1. Alain Lecomte INRIA-FUTURS (team SIGNES) & CLIPS-IMAG (Grenoble)
Proofs and Meanings
Alain Lecomte
INRIA-FUTURS (team SIGNES)
&
CLIPS-IMAG (Grenoble)

2. Joint Franco-Indian Workshop
Goal 1 : to compute sentence meaning
similar to
Goal 2 : to extract a program from a proof
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Joint Franco-Indian Workshop

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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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5. Joint Franco-Indian Workshop
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 a:A 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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Joint Franco-Indian Workshop

8. 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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Joint Franco-Indian Workshop

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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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Joint Franco-Indian Workshop

10. modifier phrase that John likes
, x:e |-- x:e , x:e |-- w: e  (e  t)
, x:e |-- (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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Joint Franco-Indian Workshop

13. 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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16. labelled with strings…
, x: A |-- ux : B x:A,  |-- 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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Joint Franco-Indian Workshop

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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 u:np|-- u:np
/john/:np |-- /john/:np /likes/:(np\s)/np, u:np |-- /likes/u:np\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)
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Joint Franco-Indian Workshop