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):(et) |-- 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