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Proof-nets and semantic applications

Proof-nets and semantic applications. Alain Lecomte ESSLLI2002. e+. t-. e-. t+. child. Semantic proof nets. child. x:e, child: e t |- child(x) : t hence : child: et |- x.child(x):e t. run :. e+. t-. x. e-. t+.  x. child. run :. e+. t-. x. e-. t+.  x. child.

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Proof-nets and semantic applications

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  1. Proof-nets and semantic applications Alain Lecomte ESSLLI2002

  2. e+ t- e- t+ child Semantic proof nets • child x:e, child: et |- child(x) : t hence : child: et |- x.child(x):et

  3. run : e+ t- x e- t+ x child

  4. run : e+ t- x e- t+ x child

  5. run : e+ t- x e- t+ x child

  6. run : e+ t- x x e- t+ x child x.child(x)

  7. run : e+ t- x x e- t+ x child x.child(x)

  8. each, every… • A determiner like every, each… decomposes into : • A quantifier, for instance :  type : (et)t • A connective, for instance : type : t(tt)

  9. needs two predicates (e  t) for obtaining one proposition (t) • A determiner is therefore of type (et)((et)t)

  10. A determiner is therefore associated with a sequent: • Its « semantic » is represented by its proof

  11. C deduction

  12. remark • With a very remarkable step : an application of the contraction rule! •  necessity of working inside Intuitionistic linear logic with exponentials • The exact sequent which encodes the determiner is : !e !e t ( t t ), ( !e t ) t ( t ) (( t ) t ) |-- --o --o --o --o --o --o --o --o

  13. Exponentials

  14. Exponentials (one-sided)

  15. Representation of the proof c   (!e –o t) –o ((!e –o t) –o t)

  16. every child c child (!e –o t) –o t  

  17. every child likes to play c likes to play t child  

  18. Application - + + A A –o B +

  19. Application A - B +

  20. A + Abstraction B -

  21. A + Abstraction B - B –o A +

  22. Syntactic proof-nets • Proof-nets for Lambek calculus • Like PN for MILL + • condition on semi-planarity

  23. every child plays 1) unfolding s+ np - s - np\s + np + (s/(np\s)) - n + s - n - child np\s - plays s + (s/(np\s))/n - every

  24. every child plays 2) links s+ np - s - np\s + np + (s/(np\s)) - n + s - n - child np\s - plays s + (s/(np\s))/n - every

  25. Attention! 2) links WRONG ! s+ np - s - np\s + np + (s/(np\s)) - n + s - n - child np\s - plays s + (s/(np\s))/n - every

  26. Parsing • through homomorphism • H(s) = t • H(np) = !e • H(n) = !e –o t • H(A/B) = H(B\A) = H(B) –o H(A)

  27. every child plays s+ np - s - np\s + np + (s/(np\s)) - n + s - n - child np\s - plays s + (s/(np\s))/n - every

  28. every child plays 3) homomorphism t !e t !e –o t + !e (!e –o t) –o t !e –o t t !e –o t child !e –o t plays t+ (!e –o t) –o ((!e –o t) –ot)) every

  29. semantic recipes • child : x.child(x) • every : P.Q.(x.(P(x)Q(x)) • plays : x.play(x)

  30. d e+ t- !e- t+ child • represented by proof-nets :

  31. e+ t- !e- t+ plays • represented by proof-nets : d

  32. every c   (!e –o t) –o ((!e –o t) –o t)

  33. plugging lexical semantic types to the homomorphic PN by cut

  34. d e+ t- !e- t+ child t !e t !e –o t + !e (!e –o t) –o t !e –o t t !e –o t child !e –o t plays t+ (!e –o t) –o ((!e –o t) –ot)) every CUT

  35. t !e t !e –o t + !e t !e (!e –o t) –o t t !e –o t plays t+ (!e –o t) –o ((!e –o t) –ot)) every d child

  36. d e+ t- !e- t+ plays t !e t !e –o t + !e t !e (!e –o t) –o t t !e –o t plays t+ (!e –o t) –o ((!e –o t) –ot)) every d child CUT

  37. t !e d t !e –o t + !e t !e (!e –o t) –o t t plays t+ (!e –o t) –o ((!e –o t) –ot)) every d child

  38. PNevery t !e d t !e –o t + !e t !e (!e –o t) –o t t plays t+ (!e –o t) –o ((!e –o t) –ot)) every d child CUT

  39. d !e t c plays t+ d   child

  40. d !e t c plays t+ d   child

  41. d !e t c plays t+ d    child

  42. d !e t c plays x t+ d    child

  43. d !e t c plays x t+ d     child

  44. d !e t c plays x plays t+ d     child

  45. d !e t c plays x plays t+ d   child   child

  46. d child(x) !e t c plays x plays t+ d   child   child

  47. d plays(x) child(x) !e t c plays x plays t+ d   child (x.((child(x),plays(x))))   child

  48. Logical synthesis:from a formula to a sentence • the reverse story: • Start : • a semantic formula  • + semantic recipes for lexical entries 1, 2, …n • Goal: • A sentence using all these recipes the meaning of which is 

  49. Usual solutions:-term unification ? s:kiss(p,m) Peter : np : p kisses : (np\s)/np: x.y.kiss(y,x) Mary : np : m GOAL

  50. np+ s-  kiss(,) np+ y.kiss(y, )  ? s:kiss(p,m) Peter : np- : p kisses : (np\s)/np: x.y.kiss(y,x) Mary : np- : m

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