GlueTag Linear Logic based Semantics for LTAG

Transcription

1 GlueTag Linear Logic based emantics or LTAG and what it teaches us about LFG and LTAG Anette Frank Language Technology Group DFKI GmbH aarbrücken, Germany Jose van Genabith Computer Applications Dublin City University Dublin, Ireland Proceedings o the LFG01 Conerence University o Hong Kong, Hong Kong Miriam Butt and Tracy Holloway King (Editors) 2001 CLI Publications

2 GlueTag Linear Logic based emantics or LTAG and what it teaches us about LFG and LTAG Abstract We review existing appoaches to semantics construction in LTAG (Lexicalised Tree Adjoining Grammar) based on the notion o derivation (tree)s. We argue that derivation structures in LTAG are not appropriate to guide semantic composition, due to a non-isomorphism, in LTAG, between the syntactic operation o adjunction on the one hand, and the semantic operations o complementation and modiication, on the other. Linear Logic based glue semantics, as developed within the LFG ramework (c. Dalrymple (1999)), allows or lexible coupling o syntactic and semantic structure. We investigate application o glue semantics to LTAG syntax, using as underlying structure the derived tree, which is more appropriate or principle-based semantics construction. We show how Linear Logic based semantics construction helps to bridge the non-isomorphism between syntactic and semantic operations in LTAG. The glue approach captures non-tree local dependencies in control and modiication structures, and extends to the treatment o scope ambiguity with quantiied NPs and VP modiiers. Finally, glue semantics applies successully to the adjunction-based analysis o long-distance dependencies in LTAG, which diers signiicantly rom the -structure based analysis in LFG. 1 Introduction In this paper we review existing appoaches to semantics construction in LTAG (Lexicalised Tree Adjoining Grammar) based on the notion o derivation (tree)s. We argue that LTAG derivation trees are not appropriate to guide semantic composition, due to a non-isomorphism, in LTAG, between the syntactic operation o adjunction on the one hand, and the semantic operations o complementation and modiication, on the other. Linear Logic based glue semantics, by now the classical approach to semantics construction within the LFG ramework (c. Dalrymple (1999)) allows or lexible coupling o syntactic and semantic structure. We investigate application o glue semantics to LTAG syntax, 1 using as underlying structure the derived tree, which seems more appropriate or principle-based semantics construction. We show how Linear Logic based semantics construction helps to bridge the nonisomorphism between syntactic and semantic operations in LTAG. Glue semantics captures non-tree local dependencies in control and modiication structures, and extends to the treatment o scope ambiguity with quantiied NPs and VP modiiers. Finally, glue semantics applies successully to the adjunction-based analysis o long-distance dependencies in LTAG, which diers signiicantly rom the -structure based analysis in LFG in terms o unctional uncertainty. On a more general perspective, the exercise is instructive in that it elucidates the role that -structure plays in LFG syntax and semantics, and helps clariy the similarities and dierences between the two rameworks. The paper is structured as ollows. In ection 2 we review basic assumptions o the LFG and LTAG rameworks to set the stage or our investigations. ection 3 examines previous approaches to semantics construction in LTAG based on derivation (tree)s, namely hieber and chabes (1990), chabes and hieber (1994), Joshi and Vijay-hanker (1999) and Kallmeyer and Joshi (1999). In We are grateul or valuable comments rom the audiences o the LFG01 conerence and the University o Konstanz, in particular Ron Kaplan, Jose Bayer and Ellen Brandner. Thanks go also to Dick Crouch and Dalrymple or comments on earlier versions o this paper. ome interesting observations could not be given ull justice in this paper, but provided important eedback or the overall conception o this work, which we hope to extend in uture research. This research was partially unded by a BMBF grant to the DFKI project whiteboard (FKZ: 01 IW 002). 1 Hepple (1999) sketches LL-based semantics or D-Trees, to overcome problems aced by categorial semantics in the analysis o quantiication. Muskens (2001) develops a description-based syntax-semantics interace or LTAG, yet with extension to tree descriptions as used in D-Trees. We briely discuss these related approaches in ection 4.7.

3 ection 4 we design LL-based semantics construction or LTAG on the basis o derived trees. In ection 4.1 we design labelling principles or LTAG elementary and derived trees as an interace to LL-based glue semantics. In ection 4.2 these principles are extended and reined to account or non-tree local dependencies and scope constraints, exempliied by modiication structures. ection 4.3 summarises the speciic assumptions introduced or glue semantics rom LTAG derived trees. ection 4.4 shows that glue semantics successully bridges the non-isomorphism between adjunction in syntax and corresponding operations in semantics. In ections 4.5 and 4.6 we consider control and long-distance constructions which, in their syntactic analysis, dier considerably rom the corresponding analyses in LFG. We show that LL-based semantics construction or LTAG straightorwardly extends to these more intricate cases. It is especially in the context o these constructions that dierences and similarities between the two syntactic rameworks emerge most clearly. This is the topic o ection 5. ection 6 concludes. 2 Basic tenets o LFG and LTAG syntax and semantics In LFG syntactic structure is represented in terms o two levels o syntactic description: c-structure and -structure (1). Context-ree P rules with -descriptions and lexical entries deine the unctional correspondence between c- and -structure. ubcategorisation and long-distance dependencies are represented in -structure, via unctional descriptions. The correspondence unction between c- and -structure also accounts or word order variation. LFG semantics is driven by Linear Logic based meaning construction rom -structure, which allows or lexible coupling o syntax and compositional semantics. Lexical entries are associated with so-called meaning constructors. These consist o a glue part, expressions in Linear Logic which reer to -structure nodes, and a meaning part. The (instantiated) meaning constructors contributed by lexical entries are assembled as premises to a Linear Logic meaning derivation, based on the glue part. Following the Curry-Howard isomorphism, a meaning is computed, in parallel, on the meaning side. We assume amiliarity with the glue semantics approach (see Dalrymple (1999), Dalrymple (2001) or more detail). (1) : pred meet ( subj)( obj) pred John NP:( subj)= g VP: subj g: num sg N:g ADV:m ( adj) VP: pers 3 pred : oten V: NP:( obj)= h obj h: num sg { pers 3 } John meets N:h adj m: pred oten tense present passive - Lexical entries with associated meaning constructors Instantiated meaning constructors John N ( pred)= John john : g σ john : σ mary : h σ N ( pred)= λy, x.meet(x, y): h σ (g σ σ) mary : σ λp, x.oten(p (x)) : (g σ σ) (g σ σ) meets V ( pred)= meet λy, x.meet(x, y): ( obj) σ (( subj) σ σ) oten ADV ( pred)= oten λp, x.oten(p (x)): ((adj ) subj) σ (adj )) ((adj ) subj) σ (adj )) Meaning derivation λy, x.meet(x, y) : h (g ) mary : h λx.meet(x, mary) : g λp, x.oten(p (x)) : (g ) (g ) λx.oten(meet(x, mary)) : g john : g oten(meet(john, mary)) :

