Polarity Computations in Flexible Categorial Grammar

Proceedings of the 7th Joint Conference on Lexical and Computational Semantics (*SEM), pages 124–129 New Orleans, June 5-6, 2018. c⃝2018 Association for Computational Linguistics Polarity Computations in Flexible Categorial Grammar Hai Hu Linguistics Department Indiana University Bloomington, IN 47405 USA huhai@indiana.edu Lawrence S. Moss Mathematics Department Indiana University Bloomington, IN 47405 USA lsm@cs.indiana.edu Abstract This paper shows how to take parse trees in CCG and algorithmically find the polarities of all the constituents. Our work uses the well- known polarization principle corresponding to function application, and we have extended this with principles for type raising and com- position. We provide an algorithm, extending the polarity marking algorithm of van Ben- them. We discuss how our system works in practice, taking input from the C&C parser. 1 Introduction The main goal of this work is to take input from text and then to automatically determine the polar- ity of all the words. For example, we aim to find the arrows in sentences like Every dog↓ scares ↑ at least two↓ cats↑, Every dog↓ and no cat↓ sleeps=, and Most rabbits= hop↑. The ↑ notation means that whenever we use the given sentence truth- fully, if we replace the marked word w with an- other word which is “≥ w,” then the resulting sen- tence will still be true. So we have a semantic inference. The ↓ notation means the same thing, except that when we substitute using a word ≤ w, we again preserve truth. Finally, the = notation means that we have neither property in general; in a valid semantic inference statement, we can only replace the word with itself rather than with some- thing larger or smaller. For example, if we had a collection of back- ground facts like cats ≤ animals, beagles ≤ dogs, scares ≤ startles, and one ≤ two, then our ↑ and ↓ notations on Every dog↓ scares ↑ at least two↓ cats↑ would allow us to conclude Every beagle startles at least one animal. The goal of the paper is to provide a computa- tional system to determine the notations ↑, ↓, = on input text to the best extent possible, either using hand-created parses, or output from a popular and freely available CCG parser C&C (Clark and Cur- ran, 2007). Using our polarity tool, we get a very easy first step on automatic inference done with little or no representation. We discuss potential applications to textual inference. Theory We extend polarity determination for categorial grammar (CG) (see S´anchez-Valencia (1991); van Benthem (1986); van Eijck (2007); Lavalle-Mart´ınez et al. (2017)). These papers only consider the Ajdukiewicz/Bar-Hillel (AB) flavor of CG, where the rules are restricted to applica- tion rules (>) and (<). There is a consensus that application rules alone are too restrictive to give wide-coverage grammars. We thus extend this work to the full set of flexible combinators used in CCG. We prove that our system is sound, in a precise sense. Further, we show how to incorpo- rate boolean reasoning (Keenan and …

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