Lexicalized Tree Adjoining Grammar
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1 Lexicalized Tree Adjoining Grammar Petr Horáček, Eva Zámečníková and Ivana Burgetová Department of Information ystems Faculty of Information Technology Brno University of Technology Božetěchova 2, Brno, CZ FRVŠ MŠMT FR97/2011/G1
2 Outline Introduction Lexicalized Tree Adjoining Grammar 2 / 53
3 Outline Introduction Tree Adjoining Grammar Lexicalized Tree Adjoining Grammar 3 / 53
4 Outline Introduction Tree Adjoining Grammar ome Important TAG Properties Lexicalized Tree Adjoining Grammar 4 / 53
5 Outline Introduction Tree Adjoining Grammar ome Important TAG Properties Lexicalized Tree Adjoining Grammar Lexicalized Tree Adjoining Grammar 5 / 53
6 Topic Introduction Tree Adjoining Grammar ome Important TAG Properties Lexicalized Tree Adjoining Grammar Lexicalized Tree Adjoining Grammar 6 / 53
7 Tree-Adjoining Grammars (TAG) Motivation Motivation is of linguistic and formal nature. Elementary objects are trees - structured objects and not strings. tructured objects are related with strong generative capacity. More relevant to linguistic description. TAG allow factoring recursion from the statement of linguistic dependencies Lexicalization of grammar formalism. Lexicalized Tree Adjoining Grammar 7 / 53
8 Tree-Adjoining Grammars (TAG) Motivation Motivation is of linguistic and formal nature. Elementary objects are trees - structured objects and not strings. tructured objects are related with strong generative capacity. More relevant to linguistic description. TAG allow factoring recursion from the statement of linguistic dependencies Lexicalization of grammar formalism. TAG is tree-generating system the set of trees constitute the object language One well known normal form of grammars - Greibach Normal Form (GNF) is a kind of lexicalization. Lexicalized Tree Adjoining Grammar 8 / 53
9 Topic Introduction Tree Adjoining Grammar ome Important TAG Properties Lexicalized Tree Adjoining Grammar Lexicalized Tree Adjoining Grammar 9 / 53
10 Tree-Adjointing Grammars (TAG) Definition Tree Adjoining Grammar (TAG) is a quintuple (T, N, I, A, ). T... a finite set of terminal symbols N... a finite set of nonterminal symbols; T N = I... a finite set of initial trees An initial tree is a phrase structure tree A... a finite set of auxiliary trees An auxiliary tree is a phrase structure tree that has a leaf nonterminal node that is the same as its root symbol... start symbol, N Lexicalized Tree Adjoining Grammar 10 / 53
11 Tree-Adjointing Grammars (TAG) Definition Tree Adjoining Grammar (TAG) is a quintuple (T, N, I, A, ). T... a finite set of terminal symbols N... a finite set of nonterminal symbols; T N = I... a finite set of initial trees An initial tree is a phrase structure tree A... a finite set of auxiliary trees An auxiliary tree is a phrase structure tree that has a leaf nonterminal node that is the same as its root symbol... start symbol, N Trees in I and A are called elementary trees. Lexicalized Tree Adjoining Grammar 11 / 53
12 Tree-Adjointing Grammars (TAG) Definition Tree Adjoining Grammar (TAG) is a quintuple (T, N, I, A, ). T... a finite set of terminal symbols N... a finite set of nonterminal symbols; T N = I... a finite set of initial trees An initial tree is a phrase structure tree A... a finite set of auxiliary trees An auxiliary tree is a phrase structure tree that has a leaf nonterminal node that is the same as its root symbol... start symbol, N Trees in I and A are called elementary trees. Parsing is done by two operations: substitution and adjunction. Lexicalized Tree Adjoining Grammar 12 / 53
13 Tree-Adjointing grammars: Example An example of an initial and an auxiliary tree Example initial tree auxiliary tree V V likes knows A nonterminal symbol marked by * is the foot node of an auxiliary tree. A nonterminal symbol marked by is a nonterminal node for substitution. Lexicalized Tree Adjoining Grammar 13 / 53
14 ubstitution ubstitution of an initial tree T 1 into a tree T 2 is to replace a substitution node in T 2 with T 1. Lexicalized Tree Adjoining Grammar 14 / 53
15 ubstitution ubstitution of an initial tree T 1 into a tree T 2 is to replace a substitution node in T 2 with T 1. Example V John V John likes likes T 1 T 2 Lexicalized Tree Adjoining Grammar 15 / 53
16 ubstitution ubstitution of an initial tree T 1 into a tree T 2 is to replace a substitution node in T 2 with T 1. Example V John V John likes likes T 1 T 2 Lexicalized Tree Adjoining Grammar 16 / 53
17 Adjunction (Adjoining) Adjoining an auxiliary tree T 1 into a tree T 2 is to inset T 1 into T 2 at the node that is the same as the root (and the foot) of T 1. Lexicalized Tree Adjoining Grammar 17 / 53