4 An LTAG grammar (Joshi 1987) consists o a set o lexicalised elementary trees (etrees), which are composed by two operations: substitution and adjunction (2). Elementary trees encode lexical syntactic properties: subcategorisation, agreement, 2 and syntactic variation in terms o tree amilies. The syntactic representation consists o the constituent tree (derived tree) built by substitution and adjunction o elementary trees (see (2), (3.a)), and a derivation tree (3.b), which records the dependencies between elementary trees as established by substitution and adjunction operations in parsing. LTAG semantics is traditionally based on the structure o derivation trees. (2) Elementary trees, substitution and adjunction NP 1 VP NP NP V NP 2 N N meets John ubstitution (3) Derived trees and derivation trees (3.a) NP 1 VP N V NP 2 John meets N VP ADV oten Adjunction VP NP 1 VP N ADV VP John oten V NP 2 meets N (3.b) meet John oten Derivation Tree Derived tree Extended Domains o Locality and Adjunction A central eature o LTAG syntax is strict lexicalisation combined with the concept o Extended Domains o Locality. Besides being strictly lexicalised, elementary trees localise all subcategorised arguments o the lexical head, representing them as substitution or oot nodes o the elementary tree. The joint assumptions o strict lexicalisation and localisation o arguments lead to adjunction as a major operation in syntactic composition. This is already evident in (2). Due to localisation o subject and object NPs in the etree meets, the derived tree or John oten meets can only be obtained by olding in the auxiliary tree or oten into the etree o meets by use o the adjunction operation. Besides optional (or recursive) modiication structures, as in (2), localisation o arguments plays a central role in the analysis o long distance dependencies. In (4) the etree or meets locally encodes a ronted object wh-phrase. In order to derive sentence (4), the etree or thinks must again be olded into the etree o meets via adjunction. Thus, joint with the concept o localisation o arguments in strictly lexicalised elementary trees, the operation o adjunction naturally leads to the concept o extended domains o locality. (4) Whom does Peter think John meets? +wh +wh NP 2 +wh Aux wh does NP 1 VP V NP 2 meet(s) e NP 1 wh VP V wh * thinks NP 2+wh Aux wh whom does NP 1 VP N V wh peter think NP 1 VP N V NP 2 2 ubject to eature constraints on nodes, not shown here. john meets e

5 Derivation trees are not dependency trees Derivation trees record the relations between elementary trees as established by substitution and adjunction operations in tree composition, and are traditionally used as the basis or semantics construction in LTAG. Derivation trees do not in general correspond to well-ormed dependency structures. This was observed by Rambow et al. (1995), and is illustrated below. In the derivation tree (4.a) or sentence (4) the dependence o think upon meet is in act inverted, as evidenced by the correct dependencies displayed in (4.b). One could argue that dependencies established by adjunction could be specially marked to account or such inverted dependencies, but more complex cases prove that this cannot, in general, lead to a well-ormed dependency tree. (5.a) displays the derivation tree or (5). 3 ince claim and seem independently adjoin to the and VP nodes o the etree adore, the derivation tree cannot represent the dependence o seem upon claim, as given in the correct dependency tree (5.b). Note urther that due to the principle o localisation o arguments, the operation o adjunction applies both to modiiers in (3) and to complementation structures such as sentence embedding verbs in (4). 4 As we shall see, this constitutes an additional complication or principle-based semantics construction rom derivation trees. (4) Whom does Peter thinks John meets? (5) picy hotdogs he claims seems to adore. (4.a) meet whom john think peter (4.b) peter think whom 3 emantics in LTAG meet john (5.a) adore hotdog claim seem spicy he (5.b) Peter claim seem adore hotdog 3.1 hieber and chabes: emantics construction with ynchronous TAG hieber and chabes (1990) associate LTAG syntax with a semantic representation in a synchronous TAG extension, where the grammar components are pairs o syntactic and semantic trees. The semantic representation, a tree-like logical orm, is built in parallel with the syntactic derivation, making use o a speciication o links between nodes in the paired tree components. On substitution o a tree t 1 into a substitution node n 1 in the syntactic tree, a parallel substitution takes place o the paired semantic tree t 2 into the node n 2 that n 1 is linked to. Ater substitution, the link being used is removed. This is illustrated or substitution o the trees or and hotdogs into α below. α : NP 2 NP 1 V VP adore NP e R adore F T T NP NP hotdogs T mary T hotdog α : NP 2 hotdogs NP 1 VP V NP adore e F spicy R T T adore mary hotdog Crucial in this approach is the speciication o links between paired syntactic and semantic trees, since they determine the attachment sites or the parallel semantic operations. Note in particular the link between the tree internal and VP nodes o adore to the single root o the associated semantic tree. It is due to these links that the more complex cases o non-isomorphic derivations vs. dependencies as in (5) can be accounted or. The syntactic tree or seems can adjoin to the tree internal VP node o α, triggering a corresponding operation on the linked root node F in the paired semantic tree, which results in the correct scoping o seem over adore. econd, the syntactic tree 3 The example is slightly changed rom Rambow et al. (1995). 4 As well as control constructions, see below.

