losers. (ii) a rank-ordering of
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This note was uploaded on 02/22/2012 for the course LINGUIST 726 taught by Professor Partee during the Spring '07 term at UMass (Amherst).

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eval - LINGUIST 726: Mathematical Methods in Linguistics...

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