Proofs%20and%20Meanings

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Proofs and Meanings

• Alain Lecomte
• INRIA-FUTURS (team SIGNES)
• CLIPS-IMAG (Grenoble)

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• Goal 1 to compute sentence meaning
• similar to
• Goal 2 to extract a program from a proof

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example

• We wish to prove
• (A ? (B ? C)) ? ((A ? B) ? (A ? C))
• We have two rules
• ? -- (A ? B) ? -- A
• ? -- B
• ?, A -- B
• ? -- (A ? B)

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proof

• The proof is the following (where ? (A ? (B
  ? C)), (A ? B), A )
• ?-- A ?-- A ? (B ? C) ? -- A ? -- (A ? B)
• ?-- (B ? C) ?-- B
• (A ? (B ? C)), (A ? B), A -- C
• (A ? (B ? C)), (A ? B) -- (A ? C)
• (A ? (B ? C)) -- ((A ? B) ? (A ? C))
• (A ? (B ? C)) ? ((A ? B) ? (A ? C))
• A proof is a tree made of successive
  appli-cations of inference rules. The roots, that
  are at its base, are the proved theorems.

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proofs as functions

• The previous proof transforms a proof of A ? (B
  ? C) into a proof of (A ? B) ? (A ? C),
• A proof of A ? B is a procedure to transform
  every proof of A into a proof of B
• If f A ? B and aA then f(a) is a proof of B
• ? -- f (A ? B) ? -- a A
• ? -- f(a) B

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• similarly
• If we get a proof of A ? B from a proof b of B
  and a hypothesis x A (by discharging it), then
  the proof of A ? B is a function ?x. b
• ?, x A -- b B
• ? -- ?x. b (A ? B)

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• ?-- z A ?-- x A ? (B ? C) ? -- z A ?
  -- y (A ? B)
• ?-- (x z) (B ? C) ?-- (y z) B
• x  (A ? (B ? C)),y (A ? B), z A -- ((x z)(y
  z)) C
• x  (A ? (B ? C)), y (A ? B) -- ?z. ((x z)(y
  z)) (A ? C)
• x  (A ? (B ? C))-- ?y. ?z.((x z)(y z)) ((A ?
  B) ?(A ? C))
• ?x.?y.?z.((x z)(y z)) (A ? (B ? C)) ?((A ? B) ?
  (A ? C))
• ?x.?y.?z.((x z)(y z)) combinator S
• Sabc ac(bc)

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what happens with language?

• program meaning
• ex John snores
• snores ? a function snore e ? t
• John ? an individual entity j of type e
• John snores ? a truth-value, snore(j) of type t

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the proof

• j e, snore e ? t -- j e j e, snore e ? t
  -- snore e ? t
• j e, snore e ? t -- snore(j) t

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modifier phrasethat John likes

• ?, xe -- xe ?, xe -- w e ? (e ? t)
• ?, xe -- (w x) (e ? t) ?, x e --
  v  e
• ?, x e -- ((w x) v) t
• ? -- ?x. ((w x) v) (e ? t)
• with ? v (John) e, w (likes) e ? (e ? t)

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that

• that ((e ? t) ? ((e ? t) ? (e ? t)))
• we get further
• ?-- ?x.((w x) v)(e?t) ?-- that((e ?t)?((e
  ?t)? (e? t)))
• ? -- (that ?x.((w x) v)) ((e ? t) ? (e ? t))

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word meanings

• John j likes ?s. ?t. ((like s) t)
• that ?P. ?Q. ?z. (P z) ? (Q z)
• We obtain as the whole meaning
• (?P. ?Q. ?z. (P z) ? (Q z)  ?x.(( ?s.?t.((like
  s) t) x) j)), which reduces to 
• (?P. ?Q. ?z. (P z) ? (Q z)  ?x.(( ?s.?t.((like
  s) t) x) j)) ?
• (?P. ?Q. ?z. (P z) ? (Q z)  ?x.(?t.((like x) t)
  j)) ?
• (?P. ?Q. ?z. (P z) ? (Q z)  ?x.((like x) j)) ?
• (?Q. ?z. (?x.((like x) j) z) ? (Q z) ) ?
• (?Q. ?z. ((like z) j) ? (Q z))  

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but words are not only meanings

• Order not free
• Peter likes Mary ? Mary likes Peter
• Resource sensitivity one occurrence of a word is
  used exactly once (words ? formulae)

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new modus ponens

• ? -- (A ? B) ? -- A
• ?, ?-- B
• A1, A2, , An -- Ak
• no longer true
• axiom instead
• A -- A

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two arrows

• ?, A -- B A, ? -- B
• ? -- B/A ? -- A\B
• ? -- A\B ? -- A
• ?, ? -- B
• ? -- B/A ? -- A
• ?, ?-- B

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labelled with strings

• ?, x A -- ux B xA, ? -- ux B
• ? -- u_B/A ? -- _u A\B
• ? -- b A\B ? -- a A
• ?, ? -- ab B
• ? -- b B/A ? -- a A
• ?, ?-- ba B

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example

• Lexicon that /that/ (n\n)/(s/np)
• John /john/ np
• likes /likes/ (np\s)/np
• /likes/(np\s)/np-- /likes/(np\s)/np
  unp-- unp
• /john/np -- /john/np /likes/(np\s)/np,
  unp -- /likes/unp\s
• /john/np, /likes/(np\s)/np, u np -- /john
  likes/u s
• /that/ (n\n)/(s/np) -- /that/
  (n\n)/(s/np) /john/np, /likes/(np\s)/np
  -- /john likes/_ s/np
• /that/(n\n)/(s/np), /john/np, /likes/(np\s)/np
  -- /that john likes/_ n\n

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problems

• Non peripheral extraction?
• the book that John gave _ to Mary
• Constituency?
• Non Associative Lambek calculus
• Structural modalities (Moortgat, 1997)