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Geometric models of meaning and compositionality

Geometric models of meaning and compositionality. Reinhard Blutner http://www.blutner.de blutner@uva.nl. Institute for Logic, Language and Computation. 0 Introduction. Perception: direct interpretat-ion of sensory input automatic, unreflective, instinctive.

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Geometric models of meaning and compositionality

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  1. Geometric models of meaning and compositionality Reinhard Blutner http://www.blutner.de blutner@uva.nl Institute for Logic,Language and Computation

  2. 0 Introduction Perception: direct interpretat-ion of sensory input • automatic, unreflective, instinctive. • normally, it is not based on some kind of reasoning. NL Comprehension: • semantic interpretation: automatic etc. • pragmatic interpretation: indirect, reflective, normally based on reasoning (inferentialism)

  3. Thesis NL interpretation (semantic + pragmatic) is basically as direct as perception. • Most current models of NL interpretation are wrong Therefore, look for alternative models. Framework of geometric representations of meaning

  4. Outlook 1 Millikan’s thesis of direct interpretation 2 Modern contextualism 3 Context-sensitivity 4 Geometric models of meaning 5 Adjectival modification 6 Compositionality for geometric models

  5. Tandem Workshop Berlin, December 11-13, 2010 Millikan’s thesis of direct interpretation • DPL-thesis • General arguments • Experimental evidence

  6. Tandem Workshop Berlin, December 11-13, 2010 Direct Perception through Language • We directly perceive a red chair here. We automatically derive that without deliberation about the reliability of thesources (reflected light). • DPL: NL interpretation is as direct as perception • Derivation of literal meaning does not proceed by conscious inference • The content of the heard utterances integrates automatically our „belief boxes“

  7. Tandem Workshop Berlin, December 11-13, 2010 Descartes vs. Spinoza • Can people comprehend assertions without believing them? • Descartes suggested that people can and should • Spinoza suggested that people should but cannot. • Burge (1993): We may invoke (conscious) justification for not believing the content of some utterance. The default position, however, is to accept such contents as true. • Evolutionary arguments (Axelrod, Millikan, Burge)

  8. Tandem Workshop Berlin, December 11-13, 2010 Gilbert’s (1993) experiment • Subjects read a crime report that contained both true and false statements. The color of critical parts of the text indicated whether a particular statement was true or false. • Some subjects performed a concurrent digit-search task as they read the critical statements [interrupted], and others did not [uninterrupted]. • Finally, subjects completed a recognition memory test for the critical sentences contained in the report.

  9. Tandem Workshop Berlin, December 11-13, 2010 Results • In the recognition test, subjects responded correctly above the change level • The participants of the interrupted group reported false information as true but not true information as false • This indicates that Spinoza is right: people automatically take presented information as true; it takes conscious attention not believing it. true / falsestatements recognized as

  10. Resumé Tandem Workshop Berlin, December 11-13, 2010 | 10 • The assumption that NL interpretation is as direct as perception (DPL-thesis) has important consequences for constructing psychologically adequate models of interpretation • I propose to take DPL as a serious challenge for computational models of NL interpretation • DPL − properly generalized − asks for a default mode of NL interpretation which is fully compositional and runs automatically.

  11. Tandem Workshop Berlin, December 11-13, 2010 Modern contextualism • The neo/post-Gricean picture of contextualism • Classification scheme • The silent assumptions

  12. The neo-/post-Gricean picture: Contextualism Tandem Workshop Berlin, December 11-13, 2010 • Using the meanings of the words plus the syntactic structure of the sentence, it is not possible to calculate the literal meaning of the sentence. Some kind of underdetermined representation can be computed only. • Semantic underdetermination and the existence of unarticulated constituents are postulated. • The mechanism of pragmatic enrichment is crucial both for determining what the speaker says and what she means.

  13. Neo-Gricean Theories (Horn, Atlas) OT-Pragmatics Relevance Theory Presumptive Meanings Tandem Workshop Berlin, December 11-13, 2010 Variants of contextualism

  14. Tandem Workshop Berlin, December 11-13, 2010 Silent assumptions • Language of thought hypothesis • Cognitive activities require a language-like representational medium • Rule-governed processes operate on representations (inferences) • Symbolic representations have a combinatorial syntax and semantics • Propositions form a Boolean lattice; Kolmogorov probabilities • This contrasts with connectionist and geometric models of semantic interpretation

  15. Disjunction puzzle Tandem Workshop Berlin, December 11-13, 2010 | 15 (A & C)  (A &C)  A (distributivity) A | C : p A | C : q A | (C  C) : between p … q Tversky and Shafir (1992) show that significantly more students report they would purchase a nonrefundable Hawaiian vacation if they were to know that they have passed or failed an important exam than report they would purchase if they were not to know the outcome of the exam.

