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Semantic Proof Nets and Their Applications in Intuitionistic Linear Logic

This paper presents a detailed exploration of semantic proof nets and their application in the framework of Intuitionistic Linear Logic. We examine the role of determiners like "every" and "each" as linked to quantifiers and connectives, elaborating on their semantic representations through proof. Notably, we apply the contraction rule to illustrate how complex semantic structures can be derived from simple proofs in logical systems. The work emphasizes the importance of syntactic proof nets and their role in parsing linguistic structures, drawing connections between proof theory and semantic representation.

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Semantic Proof Nets and Their Applications in Intuitionistic Linear Logic

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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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