6 or claim adjoins to the internal node o the resulting syntactic tree in α, leading to adjunction o its associated semantic tree to the root ormula F with relation seem. The reader may veriy that the same result is obtained or alternative derivations with claim being adjoined beore seem. α : VP α V VP* seems F R F* seem NP 2 hotdogs NP 1 VP V VP seems V NP F R F seem R T T adore mary hotdog adore e α NP 1 VP Peter V * claims NP 2 F hotdogs NP 1 VP F R T F Peter V R T F* claim peter R F claims NP 1 VP claim peter seem R T T V VP adore mary hotdog seems V NP adore e However, as pointed out in hieber and chabes (1990), the order o derivations can have an eect on the semantic representation in that dierent orders o substitution o quantiied NPs yields alternative scopings. Parsing must thereore compute all possible syntactic derivation histories, explicitly or implicitly (c. chabes and hieber (1994)) in order to capture ambiguities that are essentially semantic. This is not only problematic conceptually, but in act leads to spurious analyses in cases like (5) where the order o derivations is not distinctive or semantic interpretation. 5 Finally in view o the ollowing discussion it is important to note that in this approach it is the linking o tree internal nodes in (elementary and derived) trees that accounts or cases o non dependencylike derivations, as in (5). This, however, characterises the approach as a hybrid one, in that semantics construction is essentially built on the structure o derived trees, while accounting or scope ambiguities in terms o derivation histories as determined by syntactic analysis. 3.2 Compositional emantics rom Derivation Trees Joshi and Vijay-hanker (1999) propose compositional semantics construction rom derivation trees, ocussing primarily on predicate-argument relations. Elementary trees are associated with tripartite semantic representations. The irst part speciies the main variable o the predication, the second part states the predicate with argument variables, the third part associates variables with argument nodes in the elementary trees chabes and hieber (1994) distinguish modiier-type rom predicative auxiliary trees such as sentence embedding verbs (say, claim), where the oot node corresponds to an argument o the anchor. In contrast to the generally assumed notion o standard derivations, which excludes multiple adjunction to single nodes, chabes and hieber (1994) propose the notion o extended derivations, licensing multiple adjunction to single nodes. Allowing extended derivations or multiple adjunction o modiier-type auxiliary trees can yield alternative scopes in semantics construction, due to alternative derivation histories. For predicative auxiliary trees, however, standard derivation i.e. the constraint against multiple adjunction at single nodes is preserved. coping ambiguities are thereore correctly prohibited with cascaded sentence embeddings driven by adjunction. 6 This association is not made explicit, but could be ormalised by stating pairs o variables and the node addresses o the corresponding arguments in the elementary tree. 7 For reasons o space we can only illustrate some selected entries (see continuation next page)

7 For substitution o NP arguments the binding o variables in the associated semantic representations is straightorward. For adjunction o predicative auxiliary trees (say, think, wonder), however, the derivation structure does not mirror semantic embedding: while in (8) think takes scope over say, this dependency is inverted in the derivation tree (8.a). Joshi and Vijay-hanker (1999) propose a special treatment or predicative auxiliary trees, in that the adjunction node is basically processed as i it were a substitution node, during semantic composition. This allows or correct embedding o say by think, as well as like by wonder in the semantic representation or (8). Yet, as in (5), multiple adjunction o predicative auxiliary trees to distinct nodes o a single elementary tree (here wonder and seem into like) leads to additional complication. While the relative scope o think and say can be correctly determined in that the trees stand in a direct adjunction relation the relative scope o wonder and seem cannot be determined through variable bindings at adjunction nodes: They adjoin to distinct nodes in the verb s elementary tree. In order to derive the correct relative scopes in such conigurations Joshi and Vijay-hanker (1999) impose an ordering constraint or processing multiple (predicative) adjunctions into etrees, in a bottom-up manner: since seem adjoins to a lower node in the etree o like than wonder (node 2.2 vs. ɛ in (8.a)), it is processed irst in semantic composition, thereby taking narrow scope relative to wonder. It seems conceptually problematic to resort to specially designed ordering constraints or the traversal o derivation trees in semantic composition, especially in view o language speciic constituent structure properties which might not correspond to the structure o semantic composition. The analysis does also not explicitly deal with scope ambiguities induced by NP quantiication. (8) wondered who Peter thought John said Bill seemed to like. (8.a) α like 2.1 α Bill 2.2 β seem 2 β say ɛ β wonder 1 α who 1 α John ɛ β think 1 α 1 α Peter Kallmeyer and Joshi: Underspeciied emantics with MR Kallmeyer and Joshi (1999) develop an account or underspeciied semantics construction rom LTAG derivation trees using MR semantics. Very close to the architecture proposed in Joshi and Vijay-hanker (1999) they associate lat semantic representations with elementary trees, now adopting the ramework o Minimal Recursion emantics to deal with scope underspeciication. emantic composition is again determined by the structure o derivation trees. The paper provides an underspeciied analysis o quantiication which accounts or quantiier scope ambiguities. It then ocusses on examples o adjunct scope as in (9), where given the assumption o standard derivation allegedly must adjoin to usually, and is thereore restricted to take wide scope. Yet, given the assumption o standard derivations, scope ambiguities as in (10) can only be derived in terms o distinct derivations, i.e. distinct derivation trees. It is not clear in which way a single underspeciied representation can be constructed rom distinct derivation trees. (9) Pat allegedly usually drives a cadillac. (10) John intentionally knocked twice. 7 NP i NP VP V NP i like e wh: x 3 about: s 7 like(s 7,x 4,x 3) x 3 x 4 NP N Peter named: x 5 about: x 5 peter(x 5) V VP VP seemed about: s 8 seem(s 8,s 9) s 9 NP VP V * thought about: s 3 think(s 3,x 1,s 4) x 1 s 4

