ž Lexicalized Tree Adjoining Grammar (LTAG) associates at least one elementary tree with every lexical item.

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1 Advanced Natural Language Processing Lecture 11 Grammar Formalisms (II): LTAG, CCG Bonnie Webber (slides by Mark teedman, Bonnie Webber and Frank Keller) 12 October 2012 Lexicalized Tree Adjoining Grammar (LTAG) ž Lexicalized Tree Adjoining Grammar (LTAG) associates at least one elementary tree with every lexical item. ž Elementary trees are either initial trees or auxiliary trees. ž They combine by two operations: substitution (which is just what it says) and adjunction (which is a little harder to explain). 1 LTAG Initial ( a ) Trees! :! :! : 1 2 NA 3 P i P likes i likes "! 4 : 2 peanuts ž Down arrow # on a category X means an X must be substituted here. ž a 1 is for standard subject likes object, while a 2 is for topicalized object subject likes. LTAG Auxiliary (b) Trees! 1 : P! 2 : P* AD passionately P 3 * thinks ž Asterisk Ł on a category X means this X must adjoin at an X. ž NA on a category X means adjoining is not allowed at this X. ž NA on initial tree a 2 will prevent it from adjoining to b 2 (for example), avoiding *Bill thinks peanuts, likes.

2 A Derivation involving ubstitution and Adjunction! : 1 1 : P P P* AD passionately likes 4 The Derived Tree P P AD passionately 5! 3 :! 4 : ubstitution: likes peanuts peanuts Adjunction: ž Recall that asterisk Ł on a category X means this X must adjoin at an X. Unbounded Dependency: Topicalization 2 : i P i likes #! 2 : 6 P * thinks Unbounded dependency: Derived Tree ( D T) i peanuts P Bill 7 4 : 3 : : 5 thinks P peanuts Bill i ž Clearly we could adjoin unboundedly many auxiliary trees like b 2. likes!

3 The Raising Construction in TAG! : 6 " : 5 P P P* P P seems to walk! : 5 Result: P Bill Bill P seems P P 8 The Control Construction in TAG! : 8 : 4 P P PRO P P * to tries walk! : Result: 5 P Bill Bill tries P PRO P P 9 to to walk walk LTAG Can be Expressed as an LIG ž LTAG lexical elementary trees like a 1, a 2, and b 2, can be regarded as bearing a stack-valued feature, where the elements on the stack are the # and Ł marked leaf nodes, and where their order on the stack is defined by a leftmost-depthfirst walk along the arcs of the tree. ž ubstitution and adjunction can then be thought of as operations which pass a single such stack-valued feature to their result. ž This intuition underpins Weir s 1988 identification of the equivalence between LTAG and LIG [Joshi, 1991]. 10 Crossing Dependencies in LTAG (a n b n ) ž Consider G = (I, A) comprising the following Initial (I) and Auxiliary (A) trees:!!!" I: 1 NA P i zwemmeni A : #!!" 1 NA #!!" 2 j * P helpen j 11 * P k zag k ž een in that light its not too surprising that LTAG provides an account of Dutch crossing a n b n.

4 Crossing Dependencies: One adjunction ž These trees engender crossing dependencies between lexical sisters in (dat) Piet Marie helpen zwemmen!! 2 1!! NA #!! 1 Marie P! i zwemmeni!! 3 Piet NA! j 12 * P helpen j Crossing Dependencies: D T, one adjunction Ž (dat) Piet Marie helpen zwemmen Piet Marie P helpen j! j P! i NA zwemmen i 13 Crossing Dependencies: Two adjunctions Ž (dat) Jan Piet Marie zag helpen zwemmen NA!! 2 NA zwemmen i * P zag k P helpen j Piet! k! j #!! 4 P Jan Marie! i 14 Crossing Dependencies: D T, two adjunctions Jan Piet Marie zwemmen i helpen j P zag k! k P! j P! i NA NA 15

5 Discussion ž o LTAG provides a strongly adequate account of unbounded and crossing dependencies. ž With coordination, it can account for some structures which involve crossing: (dat) Jan Piet Marie zag helpen zwemmen en hoorde leren zingen. (dat) Jan Piet Marie [zag helpen zwemmen] en [hoorde leren zingen]. (that) Jan Piet Marie saw help swim and heard teach sing. (that) Jan saw Piet help Marie to swim and heard Piet teach Marie to sing. imple P coordination 16 Coordinate structure problems for LTAG (dat) Jan Piet Marie en ik Geert de nijlpaarden zag helpen zwemmen. ž (dat) [Jan Piet Marie] en [ik Geert de nijlpaarden] zag helpen zwemmen. (that) [Jan Piet Marie] and [I Geert the hippos] saw help swim. ) (that) Jan saw Piet help Marie (to swim) and I saw Geert help the hippos to swim. (dat) Jan Piet en ik Geert de nijlpaarden zag helpen zwemmen. ž (dat) [Jan Piet] en [ik Geert] de nijlpaarden zag helpen zwemmen. (that) [Jan Piet] and [I Geert] the hippos saw help swim. ) (that) Jan saw Piet (help the hippos to swim) and I saw Geert help the hippos to swim. 17 II: Combinatory Categorial Grammar (teedman, 2000; teedman and Baldridge, 2006) ž Like LTAG, CCG is a lexicalized grammar. ž It associates with each word in the lexicon, descriptions of syntactic structures in which it occurs. 18 The Application Rules ž (1) The functional application rules a. X/ Y Y ) X () b. Y Xn Y ) X () ž /, n mean any category can combine by this rule, even * categories (cf. next slide) 19 ž It does this with rich slashed categories that can specify both a left and right context. ž e.g. proved : = (n)/ ž Function categories are typed, shown with a subscript on slashes, which can restrict which rules can be used in combining them with other categories. ž A Derivation: (2) a. Marcel proved completeness (n)/ n Marcelprovedcompleteness P

