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Copredication in Homotopy Type
Theory
Hamidreza Bahramian1
Indiana University Bloomington
Presentation based on a joint wo...
Introduction
Copredication:
the phenomenon where two or more predicates
with different requirements on their arguments
are...
History
— Asher(2011), nouns as “dot objects”(Pustejovsky,1995)
◦ i.e., objects that can be viewed under different “aspec...
Montague semantics
Montague semantics
Montague semantics
In Luo’s framework
In Luo’s framework
In Luo’s framework
In Luo’s framework
In Luo’s framework
In Luo’s framework
In Luo’s framework
In Luo’s framework
In Luo’s framework
The problem
The problem
Homotopy Type Theory
— intensional version of Martin-Löf’s type theory [ML75]
— a proof-relevant interpretation of equal...
Homotopy Type Theory
Homotopy Theory
Homotopy Theory
Homotopy Theory
Homotopy Theory
Homotopy Theory
Homotopy Theory
Johnson,Chris. "   Path Homotopy Animation."Youtube.Youtube, 11 May 2009.Web.
9 July 2016.
Homotopy Theory
Homotopy Theory
Rowland,Todd. "Homotopic." From MathWorld--AWolframWeb Resource,created by
Eric W.Weisstein. http://mathwo...
Homotopy Theory
Homotopy Theory
CN as identifications
CN as identifications
CN as identifications
CN as identifications
Sameness
CN as identifications
Sameness Unlikeness
CN as identifications
CN as identifications
TreePT TreePT =TreePT
CN as identifications
TreePT =TreePT
TreePT TreePT
CN as identifications
TreePT =TreePT
TreePT TreePT
CN as identifications
CN as identifications
CN as identifications
CN as identifications
CN as identifications
Heim and Kratzer’s framework
Heim and Kratzer’s framework
Heim and Kratzer’s framework
Heim and Kratzer’s framework
Heim and Kratzer’s framework
Heim and Kratzer’s framework
Heim and Kratzer’s framework
Heim and Kratzer’s framework
Heim and Kratzer’s framework
Heim and Kratzer’s framework
Semantic computation using HoTT
logic
Future work
— Formal semantics of natural languages
◦ “possible world”?
— HoTT
◦ subtyping?
— Homotopy
◦ Prefiguration ...
References
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Copredication in Homotopy Type Theory: A homotopical approach to formal semantics of natural language

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This paper applies homotopy type theory to formal semantics of natural languages and proposes a new model for the linguistic phenomenon of copredication. Copredication refers to sentences where two predicates which assume different requirements for their arguments are asserted for one single entity, e.g., "the lunch was delicious but took forever". This paper is particularly concerned with copredication sentences with quantification, i.e., cases where the two predicates impose distinct criteria of quantification and individuation, e.g., "Fred picked up and mastered three books." In our solution developed in homotopy type theory and using the rule of existential closure following Heim analysis of indefinites, common nouns are modeled as identifications of their aspects using HoTT identity types, e.g., the common noun book is modeled as identifications of its physical and informational aspects. The previous treatments of copredication in systems of semantics which are based on simple type theory and dependent type theories make the correct predictions but at the expense of ad hoc extensions (e.g., partial functions, dot types and coercive subtyping). The model proposed here, also predicts the correct results but using a conceptually simpler foundation and no ad hoc extensions. The proofs in the proposal have been formalized using Agda.

Copredication in homotopy type theory (PDF Download Available). Available from: https://www.researchgate.net/publication/320704539_Copredication_in_homotopy_type_theory [accessed Oct 30 2017].

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Copredication in Homotopy Type Theory: A homotopical approach to formal semantics of natural language

  1. 1. Copredication in Homotopy Type Theory Hamidreza Bahramian1 Indiana University Bloomington Presentation based on a joint work with Narges Nematollahi2 and Amr Sabry3 1 Department of Computer Science 2 Department of Linguistics 3 Department of Computer Science
  2. 2. Introduction Copredication: the phenomenon where two or more predicates with different requirements on their arguments are applied to a single argument a) The lunch was delicious but took forever. (Asher, 2011) b) The heavy book is easy to understand. (Gotham, 2014) c) John picked up and mastered three books. (Asher 2011)
  3. 3. History — Asher(2011), nouns as “dot objects”(Pustejovsky,1995) ◦ i.e., objects that can be viewed under different “aspects” — Cooper (2011) ◦ nouns as functions from “records” to “record types” — Luo (2012), Chatzikyriakidisand Luo (2012, 2013, 2015) ◦ common nouns as types ◦ dot types + coercive subtyping — Gotham(2014) ◦ `book' denotes the set of composite objects physical+informational ◦ criteria of individuation are combined during semantic composition
  4. 4. Montague semantics
  5. 5. Montague semantics
  6. 6. Montague semantics
  7. 7. In Luo’s framework
  8. 8. In Luo’s framework
  9. 9. In Luo’s framework
  10. 10. In Luo’s framework
  11. 11. In Luo’s framework
  12. 12. In Luo’s framework
  13. 13. In Luo’s framework
  14. 14. In Luo’s framework
  15. 15. In Luo’s framework
  16. 16. The problem
  17. 17. The problem
  18. 18. Homotopy Type Theory — intensional version of Martin-Löf’s type theory [ML75] — a proof-relevant interpretation of equality — propositions-as-types principle — a full cumulative hierarchy of universes — judgmental vs propositional equality ◦ “=” vs “ ”≡
  19. 19. Homotopy Type Theory
  20. 20. Homotopy Theory
  21. 21. Homotopy Theory
  22. 22. Homotopy Theory
  23. 23. Homotopy Theory
  24. 24. Homotopy Theory
  25. 25. Homotopy Theory Johnson,Chris. "  Path Homotopy Animation."Youtube.Youtube, 11 May 2009.Web. 9 July 2016.
  26. 26. Homotopy Theory
  27. 27. Homotopy Theory Rowland,Todd. "Homotopic." From MathWorld--AWolframWeb Resource,created by Eric W.Weisstein. http://mathworld.wolfram.com/Homotopic.html
  28. 28. Homotopy Theory
  29. 29. Homotopy Theory
  30. 30. CN as identifications
  31. 31. CN as identifications
  32. 32. CN as identifications
  33. 33. CN as identifications Sameness
  34. 34. CN as identifications Sameness Unlikeness
  35. 35. CN as identifications
  36. 36. CN as identifications TreePT TreePT =TreePT
  37. 37. CN as identifications TreePT =TreePT TreePT TreePT
  38. 38. CN as identifications TreePT =TreePT TreePT TreePT
  39. 39. CN as identifications
  40. 40. CN as identifications
  41. 41. CN as identifications
  42. 42. CN as identifications
  43. 43. CN as identifications
  44. 44. Heim and Kratzer’s framework
  45. 45. Heim and Kratzer’s framework
  46. 46. Heim and Kratzer’s framework
  47. 47. Heim and Kratzer’s framework
  48. 48. Heim and Kratzer’s framework
  49. 49. Heim and Kratzer’s framework
  50. 50. Heim and Kratzer’s framework
  51. 51. Heim and Kratzer’s framework
  52. 52. Heim and Kratzer’s framework
  53. 53. Heim and Kratzer’s framework
  54. 54. Semantic computation using HoTT logic
  55. 55. Future work — Formal semantics of natural languages ◦ “possible world”? — HoTT ◦ subtyping? — Homotopy ◦ Prefiguration as theorem?
  56. 56. References

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