8 The paper is not really explicit about the distinction between semantic composition operations or adjunction versus substitution, in particular concerning the distinction between modiier and predicative auxiliary trees. We suppose that cascaded sentence embeddings can be handled along the lines o Joshi and Vijay-hanker (1999) s approach, by special conditions or variable binding on adjunction o predicative auxiliary trees. However, we see similar problems, in Kallmeyer and Joshi s account, to determine the correct embedding structure or multiple auxiliary trees adjoining to distinct nodes in a single elementary tree, as discussed or examples (5) and (8) above. 3.3 Discussion We detailed the characteristics o LTAG syntax, in particular the structure o derivation trees, and the complexities that arise or semantics construction on the basis o LTAG derivation trees. We conclude that LTAG derivation trees do not provide an appropriate structure or principle-based semantics construction. The non-isomorphism between adjunction in syntax and modiication in semantics introduces considerable complexity in semantics construction rom derivation trees. The principle o extended domains o locality, in conjunction with the adjunction operation, yields semantically inappropriate dependencies in derivation trees, as these are imposed by purely syntactic operations in the composition o strictly lexicalised elementary trees. We thereore set out to investigate semantics construction in LTAG on the basis o the derived tree, which we consider more appropriate to guide principle-based meaning composition. We apply the ramework o Linear Logic based glue semantics, which allows or considerable lexibility in the coupling o syntactic and semantic structure, while still remaining compositional in meaning construction. 4 Glue emantics or LTAG In applying glue semantics to LTAG we (i) deine semantics on the basis o derived trees, which seems more appropriate or principle-based semantics construction. (ii), the loose coupling o syntactic and semantic structures with glue allows us to bridge the gap imposed by the aorementioned nonisomorphism in LTAG. (iii), we show that the glue approach captures non-tree local dependencies in modiier and control constructions. Finally, (iv) we propose a glue-based analysis o long-distance constructions, which in LTAG are driven by tree adjunction as opposed to the -structure based analysis in LFG with unctional uncertainty. 4.1 Meaning constructors or LTAG initial elementary trees To drive LL-based semantics construction, we need to associate meaning constructors with elementary trees, the lexical units o an LTAG grammar. As an interace to glue semantics we deine principles or labelling nodes in elementary and derived trees with variables. These variables are reerred to in the glue part o the associated lexical meaning constructors, and guide meaning composition. Tree Labelling Principle I, to be stepwise reined along the way, labels argument and root nodes in LTAG initial trees with atomic eatures L t and L b, which we will call upper and lower labels: Tree Labelling Principle I Assign variables, g, h V AR to top/bottom labels L t /L b o nodes n in LTAG initial etrees α Root nodes root(α) : L b (root(α)) = x, x a new variable rom V AR Argument nodes arg i (α) : L t (arg i (α)) = x, x a new variable rom V AR (11.a) displays sample etrees with associated meaning constructors. For John meets we obtain the labelled derived tree (11.b). On substitution, eature bundles on substitution nodes and inserted root nodes are unioned, the resulting nodes display both upper and lower labels L t and L b.

9 (11.a) Labelled elementary trees (11.b) Derived tree with associated meaning constructors with assembled meaning constructors : L b = : L b = NP 1 : Lt= g VP V NP 2 : meets Lt= h h (g ) : λy, x.meet(x, y) NP: L b = q N John q : john NP: L b = m N m : mary NP 1 : Lt= g L b = q VP N V NP 2 Lt= h L b = m John meets N q : john m : mary h (g ) : λy, x.meet(x, y) The meaning constructors o the etrees used in tree composition are assembled (11.b), but in their present orm do not yield a successul proo in meaning derivation, since the variables in the glue parts are not connected. The missing equalities between variables are determined by tree composition, along the ollowing lines. Variable Equations in Tree Composition ubstitution: when substituting β into α at node n α, add equation: L t (n α ) = L b (root(β)) Adjunction: see below Using this inormation about variable equations, we could either resolve the equation system globally, or else trigger systematic variable substitutions in the set o assembled meaning constructors. We choose the latter option here, by adopting the ollowing convention: or all substitution nodes n, and all assembled meaning constructors mcs, we replace (all occurrences o) the lower label variable L b (n) = t by n s upper label variable L t (n) = t. In (11.b), this triggers the substitutions q g and m h. Variable ubstitution in (glue part o) meaning constructors mcs n mcs.((l b (n) = t L t (n) = t mcst) mcst t ) LL-derivations or meaning construction With this in place we obtain the set o meaning constructors in (12) which yields a successul proo o the meaning associated with the tree s root variable, based on the Curry-Howard isomorphism. (12) g : john h : mary h (g ) : λy, x.meet(x, y) g : john h (g ) : λy, x.meet(x, y) h : mary g : λx.meet(x, mary) : meet(john, mary) 4.2 Non-tree local dependencies I: VP modiication Up to now we were only looking at constructions involving tree-local dependencies identiied by argument substitution nodes. We now turn to constructions involving non-tree local dependencies, i.e. dependencies which are not identiied by tree-local nodes in elementary trees. Constructions that all into this class are modiiers and control constructions. Let us irst consider a VP modiying adverb like oten. In LTAG, it is represented as an auxiliary tree that adjoins to VP, as displayed in (2). We extend our tree labelling principle to modiier-type auxiliary trees, assigning identical bottom and top labels to root and oot nodes: L b (root(β)) = L t (oot(β)) = x, x a resh variable rom the set o variables V AR.

