Parsing beyond context-free grammar: Tree Adjoining Grammar Parsing I

Transcription

1 Parsng beyond context-free grammar: Tree donng Grammar Parsng I Laura Kallmeyer, Wolfgang Maer ommersemester 2009 duncton and substtuton (1) Tree donng Grammars (TG) Josh et al. (1975), Josh & chabes (1997): Tree-rewrtng system: set of elementary trees wth two operatons: aduncton: replacng an nternal node wth a new tree. The new tree s an auxlary tree and has a specal leaf, the foot node. substtuton: replacng a leaf wth a new tree. The new tree s an ntal tree Parsng beyond CFG 1 TG Parsng I Parsng beyond CFG 3 TG Parsng I duncton and substtuton (2) Overvew 1. duncton and substtuton 2. duncton constrants 3. TG dervaton trees (1) John sometmes laughs D John sometmes laughs 4. Formal propertes 5. Elementary trees for natural languages 6. CYK recognzer for TG derved tree: John D sometmes laughs Parsng beyond CFG 2 TG Parsng I Parsng beyond CFG 4 TG Parsng I

2 duncton and substtuton (3) Defnton 1 (Tree donng Grammar) Tree donng Grammar (TG) s a quadruple G = N, T, I, such that T and N are dsont alphabets, the termnals and nontermnals, I s a fnte set of ntal trees, and s a fnte set of auxlary trees. The trees n I are called elementary trees. G s lexcalzed ff each elementary tree has at least one leaf wth a termnal label. duncton constrants (2) Defnton 2 (TG wth aduncton constrants) TG wth aduncton constrants s a tuple N, T, I,, f O, f such that N, T, I, s a TG, f O : {µ µ s an nternal node or a foot node n a tree n I } {1, 0} s a functon, and f : {µ µ s an nternal node or a foot node n a tree n I } P() s a functon. Parsng beyond CFG 5 TG Parsng I Parsng beyond CFG 7 TG Parsng I duncton constrants (1) TG as defned above are more powerful than CFG but they cannot generate the copy language. In order to ncrease the expressve power, aduncton constrants are ntroduced that specfy for each node 1. whether aduncton s mandatory and 2. whch trees can be adoned. duncton constrants (3) Three types of constrants are dstngushed: node µ wth f O (µ) = 1 s sad to carry a oblgatory aduncton (O) constrant. node µ wth f O (µ) = 0 and f (µ) = s sad to carry a null aduncton (N) constrant. node µ wth f O (µ) = 0 and f (µ) and f (µ) s sad to carry a selectve aduncton () constrant. Parsng beyond CFG 6 TG Parsng I Parsng beyond CFG 8 TG Parsng I

3 duncton constrants (4) Dervaton trees (1) Example: TG for the copy language: TG dervatons are descrbed by dervaton trees: ǫ a N N a b N N b For each dervaton n a TG there s a correspondng dervaton tree. Ths tree contans nodes for all elementary trees used n the dervaton, and edges for all adunctons and substtutons performed throughout the dervaton. Whenever an elementary tree was attached to the node at address p n the elementary tree, there s an edge from to labeled wth p. Parsng beyond CFG 9 TG Parsng I Parsng beyond CFG 11 TG Parsng I duncton constrants (4) Example: (2) John seems to sleep Dervaton trees (2) Example: dervaton tree for the dervaton of (2) John seems to sleep sleep O 1 2 ohn seems John to seems sleep Parsng beyond CFG 10 TG Parsng I Parsng beyond CFG 12 TG Parsng I

4 ome formal propertes (1) Languages TG can generate: {ww w {a, b} } L 4 := {a n b n c n d n n 0} Languages TG cannot generate: {w n w {a, b} } for any n > 2. TG generated only a lmted amount of cross-seral dependences L k := {a n 1 an 2 an 3...an k n 0} for any k > 4. TG can count up to 4, not further. L := {a 2n n 0}. TG cannot generate languages whose word lengths grow exponentally. Parsng beyond CFG 13 TG Parsng I Elementary trees (1) Important features of LTG: Grammar s lexcalzed Recursve parts are put nto separate elementary trees that can be adoned (Factorng of recurson, FR) Elementary trees can be arbtrarly large, n partcular (because of FR) they can contan elements that are far apart n the fnal derved tree (Extended doman of localty) Parsng beyond CFG 15 TG Parsng I ome formal propertes (2) TGs are mldly context-senstve: TGs are slghtly more powerful than CFG, they can descrbe a lmted amount of cross-seral dependences. TGs are polynomally parsable (complexty O(n 6 )). TLs are of constant growth. Elementary trees (2) Elementary trees are extended proectons of lexcal tems. Recurson s factored away fnte set of elementary trees. The elementary tree of a lexcal predcate contans slots for all arguments of the predcate, for nothng more. Besdes lexcal predcates, there are functonal elements (complementzers, determners, auxlares, negaton) whose treatment n LTG s less clear. They can be ether n separate elementary trees (e.g., XTG grammar) or n the elementary tree of the lexcal tem they are assocated wth. Parsng beyond CFG 14 TG Parsng I Parsng beyond CFG 16 TG Parsng I

