Categorial Grammar. Algebra, Proof Theory, Applications. References Categorial Grammar, G. Morrill, OUP, 2011.

Transcription

1 Categorial Grammar Algebra, Proof Theory, Applications References Categorial Grammar, G. Morrill, OUP, The Logic of Categorial Grammars, R. Moot & Ch. Retore, Springer, FoLLI, Mehrnoosh Sadrzadeh Queen Mary University of London

2 History Panini s grammar of Sanskrit, 6th BC Ajdukiewicz s fraction calculus, 1935 Bar-Hillel s system AB, 1950 Lambek s Syntactic Calculus, 1958

3 History Ajdukiewicz s fraction calculus, 1935 basic types: names, sentences fractional types: adjectives, verbs A, B, B A one cancelation schemata: B A A => B Syntactic Connections: a string of words satisfy a syntactic connection if some ordering of types of words reduce to distinguished types.

4 History Bar-Hillel s system AB, 1950 basic types: name, sentences fractional types: adjectives, verbs A, B, B/A, A\B, two cancelation schematas B/A A => B A A\B => B

5 Example Type Assignment

6 Example Derivation

7 Example Type Assignment

8 Example Derivation

9 History Lambek s Syntactic Calculus, 1958 basic types: name, sentences implication types: adjectives, verbs A, B, A<-B, A->B, modus ponens: A<-B A => B A A->B => B

10 Philosophy Ajdukiewics and Semantics? Husserl and Russell s theory of types Lesniewski s semantic categories Frege s functorial anaysis of language Functor B/A function Argument Made precise for LC by van Benthem (1983) and Moortgat (1988).

11 Exercise (I) Derive the following sentences in Ajdukiewicz s non-directional fraction calculus: ``I set my bow. ``I have set my bow. Word I set my bow have Type N (S N) N N N N (N S) (N S)

12 Exercise (II) Derive the following sentences in Bar-Hillel s directional calculus: ``I set my bow. ``I have set my bow. Word I set my bow have Type N (N\S)/N N/N N (N\S)/(N\S)

13 Exercise (III) Derive the following sentences in Lambek s modus ponens calculus: ``I set my bow. ``I have set my bow. Word Type I N set? my? bow N have?

14 Categorial Grammar Lambek A categorial grammar is a finite relation R between the set of expressions of a language and a set of types. 2- Given a distinguished type S, the language of a categorial grammar is the set of strings s1 sn, for which there exists A1,, An, where (si, Ai) in R and we have: A1,, An => S

15 Proof System F ::= b F F F \ F F/F

16 Meta Theorems Cut Elimination. Lambek 1958 Efficient proof search Algorithms. Moot and Retore Interpolation. Roorda 1991 Equivalence to context free grammars. Pentus 1993

17 Algebra Dosen 1985 An interpretation for the Lambek Calculus with atomic var s B consists of a partially ordered semigroup P =(P,, apple) and a valuation h, defined as follows: b 2 B h(b) =# L L P h(a \ C) :={p 2 P 8p 0 2 h(a), p 0 p 2 h(c)} h(c/b) :={p 2 P 8p 0 2 h(b), p p 0 2 h(c)} h(a B) :={p 2 P 9p 1 2 h(a), 9p 2 2 h(b), p apple p 1 p 2 }

18 Validity A sequent A 1,, A n ` A is valid iff in every interpretation of the sequent calculus we have: h(a 1, A n ) h(a) where h(a 1,, A n ):=h(a 1 A n ) Proposition (Dosen, 1985) A sequent is derivable in the Lambek Calculus iff its interpretation is valid.

19 Algebra Buszkowski, 1986 An interpretation for the Lambek Calculus with atomic var s B consists of a semigroup and a valuation h, defined as before, except: b 2 B h(b) =L L P h(a B) :={p 1 p 2 2 P p 1 2 h(a), p 2 2 h(b)} Proposition A sequent is derivable in the Lambek Calculus iff its interpretation is valid.

20 Derived Properties

21 Linguistic Examples ``John loves Mary. Word John Mary Loves Type N N (N\S)/N

22 Exercise (IV) Show that LC is associate: A. (B.C) - (A.B). C Derive type-lifting in LC: A - B/(A\B) A - (B/A)\B)

23 Exercise (V) Use LC to derive: ``I set my bow. Word Type I, bow N my N/N set (N\S)/N

24 Associativity Word Hulk is green incredible Type N (N\S)/(N/N) N/N N/N Derive: ``Hulk is green. Derive: ``Hulk is incredible. Derive: ``Hulk is green incredible.