6 (3) The Conjunction Category and := (Xn X)/ X Coordination (4) Marcel conjectured and proved completeness (n)/ (Xn X)/ X (n)/ ((n)/)n ((n)/) (n)/ n 20 Forward composition (B) X/ Š Y Y/ Š Z ) B X/ Š Z Composition ž Š modality means only Š or Ð categories can combine by this rule, preventing overgeneration. ž By symmetry, CCG also has backward composition (B). ž There are also n-ary versions of forward ( B n ) and backward ( B n ) composition: X/ Š Y (Y / Š W )/ Š Z ) B 2 (X/ Š W )/ Š Z 21 Type-Raising ž Forward type-raising (T) X ) T/ i ( Tn i X) ž The subscript i on the two slashes means that they both have the same modality as the function Tn i X that the raised category is applied to. ž T is a meta-variable over categories. ž On the next slide, is type-raised to the subject of a sentence by instantiating meta-variable T as. ž ) /(n) 22 Type-Raising (5) Marcel proved and I disproved completeness (n)/ (Xn X)/ X (n)/ /(n) T T / /(n) B B / (/)n (/) / Type raising is restricted to primitive argument categories,, PP etc., and over primitive functors like verbs, resembling the traditional notion of case. 23

7 Multiple Derivations Rules for type-raising and composition lead to the possibility of multiple derivations: (6) Marcel proved completeness (n)/ n (7) Marcel proved completeness (n)/ T /(n) / B 24 Unbounded Dependencies Type-raising, composition and application provide a strongly adequate basis for analysing the unbounded character of leftward extraction. (8) a. [I think that Marcel proved] / the result. b. [I think that] /Š Marcel proved the result. c. the result [that] (NnN)/(/) [I think that Marcel proved] /. d. *the man who(m) [I think that] /Š [proved the result] n. 25 Crossing Dependencies in CCG ž Rules for crossed composition permit crossing dependencies: (9) zag:= (( + UB n)n)/ ð P zien, helpen:= (Pn)/ ð P voeren:= Pn (10) Forward Crossing Composition I ( B n ð): X/ ð Y Yn ð Z ) B 1 ð Xn ð Z X/ ð Y (Y n ð W )/ ð Z ) B 2 ð (Xn ð W )/ ð Z omdat ik Cecilia de nijlpaarden zag voeren.... because I Cecilia the hippopotamuses saw feed... because I saw Cecilia feed the hippopotamuses. Hippo entences (11) dat ik Cecilia de nijlpaarden zag voeren that I Cecilia the hippos saw feed (( + UB n)n)/ ð P Pn B ð (( + UB n)n)n ð ( + UB n)n + UB n + UB that I saw Cecilia feed the hippos 27

8 Hippo entences (Contd.) 28 CCG and LIG omdat ik Cecilia Henk de nijlpaarden zag helpen voeren.... because I Cecilia Henk the hippopotamuses saw help feed... because I saw Cecilia help Henk feed the hippopotamuses. (12) dat ik Cecilia Henk de nijlpaarden zag helpen voeren (( + UB n)n)/ ð P (Pn)/ ð P Pn B ð (Pn)n ð ((( + UB n)n)n)n ð B 2 ð (13) dat ik Cecilia Henk de nijlpaarden zag helpen voeren (( + UB n)n)/ ð P (Pn)/ ð P Pn B 2 ð ((( + UB n)n)n)/ ð P ((( + UB n)n)n)n ð B ð ž Joshi et al. (1991) were the first to observe that there is a close relation between linear indexed rules and the combinatory rules of CCG. ž CCG categories can be viewed as their result category plus a stack-valued feature identifying their arguments and the order of combination: (14) give : = (P/)/ P [,] zag helpen voeren : = (((n)n)n)n [,,,] ž As such, combinatory rules then map one-to-one to LIG rules, ž CCG is provably weakly equivalent to TAG and LIG, which means that it is nearly Context Free (aka mildly Context ensitive). 30 References Joshi, Aravind, ijay-hanker, K., and Weir, David, The Convergence of Mildly Context- ensitive Formalisms. In P ells, hieber, and T Wasow (eds.), Processing of Linguistic tructure, Cambridge MA: MIT Press teedman, Mark, The yntactic Process. Cambridge, MA: MIT Press. teedman, Mark and Baldridge, Jason, Combinatory Categorial Grammar. In Keith Brown (ed.), Encyclopedia of Language and Linguistics, Oxford: Elsevier, volume 2. econd edition,

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