5 Elementary trees (3) Example: (3) John gves a book to Mary PP gves P to Elementary trees (5) to make a comment: make and comment n the same elementary tree snce they form a lght verb constructon: to make N comment Det a Parsng beyond CFG 17 TG Parsng I Parsng beyond CFG 19 TG Parsng I Elementary trees (4) Example: (4) John expected Mary to make a comment expected selects for a subect and an nfntval sentence: John expected to make a comment The sentental obect s realsed as a foot node n order to allow extractons: Elementary trees (6) Example wth modfers: (6) the good student partcpated n every course durng the semester P good N N Det the N student (5) whom does John expect to come? Parsng beyond CFG 18 TG Parsng I Parsng beyond CFG 20 TG Parsng I

6 Elementary trees (7) CYK Recognton (2) PP partcpated P n PP P durng t each moment, we are n a specfc node n an elementary tree and we know about the yeld of the part below. Ether there s a foot node below, then the yeld s separated nto two parts. Or there s no foot node below and the yeld s a sngle substrng of the nput. We need to keep track of whether we have already adoned at the node or not snce at most one aduncton per node can occur. For ths, we dstngush between a bottom and a top poston for the dot on a node. Bottom sgnfes that we have not performed an aduncton. Parsng beyond CFG 21 TG Parsng I Parsng beyond CFG 23 TG Parsng I CYK Recognton (1) Frst presented n ay-hanker & Josh (1985), formulaton wth deducton rules n Kallmeyer & atta (2009). ssumpton: elementary trees are such that each node has at most two daughters. (ny TG can be transformed nto an equvalent TG satsfyng ths condton.) The algorthm smulates a bottom-up traversal of the derved tree. CYK Recognton (3) Item form: [, p t,, f 1, f 2, ] where I, p s the Gorn address of a node n (ǫ for the root, p for the th daughter of the node at address p), subscrpt t {, } specfes whether substtuton or aduncton has already taken place ( ) or not ( ) at p, and 0 f 1 f 2 n are ndces wth, ndcatng the left and rght boundares of the yeld of the subtree at poston p and f 1, f 2 ndcatng the yeld of a gap n case a foot node s domnated by p. We wrte f 1 = f 2 = f no gap s nvolved. Parsng beyond CFG 22 TG Parsng I Parsng beyond CFG 24 TG Parsng I

7 CYK Recognton (4) Goal tems: [α, ǫ, 0,,, n] where α I We need two rules to process leaf nodes whle scannng ther labels, dependng on whether they have termnal labels or labels ǫ: CYK Recognton (6) The rule foot-predct processes the foot node of auxlary trees β by guessng the yeld below the foot node: Foot-predct: Lex-scan: [, p,,,, + 1] l(, p) = w +1 [β, p,,,, ] β, p foot node address n β, Eps-scan: [, p,,,, ] l(, p) = ǫ (Notaton: l(, p) s the label of the node at address p n.) Parsng beyond CFG 25 TG Parsng I Parsng beyond CFG 27 TG Parsng I CYK Recognton (5) CYK Recognton (7) When movng up nsde a sngle elementary tree, we ether move from only one daughter to ts mother, f ths s the only daughter, or we move from the set of both daughters to the mother node: w Lex-scan ǫ Eps-scan Move-unary: [, (p 1),, f 1, f 2, ] [, p,, f 1, f 2, ] node address p 2 does not exst n Move-bnary: [, (p 1),, f 1, f 2, k], [, (p 2), k, f 1, f 2, ] [, p,, f 1 f 1, f 2 f 2, ] (f f = f where f = f f f =, f = f f f =, and f s undefned otherwse) Parsng beyond CFG 26 TG Parsng I Parsng beyond CFG 28 TG Parsng I

8 CYK Recognton (8) Move-unary: CYK Recognton (10) For nodes that do not requre aduncton, we can move from the bottom poston of the node to ts top poston. B B Null-adon: [, p,, f 1, f 2, ] [, p,, f 1, f 2, ] f O (, p) = 0 Parsng beyond CFG 29 TG Parsng I Parsng beyond CFG 31 TG Parsng I CYK Recognton (9) Move-bnary: CYK Recognton (11) The rule substtute performes a substtuton: B C B C ubsttute: [α, ǫ,,,, ] [, p,,,, ] l(α, ǫ) = l(, p) k k α B C Parsng beyond CFG 30 TG Parsng I Parsng beyond CFG 32 TG Parsng I

9 CYK Recognton (12) The rule adon adons an auxlary tree β at p n, under the precondton that the aduncton of β at p n s allowed: CYK Recognton (14) Complexty of the algorthm: What s the upper bound for the number of applcatons of the adon operaton? don: [β, ǫ,, f 1, f 2, ], [, p, f 1, f 1, f 2, f 2] [, p,, f 1, f 2, ] β f (, p) We have possbltes for β, I for, m for p where m s the maxmal number of nternal nodes n an elementary tree. The sx ndces, f 1, f 1, f 2, f 2, range from 0 to n. Consequently, adon can be appled at most I m(n + 1) 6 tmes and therefore, the tme complexty of ths algorthm s O(n 6 ). Parsng beyond CFG 33 TG Parsng I Parsng beyond CFG 35 TG Parsng I CYK Recognton (13) don: β f 1 f 2 f 1 f 2 Parsng beyond CFG 34 TG Parsng I