  16. Interference (Franco, Khrennikov,…) Tandem Workshop Berlin, December 11-13, 2010 | 16 • In QM the superposition of two or more states can lead to interference effects • Classical: P(A/CC’ ) = ½ P(A|C) + ½ P(A|C’ ) ,if P(C)/P(CC’ ) = P(C’ )/P(CC’ ) = ½ • (A|C+C’) = ½|A|C+C’|2= ½ |A|C|2 + ½ |A|C’|2 + |A|C||A|C’|  cos()= ½ (A|C) + ½ (A|C’ )+ interference term • A disjunction effect of -.16 (Tversky and Shafir 1992) can be described by cos  = -0.35

  17. The holistic character of decisions Tandem Workshop Berlin, December 11-13, 2010 | 17 • Decision processes are not “rational” in the sense of rational decision theory (checking different cases and calculating expected utilities) • The decision processes are led by simple but powerful global heuristics, which act in a fast and frugal way (Gigerenzer) • Models of bounded rationality can be formulated in agreement with a weak form of rationalism.

  18. Resumé Tandem Workshop Berlin, December 11-13, 2010 | 18 • Most variants of modern contextualism ignore the challenge of the DPL-thesis • Issue of performance (automatic/controlled): Inferences are controlled processes • Issue of competence: The proposed models cannot handle puzzles of bounded rationality.

  19. Tandem Workshop Berlin, December 11-13, 2010 3 Context-sensitivity • Some examples • Why a ‘tall boy’ is a big problem • The problem with ‘absolute’ adjectives • How to calculate truth-conditions

  20. Tandem Workshop Berlin, December 11-13, 2010 Some examples 1 • John ate breakfast [this morning; in the normal way] • Every boy [in the class] is seated • Peter began a novel [ to read/ to write] • I‘m parking outside [my car] • Max is tall [for a fifth grader]

  21. Tandem Workshop Berlin, December 11-13, 2010 Some examples 2 • What color is a red nose, red flag, red bean? • This apple is red [on the outside]

  22. Tandem Workshop Berlin, December 11-13, 2010 More examples • Quine (1960) was the first who noted the contrast between red apple (red on the outside) and pink grapefruit (pink on the inside). • In a similar vein, Lahav (1993) argues that an adjective such as brown doesn’t make a simple and fixed contribution to any composite expression in which it appears: In order for a cow to be brown most of its body’s surface should be brown, though not its udders, eyes, or internal organs. A brown crystal, on the other hand, needs to be brown both inside and outside. A brown book is brown if its cover, but not necessarily its inner pages, are mostly brown, while a newspaper is brown only if all its pages are brown. For a potato to be brown it needs to be brown only outside, ... (Lahav 1993: 76).

  23. Why a ‘tall boy’ is still a problem Quasi-deictic elements tall boy x [tall*(x,N) & boy(x)] (Sag, Bartsch, Bosch) Minimal proposition tall boy x N[tall*(x,N) & boy(x)] (Capellen & Lapore, Hobbs)) Underdetermination tall boy x N[tall*(x,N) & boy(x)] (Alshawi, Pinkal) * with tall(x,N)  size(x) > N Tandem Workshop Berlin, December 11-13, 2010

  24. The problem with ‘absolute’ adjectives Tandem Workshop Berlin, December 11-13, 2010 • red apple x [part(Y,x) & red(Y) & apple(x)] • Requires rather clumsy lexical entries • How much of the peel of an apple has to be red in order to call it a red peel? • This theory does not really clarify how the border line between semantics and pragmatics is ever to be determined • The best what the symbolic tradition suggests is somewhat like Montague‘s adnominal functors. red apple  x [(red(apple))(x)]

  25. Tandem Workshop Berlin, December 11-13, 2010 A theory based on adnominal functors • Montague (1970) as starting-point: adjectives as adnominal functors (worst case) • red(X) means roughly the property • (a) of having a red inner volume if X denotes fruits only the inside of which is edible • (b) of having a red surface if X denotes fruits with edible outside • (c) of having a functional part that is red if X denotes tools, … • This approach is compositional but not systematic

  26. How to calculate truth conditions? The mechanism of adnominal functors requires idiosyncratic lexical entries for fixing the interpretations of complex expressions. Alternative suggestions from Cognitive Linguistics Blending theory* (Fouconnier & Turner) Modulation (Recanati) What is the computational mechanism? A lovely notation does not yet provide a real mechanism. * In blending theory the part of a concept for which a given modification is relevant is referred to as an ‘active zone’, first discussed as such in Langacker (1991). In the case of an apple, the color is only relevant for the skin of the apple, which is its active zone. Tandem Workshop Berlin, December 11-13, 2010

  27. Resumé Tandem Workshop Berlin, December 11-13, 2010 | 27 • Compositional semantics based on standard symbolic models cannot account for systematicity • Alternative proposal: division of labor between underdetermined semantics and pragmatic theories of contextual enrichment • Inferential theories of contextual enrichment fail for reasons of explanatory adequacy.

