Checking textbook proofs

discourse entities have been studied extensively by Asher [2]. Asherdi ers between fact anaphora, event anaphora, concept anaphora and proposi-tion anaphora. For our domain, only the latter is of interest. The DRS construc-tion process we described begs some similarities to the one described in [2]. Amajor di erence is that, due to our domain, (i) we can assume the existence ofdiscourse plans that establish frames in which discourses must take place; (ii)we have only one major discourse relation, that of logical consequence, and wecan establish the relation by using automated theorem provers.The inverse of our task, translating formal proofs into natural languageproofs, is described in [7]. Facing the problem that proofs generated by theoremprovers are unstructured, tediously long and therefore unreadable, a readable,structured and short proof (omitting all the low-level details) has to be generated.Please note that being able to `understand' textbook proofs does not meanto understand the way how mathematicians reason when searching for proofs. Itallows only an understanding of how mathematicians communicate their proofsand how they build one argument on another in order to have a convincingcomplex argument structure supporting the truth for some assertion. Similarto [5], we argue that Logic is not enough to understand textbook proofs. Toverify mathematical proofs, the argumentation line of the proof author has to befollowed, which requires a high-level strategic proof understanding. The formalproof does only provide a low-level inference-based view and the informal proofto be veri ed is far away from that level of detail. Proof plans allow to capturemathematical argumentation techniques.5 ConclusionTextbook proofs represent a structured discourse of (the important and inter-esting) argumentation steps supporting the truth of some assertion, and aretherefore an ideal domain for discourse understanding. The domain has a richset of well-de ned mathematical concepts and discourse plans. Central to theissue of automatically checking textbook proofs is to describe the formalizationprocess as translation from informal proofs to formal ones. In the Automathand Mizar projects, the author of the formal proof is considered to perform thistranslation. In our approach, the computer should perform this task.We developed a three phase model to textbook proof understanding and ver-i cation: parsing, structuring and re ning. For representing textbook proofs andproof plans, we proposed to use augmented discourse representation structures.We think that an extended DRT formalism is better suited for the process of for-malizingmathematics than earlier languages proposed for this purpose [14, 15, 9]because it allows to handle phenomena that occur in natural language proofs.Being able to process textbook proofs allows us to close the gap betweenthe language of mathematicians and the language of proofs systems. Reaching this goal will enable us to build AI systems that really assist mathematicians.The research topic we identi ed might boost progress in both Natural LanguageProcessing and Automated Reasoning.References1. P. W. Abrahams. Machine veri cation of mathematical proofs. PhD thesis, MIT,1963.2. N. Asher. Reference to abstract objects in discourse. Kluwer Academic Publishers,1993.3. P. Blackburn and J. Bos. Representation and inference for natural language.Draft, 1997.4. D. G. Bobrow. 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