CATEGORIAL GRAMMARS AND THE NATURAL LANGUAGE PROCESSING Ismaïl Biskri Mathematics and Computer-Science Department University of Quebec in Trois-Rivières

HUSSERL (1913) 4 Philosophical Origins. 4 Notions of : –Categorem –Syncategorem 4 Example : –Noun : Categorem –Sentence : Categorem –Verb : Syncategorem

LESNEWSKI (1922) 4 Logical foundations. 4 Two kind of expressions : –Noun –Proposition 4 Noun : objects, class of objects. 4 Proposition : statement (describing a “state”).

LESNEWSKI (1922) 4 Nouns and Propositions are Categorems. 4 Other expressions are Syncategorems. 4 Syncategorem acts like an operator. 4 Categorem acts like an operand.

LESNEWSKI (1922) Inferential System 4 We assume that we have a set of basic types 4 The set of all types is defined recursively as follows: –Basic types are types ; –If x and y are types then Fxy is a type. (F is an applicative operator ; F is applied to an expression of type x, it yields an other expression of type y)

AJDUKIEWICZ (1935) 4 Basic expressions (categories) : –Noun (N) –Sentences (S) 4 If x and y are categories then is a category 4 Reduction rules – y  x y x y x y x

AJDUKIEWICZ (1935) Example John laughs N S N S

BAR-HILLEL (1953) 4 Basic expressions (categories) : –Noun (N) –Sentences (S) 4 If x and y are categories then x/y and x\y are categories 4 Reduction rules : –x/y y  x –y y\x  x

BAR-HILLEL (1953) Example John admires Mary N(N\S)/NN N\S S

LAMBEK (1958, 1962) 4 Lambek Calculus. 4 We will use Steedman’s notation –X/Y will be X/Y –Y\X will be X\Y 4 Many axioms 4 Many inference rules 4 Many theorems

LAMBEK (1958, 1962) Axioms 4 X  X(reflexivity) 4 (X – Y) – Z  X – (Y – Z)(associativity) 4 X – (Y – Z)  (X – Y) – Z (associativity)

LAMBEK (1958, 1962) Inference rules 4 If X  Y and Y  Z then X  Z(transitivity) 4 If X – Y  Z then X  Z/Y 4 If X – Y  Z then Y  Z\X 4 If X  Z/Ythen X – Y  Z 4 If Y  Z\Xthen X – Y  Z

LAMBEK (1958, 1962) Some Theorems 4 X  (X – Y)/Y 4 (Z/Y) – Y  Z 4 Y  Z\(Z/Y) 4 (Z/Y) – (Y/X)  Z/X 4 Z/Y  (Z/X)/(Y/X) 4 (Y\X)/Z  (Y/Z)\X

ADES, STEEDMAN (1982) 4 Combinatory Categorial Grammar 4 Two concepts : –Syntactic category –Semantic category 4 Example : the category of admires is (S : admire' np2 np1\NP : np1)/NP : np2

ADES, STEEDMAN (1982) Some rules 4 Functional application (>) : –X/Y : f – Y : y  X : f y 4 Functional composition (>B) : –X/Y : f – Y/Z : g  X/Z : z(f(gz) 4 Type Raising (>T) : –X : x  Y/(Y\X) : f(fx)

ADES, STEEDMAN (1982, 1989)) Example John-loves-Mary N: John‘(S:loves‘ np2 np1\NP: np1)/NP: np2NP: Mary' >T S : pred John'/(S: pred John'\NP: John') >B S: loves' np2 John'/NP: np > S: loves' Mary' John'

BISKRI, DESCLES (1995, 1997) 4 Applicative Combinatory Categorial Grammar. 4 Canonical association between Combinatory Categorial rules and Combinators of Combinatory Logic (Curry, Feys, 1958). 4 Combinatory Categorial rules : syntactic parsing. 4 Combinatory Logic : functional semantic parsing

BISKRI, DESCLES (1995, 1997) Combinatory Logic 4 Combinators : B, C, C *, S, etc. 4 Beta-Reduction rules : B f g x  f (g x) ; C * x f  f x 4 Combinatory expression Normal Form –B C * x y z tis not in normal form –B C * x is in normal form –x (y z)is in normal form

BISKRI, DESCLES (1995, 1997) Some rules 4 Functional application (>) : –X/Y : f – Y : y  X : f y 4 Functional composition (>B) : –X/Y : f – Y/Z : g  X/Z : B f g 4 Type Raising (>T) : –X : x  Y/(Y\X) : C * x

