1. Computing the “Fuzziness” of Scalar Quantificationin Ontological Semantics Whitney Vandiver Purdue University April 17, 2010 Purdue Linguistics Association Symposium

2. 2010 U.S. Census Residence Rule Used to determine where people should be counted as living during the 2010 Census: Followed by 21 explanations to disambiguate “most of the time” for various contexts Primary Question: If humans struggle to disambiguate, how can computational approaches handle scalar quantification? “Count people at their usual residence, which is the place where they live and sleep most of the time”1

3. Outline Scalar Quantification Need for semantic treatment Fuzziness of Membership Ontological Semantics Theory Introduction Treatment: Scales and Ranges Benefits Semantics Classification “Stationary” Scalars “Floating” Scalars Composition of Quantifiers Implementation in NLP

4. Outline Scalar Quantification Need for semantic treatment Fuzziness of Membership Ontological Semantics Theory Introduction Treatment: Scales and Ranges Benefits Semantics Classification “Stationary” Scalars “Floating” Scalars Composition of Quantifiers Implementation in NLP

5. Scalar Quantification Comparison of quantifiers… (1) He drank a little/some/a lot of coffee. (2) She bought a few/some/many books. (3) He wrote fewer/more papers this semester. (4) She consumed much less tea than John. Different quantifications a little is less than some—some is less than a lot few is less than some—some is less than many When does a little become some? And some become a lot? Evaluations of quantificational ranges differ by individual situations

6. Scalar Quantification:Need for semantic treatment Syntactic analysis2 Co-occur only with plural count nouns:(a) few, several, many, fewer Co-occur only with non-count nouns: (a) little, much, less Co-occur with both count and non-count: plenty of, a lot of, some, more Fails to distinguish between types of quantification… (5) He drank a little/a lot of coffee. (6) She bought a few/many/a lot of books. (7) She drove fewer miles than he did. (8) He explained many more problems on Friday than Thursday. Format also fails: QUANT + NOUN (a few books, less coffee, some tea) elided noun simply as QUANT (a few, less, some)

7. Scalar Quantification:Fuzziness Semantic account must offer consideration of indistinguishable boundaries Possible ranges of quantification for each scalar varies with context Creation of “fuzzy” quantifiers3 with weaker endpoints Aspect of natural language must be captured computationally Fig. 1. Fuzziness of quantification

8. Outline Scalar Quantification Need for semantic treatment Fuzziness of Membership Ontological Semantics Theory Introduction Treatment: Scales and Ranges Benefits Semantics Classification “Stationary” Scalars “Floating” Scalars Composition of Quantifiers Implementation in NLP

9. Ontological Semantics:Theory Introduction Semantic-based computational technology4,5 input of text output of text-meaning representation emulation of human understanding Ontology hierarchical relationships of concepts language independent Lexicon lexical items language dependent Formalization of semantic behavior ex. fuzziness of scalar quantification

10. Ontological Semantics: Treatment Relationship of variable ranges of quantification shown on a given scale Scale endpoint of 0 as the minimum (no/none) endpoint of 1 as the maximum (all) relative to objects being quantified Compare scales of quantifying “books required for class” and “cars in the parking lot”… Each quantifier is shown as having its own respective range on a determined scale Fig. 2. Base quantification scale

11. Ontological Semantics: Benefits Captures comparative relationship of quantifiers a little as less than a lot allows for combinations of quantification Reassessment of scale for contextual variance Allows for division of two classes based on semantic behavior Stationary scalars Floating scalars

12. Outline Scalar Quantification Need for semantic treatment Fuzziness of Membership Ontological Semantics Theory Introduction Treatment: Scales and Ranges Benefits Semantics Classification “Stationary” Scalars “Floating” Scalars Composition of Quantifiers Implementation in NLP

16. Extension of ranges with a property of relaxable rangesFig. 4. Relaxable range of some The ranges are given relaxed values for their weaker, fuzzy endpoints The support interval now becomes [0.2, 0.7]

18. Classification: Floating Scalars less, more Provide quantification along a flexible range that may be moved along a given scale A floating range may quantify any value on a scale The scale is determined by comparison to a known amount, A Therefore, quantification has no definite range and is entirely relative to A Fig. 6. Scale of Floating Scalars

19. Classification: Floating Scalars less snow than yesterday Scale of less is determined relative to A—how much snow fell yesterday A new scale is created from A downward Any value along this new scale may be quantified by less Fig. 7. Quantification of less

20. Classification: Floating Scalars more snow than yesterday Relative to how much snow fell yesterday—A A new scale is created from A upward Any value along this new scale may be quantified by more Fig. 8. Quantification of more

21. Composition of Quantifiers Scalar quantifiers may be stacked—much less or more than a few Creates a composition of two ranges on a single scale—one quantifier modifying the range of another for a more specific value Either scalar class may act on the range of its own member or of the other class: Stationary acting on stationary, i.e., a few of the many students Stationary acting on floating, i.e., much less Floating acting on stationary, i.e., more than a few Floating acting on a floating, i.e., more than just exceeding an expectation

22. Outline Scalar Quantification Need for semantic treatment Fuzziness of Membership Ontological Semantics Theory Introduction Treatment: Scales and Ranges Benefits Semantics Classification “Stationary” Scalars “Floating” Scalars Composition of Quantifiers Implementation in NLP

23. Implementation in NLP What does this provide us? Flexible ranges on an adjustable scale captures semantic behavior of natural language quantification Provides room for applications in computational semantic reasoning with fuzzy measurements Calculable properties (height, length) Relative properties (efficiency, intelligence, beauty) Proper text-meaning representation of: “Count people at their usual residence, which is the place where they live and sleep most of the time”

24. References 1 U.S. Census Bureau, Population Division. Residence Rule and Residence Situations for the 2010 Census. 14 April, 2010. http://www.census.gov/population/www/cen2010/resid_rules/resid_rules.html 2 Quirk, R., Greenbaum, S., Leech, G., and Svartvik, J. 1985. A Comprehensive Grammar of the English Language. 3Zadeh, L. 1976. A fuzzy-algorithmic approach to the definition of complex or imprecise concepts. International Journal of Man-Machine Studies, 8, 249-291. 4Nirenburg, S. & Raskin, V. 2004. Ontological Semantics. Cambridge: MIT Press. 5Raskin, V. and Taylor, J. M. The (Not So) Unbearable Fuzziness of Natural Language: The Ontological Semantic Way of Computing with Words. Proceedings of the 28th North American Fuzzy Information Processing Society Annual Conference.

25. Questions? Whitney Vandiver Linguistics Program Purdue University wvandive@purdue.edu