10 For oten, we obtain a labelled tree β as in (13). The meaning constructor or oten, as a VP modiying adverbial, should consume and produce a VP meaning, which is characterised by consuming the subject s glue variable L t (NP 1α ) o the tree α that β adjoins to, to produce the glue variable L b (root(α)) o α s root node, as sketched below. But neither o these is local to the auxiliary tree β (oten), and can thereore not be reerred to in its associated meaning constructor. (13) : VP: : L b = L b = i L t= Lt= g Lt= g Lt= i NP NP 1: VP ADV VP*: 1: VP: L b = q L b = i L b = q Lt= h oten Lt= i N ADV VP: N V NP 2: L b = m Lt= h John meets N John oten V NP 2: L b = m h (g ) : λy, x.meet(x, y) meets λp, x.oten(p (x)) : (L t(np 1α) L b (root(α))) (L t(np 1α) L b (root(α))) Labelling tree internal nodes: head projections To capture such non-tree local dependencies, we revise our Tree Labelling Principle in two ways: (i) instead o root nodes, we label the lexical anchor (head) node with some variable x. (ii) this anchor node label is projected to all non-labelled tree internal nodes, introducing a chain o variables with intermediate projection labels x, x 1, x 2,..., x, as seen in (14). 8 Yet, given LTAG s concept o lexicalised elementary trees, with direct encoding o subcategorised argments, we will keep these projection variables distinct, triggering variable substitutions only locally, i.e. at local adjunction nodes, as opposed to uniication o -structure nodes in head projection chains in LFG. Tree Labelling Principle II Anchor nodes: L b (anchor(α)) = x, x new variable rom V AR Argument nodes arg N (α): L t (arg N (α)) = x, x new variable rom V AR Modiier-type auxiliary trees β: L b (root(β)) = L t (oot(β)) = x, x new variable rom V AR Projecting anchor node variable to all non-labelled tree internal nodes, introducing intermediate projection variables x x 1 x 2... x On adjunction o some auxiliary tree β to a node n in α, n s label eatures are split: the top label eature L t o n is assigned as the top label eature L t o root(β), and the bottom label eature L b o n is assigned as the bottom label eature L b o β s oot node. We obtain the derived tree in (14). : (14) : NP 1: g V: 1 meets VP: 1 NP 2: h h (g ):λy, x.meet(x, y) VP: i k i ADV: VP*: k oten 1 John oten V: λp, x.oten(p (x)) : (L t (NP 1α ) i) (L t (NP 1α ) i) meets N g NP 1: q VP: i k i N ADV: VP: k 1 h NP 2: m 8 In the ollowing we omit the eature names L t and L b, to avoid conusion with the upper and lower bounds o projected labels x x 1 x 2... x. N

11 We now extend the conditions or equating variables in tree composition to the case o adjunction, and generalise the conditions or variable substitutions in meaning constructors to account or both substitution and adjunction. With these extensions and the labelling o tree internal nodes by projected anchor variables, the variable i in the meaning constructor or oten will ater variable substitution successully reer to the sentence s root node variable. 9 Variable Equations in Tree Composition ubstitution: when substituting β into α at node n α, add equation: L t (n α )=L b (root(β)) Adjunction: when adjoining β to α at node n α, add equation: L t (n α ) = L b (root(β)) Variable ubstitutions in meaning constructors mcs using set o equations EQ (inal) n ( (L t (n) = x and x = y EQ) mcs : mcsy x) Labelling arguments as arguments o lexical heads With these changes it is still not possible to reer to the non-tree local variable or the subject L t (NP 1α ) in the meaning constructor o oten. We thereore urther revise Tree Labelling Principle (II) by encoding argument nodes in elementary trees as arguments o their lexical head, using the local tree s anchor label and a grammatical unction identiier. Rather than using identiiers like NP 1, NP 2, etc., which in LTAG encode grammatical unctions, we make use o grammatical unction labels subj, obj, comp, etc., similar to those used in LFG. 10 o, i is the label o the lexical anchor, the subject node NP 1 will be labelled :subj, the object node NP 2 :obj. As a result, the labelled trees look more LFG-like, but do not introduce additional linguistic assumptions into LTAG: Note that indices on NPs, such as NP 1, NP 2 do in act encode grammatical unctions in LTAG. This is brought out by looking at pairs o trees in the passive relation change. 11 Lexical heads are identiied as primary anchors o elementary lexicalised trees, and head projection lines rom these anchors emerge naturally as the complement o the set o nodes which are marked as argument or paired root/oot nodes o the local tree. Finally, LTAG s principle o extended domains o locality requires all arguments o a lexical item to be encoded within its elementary tree. Encoding arguments as arguments o their head is thus in line with LTAG s basic assumptions. Tree Labelling Principle III (clause (ii)) (revised rom (II)) Argument nodes: L t (arg N (α)) = L b (anchor(α)):gf N, where GF N is the grammatical unction corresponding to arg N Elementary trees and meaning constructors or John oten sees are now revised according to the new conventions (see (15)). The meaning constructor or oten reers to the non-tree local subject NP variable in terms o the local variable i: (i:subj i) (i:subj i). In tree composition we establish variable equations, which trigger variable substitutions in the assembled set o meaning constructors (16.a). The resulting set o (resolved) meaning constructors is (16.b). The glue ormula contributed by oten does now reer to the verb s subject node label as :subj and to the root node label as. 9 We will display the analysis in (15) and (16), ater a urther revision to our Tree Labelling Principle. 10 ee also extensions in FTAG, which makes use o grammatical unction eatures like subj, obj, etc. 11 The subject NP in the active tree is marked NP 1, the object NP 2. In the passive, the logical object is marked as the passive subject, again by index 1. The demoted logical subject, i present, is labelled NP 0. NP 1 VP NP 1 VP V NP 2 Aux VP loves V PP loved P by NP 0