BISKRI, DESCLES (1995, 1997) Example 1 1 [N:John]-[(S\N)/N:loves]-[N:Mary] Typed concatenated structure 2[S/(S\N):(C* John)]-[(S\N)/N:loves]-[N:Mary](>T) 3 [S/N:(B (C* John) loves)]-[N:Mary](>B) 4 [S:((B (C* John) loves) Mary)](>) Typed applicative structure 5[S : ((B (C* John) loves) Mary)] 6[S : ((C* John) (loves Mary))](B) 7[S : ((loves Mary) John)](C * )

BISKRI, DELISLE (2000) Example 2 : [N/N:la]-[N:liberté]-[(S\N)/N:renforce]-[N/N:la]-[N:démocratie] 2. [N:(la-liberté)]-[(S\N)/N:renforce]-[N/N:la]-[N:démocratie](>) 3. [S/(S\N):(C * (la liberté))]-[(S\N)/N: renforce]-[N/N: la]-[N: démocratie](>T) 4. [S/N : (B (C * (la liberté)) renforce)]-[N/N : la]-[N : démocratie](>B) 5. [S/N : (B (B (C * (la liberté)) renforce) la)]-[N : démocratie](>B) 6. [S : ((B (B (C * (la liberté)) renforce) la) démocratie)](>) 7. [S : ((B (B (C * (la liberté)) renforce) la) démocratie)] 8. [S : ((B (C * (la liberté)) renforce) (la démocratie))]B 9. [S : ((C * (la liberté)) (renforce (la démocratie)))]B 10. [S : ((renforce (la démocratie)) (la liberté)))] C* 11. [S : renforce (la démocratie) (la liberté)]

BISKRI, DELISLE (2000) Example 3 1. [(S/N1)/N2:thoudaiimou]-[N1:elhouriyathou]-[N2:eddimouqratiyatha] 2. [(S/N1)/N2:thoudaiimou]-[S\(S/N1):(C*elhouriyathou)]- [N2:eddimouqratiyatha](<T) 3. [S/N2 : (B (C * elhouriyathou) thoudaiimou)]-[N2 : eddimouqratiyatha] (<Bx) 4. [S : ((B (C * elhouriyathou) thoudaiimou) eddimouqratiyatha)](>) 5. [S : ((B (C * elhouriyathou) thoudaiimou) eddimouqratiyatha)] 6. [S : ((C * elhouriyathou) (thoudaiimou eddimouqratiyatha))]B 7. [S : ((thoudaiimou eddimouqratiyatha) elhouriyathou)]C * 8. [S : thoudaiimou eddimouqratiyatha elhouriyathou]

BISKRI, DESCLES (1995) The Backward Modifier : Example 4 1[N : John]-[(S\N)/N : loves]-[N : Mary]-[(S\N)\(S\N) : madly] … 4[S : ((B (C* John) loves) Mary)]-[(S\N)\(S\N) : madly] 5[S : ((C* John) (loves Mary))]-[(S\N)\(S\N) : madly](B) 6[S/(S\N) : (C* John)]-[S\N : (loves Mary)]-[(S\N)\(S\N) : madly] (>dec) 7[S/(S\N) : (C* John)]-[S\N : (madly (loves Mary))](<) 8[S : ((C* John) (madly (loves Mary)))](>) 9[S : ((C* John) (madly (loves Mary)))] 10[S : ((madly (loves Mary)) John)](C*)

BISKRI, DESCLES (1995) Coordination a) Two segments of the same kind, with the same structure and contiguous to AND : [John loves]S/N and [William hates]S/N these pictures b) Two segments into an elliptic construction : John loves [Mary madly] and [Jenny wildly] [John] loves [Mary] and [William Jenny] c) Two segments of different structures : Mary walks [slowly] and [with happiness]. John [sings] and [plays the violin]. d) Two segments without distributivity : The flag is [white] and [red] (≠ The flag is white and the flag is red).

BISKRI, DESCLES (1995) Example 5 1[N:John]-[(S\N)/N:loves]-[N:Mary]-[CONJD:and]-[(S\N)/N:hates]- [N:Jenny]... 4[S:((B (C* John) loves) Mary)]-[CONJD:and]-[(S\N)/N:hates]-[N:Jenny] 5[S:((B (C* John) loves) Mary)]-[CONJD:and]-[S\N:(hates Jenny)] (>) 6[S:((C* John) (loves Mary))]-[CONJD:and]-[S\N:(hates Jenny)] (B) 7[S/(S\N):(C* John)]-[S\N:(loves Mary)]-[CONJD:and]-[S\N:(hates Jenny)] (>dec) 8[S/(S\N):(C* John)]-[S\N:(  and (loves Mary) (hates Jenny))] ( ) 9 [S:((C* John) (  and (loves Mary) (hates Jenny)))] (>) 10[S : ((C* John) (  and (loves Mary) (hates Jenny)))] 11[S : ((  and (loves Mary) (hates Jenny)) John)](C*) 12[S : (and ((loves Mary) John) ((hates Jenny) John))](  )