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

Joint Franco-Indian Workshop Goal 1 : to compute sentence meaning similar to Goal 2 : to extract a program from a proof 19/12/2003 Joint Franco-Indian Workshop

Joint Franco-Indian Workshop example We wish to prove: (A  (B  C))  ((A  B)  (A  C)) We have two rules:  |-- (A  B)  |-- A  |-- B , A |-- B  |-- (A  B) 19/12/2003 Joint Franco-Indian Workshop

Joint Franco-Indian Workshop 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. 19/12/2003 Joint Franco-Indian Workshop

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 19/12/2003 Joint Franco-Indian Workshop

Joint Franco-Indian Workshop 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) 19/12/2003 Joint Franco-Indian Workshop

Joint Franco-Indian Workshop |-- 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) 19/12/2003 Joint Franco-Indian Workshop

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 19/12/2003 Joint Franco-Indian Workshop

Joint Franco-Indian Workshop 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 19/12/2003 Joint Franco-Indian Workshop

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)] 19/12/2003 Joint Franco-Indian Workshop

Joint Franco-Indian Workshop 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)) 19/12/2003 Joint Franco-Indian Workshop

Joint Franco-Indian Workshop 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))   19/12/2003 Joint Franco-Indian Workshop

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) 19/12/2003 Joint Franco-Indian Workshop

Joint Franco-Indian Workshop new modus ponens  |-- (A  B)  |-- A , |-- B {A1, A2, …, An} |-- Ak no longer true axiom instead: A |-- A 19/12/2003 Joint Franco-Indian Workshop

Joint Franco-Indian Workshop two arrows , A |-- B A,  |-- B  |-- B/A  |-- A\B  |-- A\B  |-- A ,  |-- B  |-- B/A  |-- A , |-- B 19/12/2003 Joint Franco-Indian Workshop

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 19/12/2003 Joint Franco-Indian Workshop

Joint Franco-Indian Workshop 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 19/12/2003 Joint Franco-Indian Workshop

Joint Franco-Indian Workshop problems Non peripheral extraction? the book that John gave _ to Mary Constituency? Non Associative Lambek calculus Structural modalities (Moortgat, 1997) 19/12/2003 Joint Franco-Indian Workshop