  28. Tandem Workshop Berlin, December 11-13, 2010 4 Geometric models of meaning • Geometric models of meaning • Possible worlds and conceptual states • Comparing models of meaning • Prototype semantics

  29. Geometric Models of Meaning 1 Basic claim: An understanding of problem solving, categorization, memory retrieval, inductive reasoning, and other cognitive processes requires that we understand how humans assess similarity. W. S. Torgerson (1965): Multidimensional scaling of similarity. Psychometrika 30: 379–393. Tandem Workshop Berlin, December 11-13, 2010 | 29

  30. Tandem Workshop Berlin, December 11-13, 2010 Geometric Models of Meaning 2 • A. Tversky (1977): Features of similarity. Psychological Review 84: 327–352. • P. Gärdenfors: The Geometry of Thought (2000)Concepts as convex spaces • D. Widdows: Geometry and Meaning (2004)Distributional semantics

  31. Tandem Workshop Berlin, December 11-13, 2010 Possible worlds and conceptual states • Possible worlds: Isolated entities which are used for modeling propositions (sets of possible worlds) • Conceptual states: geometrical objects which form vector spaces. The addition of two vectors is an operation which describes the superposition of states interference phenomena

  32. Comparing Models of Meaning Tandem Workshop Berlin, December 11-13, 2010

  33. Tandem Workshop Berlin, December 11-13, 2010 Conceptual states: sim & prob • A conceptual state is a vector indicating how diagnostic each component (instance, feature) is for the whole state (frozen statistics) • sim: The scalar product of two normalized vectors is a measure for their similarity • prob: If designates a component, then gives the probability that the conceptual state triggers component . [Born rule]

  34. Resumé Tandem Workshop Berlin, December 11-13, 2010 | 34 • Geometric models of meaning can account for similarity ratings and ratings of probability and typicality • Geometric approaches to meaning have problems with handling compositionality.

  35. Tandem Workshop Berlin, December 11-13, 2010 5Adjectival modification • Three kinds of adjectives • Conjunction puzzle • The modification rule • Examples

  36. Three kinds of adjectives (based on B. Partee) Tandem Workshop Berlin, December 11-13, 2010 | 36 • Intersective • Adj’(Q)(x)  PAdj’(x) & Q(x) • Subsective • Adj’(Q)(x)  Q(x) • Privative • Adj’(Q)(x)  Q(x)

  37. Natural classification? (B. Partee) intersective subsective privative modal Tandem Workshop Berlin, December 11-13, 2010 | 37 blond, rectangular, French modal tall, good, typical, recent alleged, potential, arguable alleged, potential, arguable blond, rectangular, French spurious, imaginary, fake. spurious, imaginary, fake.

  38. The problem Tandem Workshop Berlin, December 11-13, 2010 | 38 • to give a uniform account for each of the big syntactic classes • to explain the pragmatic differences within the classes • to account for both • graded membership function • typicality function • to explain the puzzles of typicality • to conform the DPL thesis.

  39. Conjunction puzzle Tandem Workshop Berlin, December 11-13, 2010 | 39 Let (a) be a particular apple. There is no doubt that it is psychologically less proto-typical of an apple (whose prototype looks more like (b)) than of an apple-with-stripes; hence cstriped apple (a) > capple (a) CE = cstriped apple (x)  capple (x) (Osherson & Smith 1981) (a) (b) (c)

  40. Typicality ratings Tandem Workshop Berlin, December 11-13, 2010 | 40 Smith & Osherson 1984: Conceptual combination with prototype concepts (11 point scale 0-10)

  41. The modification rule How does a vector modify a vector ? The answer depends on the nature of the vectors: A. Vectors as superpositions of instances B. Distributional semantics. Vectors as document-based word-vectors (Schütze) Many proposals in the literature: Aerts, Zadeh, Plate, Smolensky, … Tandem Workshop Berlin, December 11-13, 2010 | 41

  42. A. Prototypes as superposed instances Tandem Workshop Berlin, December 11-13, 2010 | 42 • Even if the prototype is not one of the presented instan-ces it is recognized as such. • Modification rule + recalibrating to unit length

  43. B. Distributional Semantics Document 1 is about music instruments, document 2 about fishermen, and document 3 about financial institutions (applying LAS as pre-processing) Modification rule*: * See Mitchell & Lapata (2008): Vector-based models of semantic composition. (Using circular convolution) Tandem Workshop Berlin, December 11-13, 2010 | 43

  44. Modification: Tandem Workshop Berlin, December 11-13, 2010 • Build the tensor product • Apply a linear operator for reducing the dimension by 1, e.g. •  [ ] =

  45. Conjunction effect fuzzy logic Tandem Workshop Berlin, December 11-13, 2010 apple striped striped apple

  46. Tandem Workshop Berlin, December 11-13, 2010 Striped apple in 2D Apple striped Form Texture

  47. Red and White Beans Tandem Workshop Berlin, December 11-13, 2010 Color Distribution Red Beans Color Distribution Beans General Distribution Red

  48. Red and White Beans Tandem Workshop Berlin, December 11-13, 2010 General Distribution White Color Distribution White Beans Color Distribution Beans

  49. Tall Boy Tandem Workshop Berlin, December 11-13, 2010 tallboy tall boy

  50. Red apple: color of peel Tandem Workshop Berlin, December 11-13, 2010 redapple red apple Kullback-Leibler information = 0.25

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