On the Relationship between Structure and Reference in Mathematical Discourse

We study the relationship between the structure of discourse and the use of referring expressions in the mathematical domain. We address linguistic, algorithmic as well as representation issues. We show how referential expressions refer to mathematical statements and how a knowledge intensive approach, domain reasoning with the use of proof plans, are used for discourse understanding. We propose to represent discourse plans as underspeciied discourse representation structures being selected and instantiated during discourse processing. Our main emphasis is on the handling of abstract discourse entities. 1 Motivation We have the following practical application in mind: the automatic veriication of mathematical textbook proofs. Imagine a program that understands mathematical discourse. Such a device reads proofs, say mathematical arguments taken from textbooks on elementary mathematics, and is then able to communicate its knowledge about what it has read and analyzed. It answers questions about the proof accurately and is capable to identify gaps or aws in the argumentation line. The proof system ooers a high-level analysis of the proof, including the ability to reorganize the proof structure, as well as a technical low-level access to details of the proof. We believe that building such a proof under-stander is a feasible task, and certainly much easier than building systems that understand everyday discourse. This is because textbook proofs are a highly structured form of discourse: a proof per cases consists of at least two cases where each case might be a complex mathematical argument itself; a proof by induction should consist of at least three segments: base case, induction hypothesis and induction step. Understanding textbook proofs consists of extracting the structural and logical information which is contained in the proof. This includes the identi-cation of (i) the proof method being applied; (ii) sequences of sentences that constitute subproofs; (iii) the contribution of each of the proof segments to the overall goal to decrease the proof obligations ; (iv) the scope and quantiication of the variables ; (v) the logical/structural relations between sentences/segments. Being able to capture how mathematicians communicate their arguments is a prerequisite for argument veriication. Therefore, from the proof understanding point of view, it is clear that we cannot check the mathematical argument incrementally, i.e., sentence by sentence. Contrarily, a good model of segmentation is essential. From the linguistic point of view, being able to segment a text ooers a more accurate treatment of anaphora and ellip-sis. Beside the fact that …