12 (15) : NP 1: :subj VP: V: 1 1 NP 2: :obj meets :obj ( :subj ) : λy, x.meet(x, y) (16.a) john : q mary : m λy, x.meet(x, y): :obj ( :subj ) λp, x.oten(p (x)) : (i :subj i) (i :subj i) VP: k i ADV: VP*: k oten (16.b) john : :subj mary : :obj λy, x.meet(x, y): :obj ( :subj ) λp, x.oten(p (x)) : ( :subj ) ( :subj ) i λp, x.oten(p (x)):(i:subj i) (i:subj i) : :subj NP 1: q VP: i k i ubstitutions: q :subj, m :obj, i N ADV: VP: k 1 1 : obj John oten V: NP 2: m However, the set o meaning premises (16.b) does not yield a successul meaning derivation. The resource :obj (rom ) can be consumed by :obj ( :subj ), to produce the resource :subj corresponding to the VP meaning λx.meet(x, mary). But this latter resource cannot be consumed by the meaning constructor or oten, which expects a VP constructor ( :subj ). This latter glue ormula resulted rom variable substitutions in the meaning constructor o oten, which can only reer to its local variable i in root and oot node. In tree composition we established equality o this variable with the connecting adjunction node s top label Labelling arguments as arguments o local head projection The LFG instructed reader will now suggest that we give up the distinction between head projection variables x... x, by triggering global variable substitutions on head projection labels. However, we want to introduce as little additional assumptions in LTAG-based semantics construction as needed. We will show that we can continue to restrict ourselves to local variable substitutions at adjunction nodes, by a weaker extension o our labelling principle, which will encode the attachment o arguments as attachments to the respective level o the local head projection. And as we shall see later, in the discussion o long-distance dependencies, it is by this move as opposed to global variable substitutions in head projections that we can correctly deine scope constraints in longdistance constructions, given LTAG s principle o extended domains o locality. Our inal version o the Tree Labelling Principle does now encode argument nodes arg N (α) as arguments o their local head projection, by reerring to the projection label o the argument s mother node (a tree internal node). That is, the projection index y in labels x y :GF reers to the projection variable o the argument s mother (head projection) node. Tree Labelling Principle (Final Version) Anchor nodes: L b (anchor(α)) = x, x new variable rom V AR Modiier-type auxiliary trees: L b (root(β)) = L t (oot(β)) = x, x new variable rom V AR Projecting anchor node variable to all non-labelled tree internal nodes, introducing intermediate projection variables x x 1 x 2... x Argument nodes arg N (α): L t (arg N (α)) = L b (mother(arg N (α))): GF N meets N

13 The labelling o elementary trees diers only slightly rom the previous version, the subject NP o meets being labelled :subj as beore, since it attaches to the highest projection o the elementary tree, whereas the object NP attaches to projection level 1, and is thus labelled 1 :obj. With variable substitutions q :subj, m 1 :obj, and i the premises to meaning construction in (18.b) yield a successul proo in meaning derivation. (17) : NP 1: :subj VP: V: 1 1 NP 2: 1 :obj meets 1 :obj ( :subj ) : λy, x.meet(x, y) NP 1: : :subj q VP: i k i N ADV: VP: k 1 1 1: obj John oten V: NP 2: m (18.a) john : q mary : m (18.b) john : :subj mary : 1 :obj λy, x.meet(x, y): 1 :obj ( :subj ) λp, x.oten(p (x)) : (i :subj i) (i :subj i) λy, x.meet(x, y): 1 :obj ( :subj ) λp, x.oten(p (x)) : ( :subj ) ( :subj ) meets N λy, x.meet(x, y): 1:obj ( :subj ) mary: 1:obj λx.meet(x, mary): :subj λp, x.oten(p (x)):( :subj ) ( :subj ) λx.oten(meet(x, mary)): :subj john: :subj Deriving scope ambiguities oten(meet(john, mary)): With our Tree Labelling Principle in place, we will now illustrate that LTAG semantics construction based on derived trees accounts or scope ambiguities induced by modiiers and NP quantiiers. By identiying root and oot labels o modiiers, and due to local variable substitutions in tree composition, we account or the scoping behaviour o modiiers to take scope over other modiiers within their clausal projection. In particular, proper labelling conditions o argument nodes ensures that the meaning constructor s non-local variable stays local to the clause nucleus. 12 In (19) we consider a case o modiier scope ambiguity, with one o the adverbs adjoining to, the other to VP. 13 Ater substitutions, the set o premises allows or derivation o alternative meanings, by either irst consuming the meaning constructor or twice, and then intentionally, or vice versa. That is, rom the single derived tree we obtain ambiguous semantic analyses, with alternative modiier scopes: intentionally(john, twice(call(john, mary))), and twice(intentionally(john, call(john, mary))). 12 ee ection 4.4 or the analysis o predicative auxiliary trees. 13 Example (19) brings us to the special case where modiier adjunction applies to the root node, which doesn t speciy an upper label. For this case we need to reine the conditions or adjunction in tree composition by adopting clause (i), which provides an upper label or modiier adjunction to root nodes. Clause (ii) can then apply as beore. Clause (ii) will also cover predicative auxiliary trees (see below). (i) On adjunction o a modiier auxiliary tree β to a node n in α, i n does not speciy an upper label, we instantiate an upper label, assigning it the value o n s lower label. (ii) On adjunction o some auxiliary tree β to a node n in α, n s label eatures are split: the top label eature L t o n, i instantiated, is assigned as the top label eature L t o root(β), and the bottom label eature L b o n is assigned as the bottom label eature L b o β s oot node. We urther assume that or meaning derivation it is the lower label o the sentence s root node that constitutes the target o the proo in meaning derivation (resp. the variable it is substituted with).

