An algorithm for generating the possible quantitier scopings for a sentence, in order of preference, is outlined. The scoping assigned to a quanttrier is determined by its interactions with other quanttriers, toodais, negation, and certain syntacticconstituent boundaries. When a potential scoping is logically equivalent to another, the less preferred one is discarded.

this paper, we outline a theory of linguistic structure which talks about talking about trees; we call this theory Description theory (D-theory). While important issues must be resolved before a complete picture of D-theory emerges (and also before we can build programs which utilize it), we believe that this theory will ultimately provide a framework for explaining the syntax and semantics of natural language in a manner which is intrinsically computational. This paper will focus primarily on one set of motivations for this theory, those engendered by attempts to handle certain syntactic phenomena within the framework of deterministic parsing

• The universals every and each are true generalized quantifier determiners: (1) every, each: = NP ↑ 3SG /⋄N3SG: λpλqλ...∀x[px → qx...] • NP ↑ schematizes over all NP types raised over functions of the form T |NP, which we can spell out for the determiner case as: (2) a. every, each: = (S/(S\NP3SG))/⋄N3SG: λpλq.∀x[px → qx] b. every, each: = (S\(S/NP)) / ⋄N3SG: λpλq.∀x[px → qx] c. every, each: = ((S\NP)\((S\NP)/NP)) / ⋄N3SG: λpλq.λy.∀x[px → qxy] d. every, each: = (((S\NP)/NP)\(((S\NP)/NP)/NP)) / ⋄N3SG: λpλq.λyλz.∀x[px → qyxz] e. etc. 1

Previous work on quantifier scope annotation focuses on scoping sentences with only two quantified noun phrases (NPs), where the quantifiers are restricted to a predefined list. It also ignores negation, modal/logical operators, and other sentential adverbials. We present a comprehensive scope annotation scheme. We annotate the scope interaction between all scopal terms in the sentence from quantifiers to scopal adverbials, without putting any restriction on the number of scopal terms in a sentence. In addition, all NPs, explicitly quantified or not, with no restriction on the type of quantification, are investigated for possible scope interactions. 1

iv guages is determined only in part by the synchronization. The base formalism chosen can offer greater or lesser opportunity for divergence in the derived structures. My choice of a base formalism is motivated directly by research into applications of synchronous TAG-based grammars to two natural language applications: semantic interpretation and natural language translations. I first examine a range of TAG variants that have not previously been studied in this level of detail to determine their computational properties and to develop algorithms that can be used to process them. Original results on the complexity of these formalisms are presented as well as novel algorithms for factorizing grammars to reduce the time required to process them. In Part II, I develop applications of synchronous Limited Delay Tree-Local Multicomponent TAG to semantic interpretation and probabilistic synchronous Tree Insertion Grammar to statistical natural language translation. Contents

We are going to present what we believe to be conclusive evidence for a semantic ambiguity in indefinite noun phrases, over and above any scope ambiguities they may exhibit in appropriate contexts. Sentence (I), for example, contains no opaque verbs, no modals or negation, and no other