Computing the "Fuzziness" of Scalar Quantification in Ontological Semantics

Computing the “Fuzziness” of Scalar Quantificationin Ontological SemanticsWhitney VandiverPurdue UniversityApril 17, 2010Purdue Linguistics Association Symposium

2010 U.S. Census Residence RuleUsed 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 contextsPrimary 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

OutlineScalar QuantificationNeed for semantic treatmentFuzziness of MembershipOntological SemanticsTheory IntroductionTreatment: Scales and RangesBenefitsSemantics Classification“Stationary” Scalars“Floating” ScalarsComposition of QuantifiersImplementation in NLP

Scalar QuantificationComparison 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 quantificationsa little is less than some—some is less than a lotfew is less than some—some is less than manyWhen does a little become some? And some become a lot?Evaluations of quantificational ranges differ by individual situations

Scalar Quantification:Need for semantic treatmentSyntactic analysis2Co-occur only with plural count nouns:(a) few, several, many, fewerCo-occur only with non-count nouns: (a) little, much, less Co-occur with both count and non-count: plenty of, a lot of, some, moreFails 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)

Scalar Quantification:Fuzziness Semantic account must offer consideration of indistinguishable boundariesPossible ranges of quantification for each scalar varies with contextCreation of “fuzzy” quantifiers3 with weaker endpointsAspect of natural language must be captured computationallyFig. 1. Fuzziness of quantification

Ontological Semantics:Theory IntroductionSemantic-based computational technology4,5input of textoutput of text-meaning representationemulation of human understandingOntologyhierarchical relationships of conceptslanguage independentLexiconlexical itemslanguage dependentFormalization of semantic behaviorex. fuzziness of scalar quantification

Ontological Semantics: TreatmentRelationship of variable ranges of quantification shown on a given scaleScaleendpoint of 0 as the minimum (no/none)endpoint of 1 as the maximum (all)relative to objects being quantifiedCompare 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 scaleFig. 2. Base quantification scale

Ontological Semantics: BenefitsCaptures comparative relationship of quantifiersa little as less than a lotallows for combinations of quantificationReassessment of scale for contextual varianceAllows for division of two classes based on semantic behaviorStationary scalarsFloating scalars

Classification: Stationary Scalars(a) little, few, some, many, mostProvide quantification along a fixed range of a scaleFig. 3. Definite range of someDefinite range of some as the crisp set [0.3, 0.6]

Value of 1 is relative to the objects being quantifiedClassification: Stationary ScalarsMust account for overlap of possible quantificational rangesExtension of ranges with a property of relaxable rangesFig. 3. Definite range of some

Classification: Stationary ScalarsMust account for overlap of possible quantificational ranges

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

Classification: Stationary ScalarsRelaxable ranges allow for overlap of quantification(a) little (a) fewsomemany muchmostFig. 5. Overlap of relaxable ranges

Classification: Floating Scalarsless, moreProvide quantification along a flexible range that may be moved along a given scaleA floating range may quantify any value on a scaleThe scale is determined by comparison to a known amount, ATherefore, quantification has no definite range and is entirely relative to AFig. 6. Scale of Floating Scalars

Classification: Floating Scalarsless snow than yesterday Scale of less is determined relative to A—how much snow fell yesterdayA new scale is created from A downwardAny value along this new scale may be quantified by lessFig. 7. Quantification of less

Classification: Floating Scalarsmore snow than yesterdayRelative to how much snow fell yesterday—AA new scale is created from A upwardAny value along this new scale may be quantified by moreFig. 8. Quantification of more

Composition of QuantifiersScalar quantifiers may be stacked—much less or more than a fewCreates a composition of two ranges on a single scale—one quantifier modifying the range of another for a more specific valueEither 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 studentsStationary acting on floating, i.e., much lessFloating acting on stationary, i.e., more than a fewFloating acting on a floating, i.e., more than just exceeding an expectation

Implementation in NLPWhat does this provide us?Flexible ranges on an adjustable scale captures semantic behavior of natural language quantificationProvides room for applications in computational semantic reasoning with fuzzy measurementsCalculable 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”

References1 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.html2 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.

Computing the "Fuzziness" of Scalar Quantification in Ontological Semantics

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