14 (19) john : q mary : m λy, x.call(x, y): 1 :obj ( :subj ) λp, x.intent(x, P (x)): (j:subj j) (j:subj j) λp, x.twice(p (x)): (i:subj i) (i:subj i) john : :subj mary : 1 :obj λy, x.call(x, y): 1 :obj ( :subj ) λp, x.twice(p (x)): ( :subj ) ( :subj ) λp, x.intent(x, P (x)): ( :subj ) ( :subj ) : j n ADV: j : n intentionally NP :subj 1: ubstitutions: q :subj, m 1 :obj, i, j q VP: V: John 1 called i i k VP: ADV: 1 k 1: obj NP 2: m twice Yet, or let and right adjoining VP modiiers as in John intentionally called twice, assuming standard derivation, we obtain alternative derived trees with identical, ambiguous meanings. We thereore adopt chabes and hieber (1994) s extended derivations or modiier-type auxiliary trees, allowing multiple adjunction to single nodes. Assuming urther that simultaneous let and right adjunction to a single node produces a shared adjunction root node, as described in (chabes and Waters 1995), we obtain a single derived tree, which yields the same ambiguity as in (19). Finally, we need to account or scope restrictions. In Crouch and van Genabith (1999) scope restrictions are deined by scope constaints, which restrict the order o derivations in meaning construction, to yield corresponding scope meanings. cope constraints reer to variables in glue expressions. But modiiers attaching to the same clause exhibit identical glue variables (ater equality resolution) they cannot be distinguished in the glue part. This problem could be solved by allowing scope constraints to reer to glue : meaning pairs, i.e. by exploiting the Curry-Howard isomorphism. In John called intentionally twice the governing modiier could introduce a constraint intentionally twice an instruction to consume/apply the glue : meaning pair associated with intentionally beore the one associated with twice, corresponding to wide scope o twice. cope ambiguities with NP quantiiers are accounted or in the deinition o the associated meaning constructors, in line with standard glue semantics (c. Dalrymple (1999)). In (20) we display the entries or quantiied pronominal NPs. For Everyone meets someone, we obtain ater substitutions a set o instantiated meaning constructors which allows or alternative meaning derivations, corresponding to alternative quantiier scopes. (20) NP: g everyone NP h someone λp. x(person(x) P (x)): λp. x(person(x) P (x)): (g X) X (h X) X 4.3 Taking stock Instantiated mcs or Everyone meets someone λp. x(person(x) P (x)):( :subj X) X λp. x(person(x) P (x)):( 1 :obj X) X λy, x.meet(x, y): 1 :obj :subj Let us review the basic assumptions in our design o LL-based semantics construction or LTAG. As in glue semantics applied to LFG, semantics construction is lexicon driven, here by associating meaning constructors with lexicalised elementary trees. While glue semantics in LFG is based on -structure variables in the glue part o meaning constructors reer to instantiated -structure nodes (or their semantic σ projections) glue semantics or LTAG is based on labelled derived trees. The principle or labelling elementary trees was introduced stepwise, in order to introduce as little additions to the LTAG ramework as needed, and to motivate speciic conceptual moves and machinery. Labelling o trees with eatures assigning upper and lower variables is a necessary extension to LTAG i semantics construction is based on derived as opposed to derivation trees, and we have presented evidence or serious shortcomings o the latter approach in ection 3. It

15 is especially the labelling o tree internal nodes by (projected) anchor labels that extends basic assumptions o LTAG, which in its basic orm only talks about root, substitution and adjunction nodes in elementary trees. As we have seen, in semantics construction rom derived trees, reerence to tree internal nodes is crucial or connecting variables in adjunction structures or modiication. Further we have seen that non-tree local dependencies as in the case o VP modiying adverbs (and similarly or control verb constructions, see below) can only be captured by (i) labelling arguments as arguments o their lexical head, and (ii) association with grammatical unction labels (or similar naming conventions). Finally, (iii), the labelling o arguments needs to speciy the head projection level where attachment takes place in order to allow or a principled solution to the scoping o adverbs and as we shall see wh-elements in long-distance constructions. In tree composition we establish equations between variables in upper and lower labels, based on two basic principles or substitution and adjunction. Variable substitutions in meaning constructors are uniquely based on these equations. That is, only those variables that are local to substitution or modiier root nodes play a role in establishing connections between the isolated variables assigned in elementary trees. Variables o intermediate projection nodes which are not touched by adjunction or substitution operations can be saely ignored in meaning construction. We consider head projection labelling as the most crucial addition to LTAG s standard assumptions. It is a necessary extension or semantics construction rom derived trees, and a crucial assumption to account or the most characteristic and diicult aspects o LTAG syntax and its syntax-semantics interace, namely adjunction and its interplay with semantic composition and the deinition o scope. This is evident in the analysis o modiiers and scope ambiguity, and required or long distance dependencies in ection Bridging the syntax-semantics non-isomorphism in LTAG As outlined in ection 2, a main characteristic o LTAG syntax is that due to the principle o extended domains o locality and the existence o long distance phenomena sentential embedding is necessarily driven by adjunction in syntax, that is, by the same syntactic operation that applies to optional (and recursive) modiiers. This results in a non-isomorphism between adjunction in syntax on the one hand, and the semantic operations o complementation vs. modiication on the other. As we saw in discussion o Joshi and Vijay-hanker (1999), this generates inverted embedding structures in derivation trees, as opposed to correct embeddings in derived tree structures, and leads to complications in semantics construction rom derivation trees. One would thereore expect that in our approach, where the syntax-semantics interace is built on the derived tree, the problem o this non-isomorphism is automatically circumvented. But this is not the case. The problem arises in our approach as well, yet in a slightly dierent way. Let us take a look at a sentence embedding verb like think in a simple embedding coniguration (21). entence embedding verbs (and similarly control verbs) are represented as auxiliary trees, yet dier rom modiier-type auxiliary trees in that their oot node corresponds to an argument o the (21) : j : j j NP :subj j 1: j NP 1: p VP: j 1 :subj j VP: j j 1 Peter V: 1 j1:comp : j j1 j1:comp V: *: j thinks NP :subj 1: q VP: 1 think(s) λp, x.think(x, P ): j 1 :comp (j :subj j ) John V: 1 meets 1:obj NP 2: m

