Deriving syntax-semantics mappings: node linking, type shifting and scope ambiguity Dennis Ryan Storoshenko Robert Frank Yale University Yale University Department of Linguistics Department of Linguistics 370 Temple Street 370 Temple Street New Haven, CT 06511 New Haven, CT 06511 [email protected] [email protected] Abstract 2 Tree Shapes and Type Shifting In this paper, we introduce a type-shifting We adopt a traditional view of syntactic elemen- operation which provides a principled tary trees as the realization of a single lexical means of describing the derivational links predicate and its grammatical “associates” (cf. the required in Synchronous TAG accounts of Condition on Elementary Tree Minimality and quantification. No longer do links appear on root nodes of predicates on an ad hoc Theta Criterion of Frank (2002)). The corre- basis, rather they are generated as a part of sponding semantic objects are composed from the a type-shifting mechanism over arguments meaning assignments for the lexical anchor to- of the predicate. By introducing to the sys- gether with the meanings associated with non- tem a set of temporal variables, we show projected non-terminals. Substitution nodes are how this operation can also be used to ac- interpreted as typed variables (with types deter- count for the scope interactions of clausal mined by a bijection from syntactic categories to embedding. We then move on to consider semantic types: DP to type e, NP to type e, t , additional cases of multiple clausal embed- h i ding and coordination. CP, TP and VP to type t, etc. We follow Pogodalla (2004) in assuming that such variables are bound by (linear) lambda operators, and take syntactic 1 The Issue substitution of S into T to correspond to (seman- Investigations of the syntax-semantics interface in tic) function application of T to S. For a syntac- Tree Adjoining Grammar, particularly those mak- tic node N targeted for adjoining, we assume that ing use of Synchronous TAG, grapple with the the corresponding node in the semantic represen- limitations imposed by the restrictiveness of tree- tation is embedded beneath an abstracted function or set-local MCTAG. To the degree that they suc- variable (with type α, α where α is the type de- h i cessfully treat the mapping between syntax and termined by the category bijection for N). Ad- semantics in this restricted setting, this provides joining of auxiliary tree A to tree T corresponds to evidence in favor of Joshi’s hypothesis that the application of T to A. We assume that adjoining mild context-sensitivity of TAG is a fundamen- always applies at nodes to which it may; when no tal property of grammar. Nonetheless, the anal- content is added, a semantic identity function is yses that have been put forward are at times ad applied. Some linkages between the syntactic and hoc. One wonders why a certain semantic ob- semantic trees are straightforward: non-projected ject is associated with some piece of syntax, and non-terminals are linked to the lambda operators why certain nodes of the syntactic representations binding their associated variables, while projected are linked to the semantics in one manner as op- nodes in the syntax are linked to lambda operators posed to another. In this paper, we report on binding variables of the appropriate α, α type. h i our first efforts to formulate principles govern- This gives rise to a pairing of the sort in Figure 1 ing STAG pairings, in an effort to provide a more for the transitive verb love. restrictive framework for characterizing STAG- What is less clear is how to establish the non- derivable syntax-semantics mappings. bijective linkages between sites for syntactic at- 10 Proceedings of the 11th International Workshop on Tree Adjoining Grammars and Related Formalisms (TAG+11), pages 10–18, Paris, September 2012. TP age between the syntactic position and the seman- DP 1 T a(t) tic argument slot is basic, as given in Figure 1. i↓ ′ Once these are established, semantic trees can un- T VP 3 e, t y h i dergo an operation that creates multiple linkages λx λy λa 2 1 3 in a systematic fashion. Specifically, we make use DP V′ e e, t x h h ii of an operation similar to argument raising from ti V DP 2 loves Hendriks (1988). In Hendriks’ operation, the type ↓ of an argument is lifted (Partee and Rooth, 1983) loves from its basic e type to the generalized quantifier Figure 1: Syntactic and Semantic Tree Pair for loves e, t ,t type, allowing a raised argument to ef- fectivelyhh i i take scope over the predicate. Applica- t tion of this operation to the internal argument of a DP two-place predicate is shown in (1). x t t* e ∀ b D NP (1) e, e, t : f e, t xb xb h h ii ⇒ h i e, e, t ,t ,t : every N h hhh i i i λy.boy(y) λxe.