18 Adjunction (Adjoining) Adjoining an auxiliary tree T 1 into a tree T 2 is to inset T 1 into T 2 at the node that is the same as the root (and the foot) of T 1. Example PP John V walks PREP in John V PP PREP T 2 T 1 walks in T 1 is adjoined at in T 2 Lexicalized Tree Adjoining Grammar 18 / 53
19 Adjunction (Adjoining) Adjoining an auxiliary tree T 1 into a tree T 2 is to inset T 1 into T 2 at the node that is the same as the root (and the foot) of T 1. Example PP John V walks PREP in John V PP PREP T 2 T 1 walks in Lexicalized Tree Adjoining Grammar 19 / 53
20 Adjunction (Adjoining) Adjoining an auxiliary tree T 1 into a tree T 2 is to inset T 1 into T 2 at the node that is the same as the root (and the foot) of T 1. Example PP John V walks PREP in John V PP PREP T 2 T 1 walks in T 1 is adjoined at in T 2 Lexicalized Tree Adjoining Grammar 20 / 53
21 Adjunction (Adjoining) Any adjunction on a node marked for substitution is disallowed. Adjoining Constraints to have more precision for specifying which auxiliary trees can be adjoined at a given node. 1 elective Adjunction ( A(T)) - only members of a set T A of auxiliary trees can be adjoined on the given node, the adjunction of an auxiliary is not mandatory on the given node. 2 Null Adjunction (N A) - disallows any adjunction on the given node. 3 Obligatory Adjunction (O A(T)) - an auxiliary tree member of the set T A must be adjoined on the given node. These constraints on adjoining are needed for formal reasons in order to obtain some closure properties. Lexicalized Tree Adjoining Grammar 21 / 53
22 Derivation in TAG Ad yesterday D N V a man saw N Mary Lexicalized Tree Adjoining Grammar 22 / 53
23 Derivation in TAG Ad yesterday D N V a man saw N Mary This tree yields the sentence Yesterday a man saw Mary and is derived from the following elementary trees: Lexicalized Tree Adjoining Grammar 23 / 53
24 Derivation in TAG Ad yesterday D N V a man saw N Mary This tree yields the sentence Yesterday a man saw Mary and is derived from the following elementary trees: Ad yesterday Lexicalized Tree Adjoining Grammar 24 / 53
25 Derivation in TAG Ad yesterday D N V a man saw N Mary This tree yields the sentence Yesterday a man saw Mary and is derived from the following elementary trees: D Ad yesterday a Lexicalized Tree Adjoining Grammar 25 / 53
26 Derivation in TAG Ad yesterday D N V a man saw N Mary This tree yields the sentence Yesterday a man saw Mary and is derived from the following elementary trees: D Ad a D N yesterday man Lexicalized Tree Adjoining Grammar 26 / 53
27 Derivation in TAG Ad yesterday D N V a man saw N Mary This tree yields the sentence Yesterday a man saw Mary and is derived from the following elementary trees: D Ad a D N 0 yesterday man V 1 saw Lexicalized Tree Adjoining Grammar 27 / 53
28 Derivation in TAG Ad yesterday D N V a man saw N Mary This tree yields the sentence Yesterday a man saw Mary and is derived from the following elementary trees: D Ad a D N 0 N yesterday man V 1 Mary saw Lexicalized Tree Adjoining Grammar 28 / 53
29 Derivation in TAG β yest : α a : α man : α saw : α Mary : D Ad a D N 0 N yesterday man V 1 Mary saw Lexicalized Tree Adjoining Grammar 29 / 53
30 Derivation in TAG β yest : α a : α man : α saw : α Mary : D Ad a D N 0 N yesterday man V 1 Mary saw Derivation tree for Yesterday a man saw Mary. α saw α man α Mary β yest α a Lexicalized Tree Adjoining Grammar 30 / 53
31 Derivation in TAG β yest : α a : α man : α saw : α Mary : D Ad a D N 0 N yesterday man V 1 Mary saw Derivation tree for Yesterday a man saw Mary. α saw α man α Mary β yest α a The order in which the derivation tree is interpreted has no impact on the resulting derived tree. Lexicalized Tree Adjoining Grammar 31 / 53
32 Derivation in TAG Derived Tree A tree built by composition of two others trees. the derived tree does not give enough information to determine how it was constructed adjunction and substitution are considered in a TAG derivation Lexicalized Tree Adjoining Grammar 32 / 53
33 Derivation in TAG Derived Tree A tree built by composition of two others trees. the derived tree does not give enough information to determine how it was constructed adjunction and substitution are considered in a TAG derivation Derivation Tree It is an object that specifies uniquely how a derived tree was constructed. Lexicalized Tree Adjoining Grammar 33 / 53
34 Topic Introduction Tree Adjoining Grammar ome Important TAG Properties Lexicalized Tree Adjoining Grammar Lexicalized Tree Adjoining Grammar 34 / 53
35 Tree ets and tring Languages Tree et of a TAG T G Defined as the set of completed initial trees derived from some -rooted initial trees. T G = {t t is derived from some -rooted initial tree} Note that completed initial tree is an initial tree with no substitution nodes. Lexicalized Tree Adjoining Grammar 35 / 53