16 lexical head. In LTAG this dierence is captured by the distinction between modiier and predicative auxiliary trees. In our Tree Labelling Principle labelling o root and oot nodes with identical variables was restricted to modiier auxiliary trees. The oot node o a predicative auxiliary tree, by contrast as argument node o the lexical anchor alls under the conditions or argument nodes, and is assigned a composed label, consisting o a projection variable and a grammatical unction label. 14 For the predicative auxiliary tree think we thus obtain a labelled elementary tree as displayed in (21). In contrast to modiier trees, root and oot node labels are distinct. In tree composition, the bottom label o the oot node is instantiated with the bottom label o the node to which adjunction applies ( in (21)), in accordance with our principles or assigning top and bottom labels in tree composition. 15 Although the derived tree does not relect any crucial dierence between adjunction and substitution, we need to adjust the conditions or variable equations in tree composition to account or the special aspects o semantic composition with predicative auxiliary trees, as opposed to modiier auxiliary trees. On adjunction o modiiers root and oot node variables need to be equated with the projection variable o the adjunction node, to allow modiiers to take scope within the maximal projection. This we obtained by equating upper and lower labels at the modiier s root node in the derived tree. For predicative auxiliary trees, by contrast, we need to link, i.e. equate, the labels o argument oot node and adjunction node, parallel to the standard case o argument substitution. All other things being equal, then, we account or the non-isomorphism o adjunction in semantics construction through reinement o variable equations in tree composition: We distinguish between adjunction o modiier and predicative auxiliary trees, the latter being deined along the lines o standard cases o argument substitution. Thus, in variable equation, on adjunction o predicative auxiliary trees the oot node plays the role the substitution node n α plays in substitution, and the adjunction node n α plays the role o the root node o the inserted tree in substitution. Variable Equations in Tree Composition (Final Version) ubstitution: When substituting β into α at node n α, add equation: L t (n α )=L b (root(β)) Adjunction: When adjoining β to α at node n α, i β is a modiier auxiliary tree, add equation: L t (n α ) = L b (root(β)) i β is a predicative auxiliary tree, add equation: L t (oot(β)) = L b (n α ) With these adjustments we trigger substitution o by j 1 :comp in the set o meaning constructors (22) assembled or (21). 16 Ater substitution o variables, complex terms such as j 1 :comp:subj are treated as atoms in matching producers and consumers in Linear Logic meaning derivation. (22) ubstitutions: p j :subj, q :subj, m 1 :obj, j 1 :comp peter : p peter : j :subj john : q john : j 1 :comp:subj mary : m mary : 1 :obj λy, x.meet(x, y): 1 :obj ( :subj λy, x.meet(x, y): 1 :obj (j 1 :comp:subj j 1 :comp) ) λp, x.think(x, P ):j λp, x.think(x, P ):j 1 :comp (j :subj j 1 :comp (j :subj j ) ) think(peter, meet(john, mary)) 14 We assume a mapping rom argument oot nodes with discriminating eatures (such as ±in etc.) to corresponding grammatical unctions comp and xcomp. 15 ince the upper label o the adjunction node is not instantiated, the upper label o the root node is instantiated to (see the reined deinition or adjunction in n. 13). The target variable in the meaning proo is L b (root) = j. 16 Note that in this example the order in which substitutions are triggered is critical. In general, though, we can avoid speciication o some canonical order or substitutions by repeatedly applying substitutions to the modiied premises in the same, arbitrarily ixed order, until no more modiications are triggered in a complete run. Alternatively, one could adopt a general resolution algorithm, along the lines o Kaplan and Bresnan (1995).

17 4.5 Non-tree local dependencies II: Control constructions We now show how LL-based semantics construction on the basis o labelled LTAG trees accounts or non-tree local dependencies in control constructions Control constructions in LFG We irst take a quick look at the treatment o control in LFG syntax and LL-based glue semantics. Lexical entries o control verbs speciy the relation between controller and embedded subject in terms o unctional equations. In the resulting -structure the matrix subj o tries is uniied with the subj o the embedded verb, that is, they are instantiated to some unique -structure node (g in (23)). The meaning constructor associated with tries 17 consumes the VP meaning o the embedded verb ( xcomp subj σ ) ( xcomp σ ) to produce the VP meaning o tries, which then consumes the overt matrix subject to produce the sentence meaning (( subj σ ) σ ). Parallel to linear logic derivation based on the glue ormulae the meaning is computed in the associated meaning terms. (23) tries V, ( pred)= try ( subj) ( xcomp) ( subj)= ( xcomp subj) λp, x.try(x, P ): (( xcomp subj σ ) ( xcomp σ )) (( subj σ ) σ ) Mcs or: John tries to meet john : g mary : m λy, x.meet(x, y):m (g h) λp, x.try(x, P ):(g h) (g ) λy, x.meet(x, y) : m (g h) mary : m λx.meet(x, mary) : g h λp, x.try(x, P ) : (g h) (g ) λx.try(x, λx.meet(x, mary)) : g try(john, λx.meet(x, mary)) : john : g Control constructions in LTAG In LTAG syntax control verbs are encoded as predicative auxiliary trees, since they can occur in longdistance dependencies. Ininite verbs embedded under control verbs do not display an overt subject. The corresponding elementary tree thereore encodes a PRO subject node. Due to adjunction, the derivation tree or John tries to meet incorrectly represents the control verb as dependent on the embedded verb, meet. Moreover, since no substitution takes place into the subject argument position o meet the derivation tree does not represent John as an argument o meet. (24) NP 1 VP V to * tries PRO VP V NP 2 meet meet try mary john Glue emantics or LTAG control contructions Applying our Tree Labelling Principle to the predicative auxiliary tree tries (25.a), we label the subject NP and the adjunction oot node * as subj and xcomp arguments, anchored to the respective head projection variables j and j 1. Labelling and meaning constructor or the embedded ininitive etree meet (see (24)) is identical to (17), except or the presence o a subject PRO argument, which is not a substitution node. For John tries to meet we obtain the labelled derived tree (25.b). With variable substitutions in the set o assembled meaning constructors, we obtain a successul proo o the sentence meaning, similar to the meaning derivation in the LFG analysis. 17 We ollow the analysis given in Asudeh (2000) and Asudeh (2001).