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In this paper I introduce a formalism for natural language understandingbased on a computational implementation of Discourse RepresentationTheory. The formalism covers a wide variety of semantic phenomena(including scope and lexical ambiguities, anaphora and presupposition),is computationally attractive, and has a genuine inference component. Itcombines a well-established linguistic formalism (DRT) with advancedtechniques to deal with ambiguity (underspecification), and isinnovative in the use of first-order theorem proving techniques.The architecture of the formalism for natural language understandingthat I advocate consists of three levels of processing:underspecification, resolution, andinference. Each of these levels has a distinct function andtherefore employs a different kind of semantic representation. Themappings between these different representations define the interfacesbetween the levels.I show how underspecified semantic representations can be built in acompositional way (for a fragment of English Grammar) using standardtechniques borrowed from the λ-calculus, how inferences can becarried out on discourse representations using a translation tofirst-order logic, and how existing research prototypes (discourseprocessing and spoken-dialogue systems) implement the formalism.