25 Non-Associative Calculus Formulae F ::= b (F F) (F \ F) (F/F) Antecedent Terms A ::= F (A, A) Ant terms with a hole Contexts C ::= [ ] (C, A) (A, C) Substitution subst([ ], ) := subst(, 0 [], ) := (,subst( 0 [], )) subst( [], 0, ) := (subst( [], ), 0 )

26 Non-Associative Calculus

27 Meta Theorems Cut Elimination. Lambek 1961, Kandulski 1988 Decidability. Polynomial proof search. De Groote 1999 Nested or tree or deep Calculi. Kashima 1994

28 Kripke Semantics An interpretation for the Non-Assoc. Lambek Calculus with atomic var s B is a Kripke model M = (W, R, V) for R a ternary relation and a valuation h, defined: h(b) =(M, w) iff w 2 V(b) h(a B) =(M, w) iff 9w 0, w 00 2 W, R(w, w 0, w 00 ) h(a) =(M, w 0 ), h(b) =(M, w 00 ) h(a \ B) =(M, w) iff 8w 0, w 00 2 W, R(w 00, w 0, w) h(a) =(M, w 0 ), h(b) =(M, w 00 ) h(a/b) =(M, w) iff 8w 0, w 00 2 W, R(w 00, w, w 0 ) h(a) =(M, w 0 ), h(b) =(M, w 00 )

29 Validity A sequent of non-assoc. Lambek calculus is valid iff it is true in all worlds of all Kripke models under all valuations. Proposition (Dosen, 1992) A sequent of non-assoc. Lambek Calculus is derivable iff it is valid.

30 Exercise (VI) Show how the following is not derivable in non-associative LC. (N/N). (N/N) - N/N Compare it with the corresponding derivation in LC, i.e. note where does the derivation fail. Show that ``Hulk is very green is derivable in non-assoc LC (take very to have type (N/N)/(N/N).

31 Multi-Modal LC The fundamental idea of multi-modal LC is that one uses different families of connectives, distinguished by means of indices or modes. Each family of connectives has its own structural rules. These apply to different modes the use of which depends where it is most advantageous. First done by - Oehrle and Zhang (1989) - Moortgat and Morrill (1991) What I will present: - Kurtonina and Moortgat (1996)

32 MMLC Formulae F ::= p 2 j F j F (F \ i F) (F/ i F) (F i F) Antecedent Terms A ::= F hai i (A, A) i

33 Logical Rules + axiom + cut

34 What is this Modal Logic? not S4 - T not K not S5

35 Multimodal Tense Logic Prior 1967 Adjunction i a2 i Future Possibility F Past Necessity H It will at some point be the case It has always been the case Kurtonina 1995, 1998

36 ternary Kripke Models binary Kurtonina and Moortgat 1997 M = (W, R, R j, V) h( j A)=(M, w) iff h(2 j A)=(M, w) iff 9w 0 2 W, R 0 j (w, w0 ), h(a) =(M, w 0 ) 8w 2 W, (R 0 j (w0, w) =) h(a) =(M, w 0 ))

37 Meta Theorems Cut Elimination. Moortgat 1996 PSPACE complete. Moot and Retore Interpolation?

38 Applications Linguistic Features, Heylen 1999 Quantifier Scope, Bernardini 2002 Pronouns, Moot and Retore 2006

39 Exercise (VII) Derive the adjunction property in MMLC And use it to analyse: <i>[i] F - F - [i]<i> F he likes Mary * her likes Mary * Mary likes he * Mary likes her Word he her Mary likes Type [n]<n>n [a]<a>n N ([n]<n> N \ S)/[a]<a>N

40 Applications Pronouns, Moot and Retore 2006 he likes Mary Derivable since 1- Mary has type N 2- N - [a]<a>n

41 Applications Pronouns, Moot and Retore 2006 Mary likes her Derivable since 1- Mary has type N 2- N - [n]<n>n

42 Applications Pronouns, Moot and Retore 2006 her likes Mary Underivable since 1- her has type [a]<a>n 2- [a]<a>n can only be object of likes

43 Structural Modal Rules Indices also allow us to reintroduce structural rules, albeit in an indexed way, applicable to certain modes.

44 Structural Modal Rules Indices also allow us to reintroduce structural rules, albeit in an indexed way, applicable to certain modes.

45 Applications Dutch verb clusters. Moortgat and Oehrle 1994 (dat) Marie Jan Plaagt (that) Marie Jan teases * (dat) Marie Jan wil Plagen * (that) Marie Jan wants to tease

46 MM Types Word Marie Jan wil plagen Type N N [1]((N\0 S)/4 Inf) [1](N\0 Inf)

47 Structural Rules K2 for index 0 K for indices 1 and 4 Inclusion from 0 to 1

48 Indices Combining Logics Index Rules n - a nc c Associativity Commutativity Associativity, Commutativity

49 Structural Rules

50 Structural Rules

51 Structural Rules

52 Remains to explore Semantics: van Benthem 1983, will discuss on Tue Proof Theory: Dyckhoff and Sadrzadeh, Adjoint modalities on positive logic base RSL 2010, Adjoint modalities on intuitionstic base, ToCL 2013 Applications: Intensionality