36 Tree ets and tring Languages Tree et of a TAG T G Defined as the set of completed initial trees derived from some -rooted initial trees. T G = {t t is derived from some -rooted initial tree} Note that completed initial tree is an initial tree with no substitution nodes. Tree tring language of a TAG L G Defined as the set of yields of all trees in the tree set. L G = {w w is the yield of some t in T G } Lexicalized Tree Adjoining Grammar 36 / 53
37 ome Properties of the Tree ets and tring Languages All closure properties of context-free languages (CFL) also hold for tree-adjoining languages (TAL). CFL TAL TAL can be parsed in polynomial time. Tree-adjoining grammars generate some context-sensitive languages. Lexicalized Tree Adjoining Grammar 37 / 53
38 ome Properties of the Tree ets and tring Languages: Example 1 Example Consider following TAG G 1 = ({a, e, b}, {}, {α 6 }, {β 2 }, ) α 6 : β 2 : NA e a NA b Lexicalized Tree Adjoining Grammar 38 / 53
39 ome Properties of the Tree ets and tring Languages: Example 1 Example Consider following TAG G 1 = ({a, e, b}, {}, {α 6 }, {β 2 }, ) α 6 : β 2 : NA e a NA b G 1 generates the language L 1 = {a n eb n n 1} Lexicalized Tree Adjoining Grammar 39 / 53
40 ome Properties of the Tree ets and tring Languages: Example 2 Example Consider following TAG G 1 = ({a, b, c, d, e}, {}, {α 6 }, {β 3 }, ) α 6 : β 3 : NA e a d b NA c Lexicalized Tree Adjoining Grammar 40 / 53
41 ome Properties of the Tree ets and tring Languages: Example 2 Example Consider following TAG G 1 = ({a, b, c, d, e}, {}, {α 6 }, {β 3 }, ) α 6 : β 3 : NA e a d b NA c G 1 generates the language L 1 = {a n b n ec n d n n 1} Lexicalized Tree Adjoining Grammar 41 / 53
42 ome Properties of the Tree ets and tring Languages: Example 2 Example ome derived trees of G 2 NA a d b NA e c Lexicalized Tree Adjoining Grammar 42 / 53
43 ome Properties of the Tree ets and tring Languages: Example 2 Example ome derived trees of G 2 NA NA a d b NA c a NA d e a d b NA c b NA e c Lexicalized Tree Adjoining Grammar 43 / 53
44 Topic Introduction Tree Adjoining Grammar ome Important TAG Properties Lexicalized Tree Adjoining Grammar Lexicalized Tree Adjoining Grammar 44 / 53
45 Lexicalized Grammars Lexicalized Grammar Each elementary structure is associate with a lexical item. The grammar consists of lexicon, where: each lexical item is associated with a finite number of structures and there are operations which tell how these structures are composed. Lexicalized Tree Adjoining Grammar 45 / 53
46 Lexicalized Tree Adjoining Grammar (LTAG) Definition A grammar is lexicalized if it consists of a finite set of structures each associated with a lexical item. Each lexical item is called the anchor of the corresponding structure. Grammar contains an operation or operations for composing the structure. LTAG is a TAG in which every elementary (initial and auxiliary) tree is anchored with a lexical item. Lexicalized Tree Adjoining Grammar 46 / 53
47 Lexicalized Tree Adjoining Grammar (LTAG) Definition A grammar is lexicalized if it consists of a finite set of structures each associated with a lexical item. Each lexical item is called the anchor of the corresponding structure. Grammar contains an operation or operations for composing the structure. LTAG is a TAG in which every elementary (initial and auxiliary) tree is anchored with a lexical item. Notes The anchor must be overt (= not empty string). The structures defined by the lexicon are called elementary structures. tructures built up by combination of others are called derived structures. Lexicalized Tree Adjoining Grammar 47 / 53
48 Lexicalized Tree Adjoining Grammar (LTAG) The definition of Lexicalized Grammar implies the following proposition: Proposition Lexicalized grammars are finitely ambiguous. Lexicalized Tree Adjoining Grammar 48 / 53
49 Lexicalized Tree Adjoining Grammar (LTAG) The definition of Lexicalized Grammar implies the following proposition: Proposition Lexicalized grammars are finitely ambiguous. Further, this closure property holds: Closure under lexicalization TAGs are closed under lexicalization. Lexicalized Tree Adjoining Grammar 49 / 53
50 Conclusion Notes Lexicalization of grammars is of linguistic and formal interest. Rules should not be separated from their lexical realization. By using TAGs we can lexicalize the CFGs. ubstitution and adjunction gives this possibility to lexicalize CFG. Lexicalized Tree Adjoining Grammar 50 / 53
51 References James Allen: Natural Language Understanding, The Benjamin/Cummings Publishing Company. Inc., 2005 Yuji Matsumoto: yntax and Parsing: Phrase tructure and Dependency Parsing Algorithms, LT, 2011 Yves chabes, Aravind K. Joshi: Tree-Adjoining Grammars and Lexicalized Grammars, 1991 Lexicalized Tree Adjoining Grammar 51 / 53
52 Thank you for your attention!
53 End
ž Lexicalized Tree Adjoining Grammar (LTAG) associates at least one elementary tree with every lexical item.
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