λT e,t ,t .T (λye.f(y)(x)) hh i i boy The lambda gymnastics involved here are sub- Figure 2: Tree Set for every boy (Schema for all Gen- stantial. We can accomplish a similar effect with eralized Quantifiers) the paired STAG structure in Figure 1 in a sim- pler way, if we allow one of the e-type arguments to be linked to a new functional α, α variable. h i tachment and semantic composition. Originating We represent this linkage as the combination of in Shieber and Schabes (1990), and continuing in two variables under the scope of a single lambda all of the subsequent STAG-based work on scope operator, as shown in Figure 3. We take such we are aware of, it is assumed that the DP posi- set-valued lambda operators to encode the fact tion in, say, the subject of a transitive verb-headed that the arguments must be introduced in a single elementary tree is linked to both the e-type exter- derivational step, via combination with a MCS. nal argument of the predicate and the t-type root In order to ensure that the newly introduced func- of the tree. This dual-linkage is mirrored in non- tional variable Q does not disturb the surround- STAG accounts of quantification, such as the Hole ing semantic combinations, it is crucial that Q be Semantics-based account in Kallmeyer and Joshi type-preserving (i.e., of type α, α for some α). (2003), and subsequent works in that tradition. h i No matter which type of semantic account the an- a(t) a(Q(t)) alyst prefers, it is widely accepted that quantifica- e, t y e, t y h i h i tion requires this simultaneous access to both an λxλyλa λ Q,x λyλa ⇒ { } e e, t x e e, t x argument position and the root of a tree. Deriva- h h ii h h ii tionally, this is of course simply a matter of tree- local MCS combination, but in STAG, there is P P the additional wrinkle of derivational links. Such Figure 3: Schematic Example of Type Lifting in Trees multiple linkages are crucial for the establishment (Shown for Internal Argument) of scope for quantificational DPs, represented as multi-component sets (MCSs) in the semantics, The linkage that has been widely exploited to but not the syntax, as in Figure 2. The variable handle quantifier interpretation fits this pattern: component of this MCS substitutes into the e-type the e-type argument is linked with a t,t function h i argument slot, while the t-recursive scope auxil- variable, which will host its scope, shown in Fig- iary tree adjoins at the semantic predicate’s t root. ure 4. Whereas earlier accounts derived quantifier It is difficult to see what within the verbal pred- scope ambiguity through underspecified ordering icate itself directly motivates a link between the of multiple adjoining at the root t-node of a ver- DP syntactic position and the t adjoining site in bal predicate’s semantics, we derive the same am- the semantics. We will assume that only the link- biguity through underspecified ordering of type 11 TP (2) Someone wants to visit every European DP 1 T a(Q(t)) city. (want > , > want) i↓ ′ ∀ ∀ T VP 3 e, t y The example is derivable using the tree set for h i λ Q,x λy λa { } 2 1 3 the control predicate in Figure 6, along with trees DP V′ e e, t x h h ii for the embedded clause and for the quantifiers, all lexical variants of the trees in Figures 1 and ti V DP 2 loves ↓ 2. The embedded predicate is shown in Figure loves 5, with type-shifting having applied in the order which yields surface scope. Recall though that Figure 4: Syntactic and Semantic Tree Pair for loves (Type-lifted Internal Argument) inverse scope is equally possible, as we place no restriction on the order of the applications of type- shifting. The derivation proceeds as in Figure 7, lifting operations, one for each of the predicate’s with the order of the two instances of type lifting arguments. These iterations of type-shifting take over the arguments of to visit left unspecified. place after the construction of an elementary tree, TP but before the tree enters into any TAG combi- DP 1 T a(P (Q(t))) natory operations. That is, the links (and their i↓ ′ relative scopes) are all in place before any sub- T VP 3 e, t y h i stitution or adjoining operations take place.