Analysis of Mixed Natural and Symbolic Input in Mathematical Dialogs

Analysis of Mixed Natural and Symbolic Language Input in Mathematical Dialogs Magdalena Wolska Ivana Kruijff-Korbayov´a Fachrichtung Computerlinguistik Universit¨at des Saarlandes, Postfach 15 11 50 66041 Saarbr¨ucken, Germany � magda,korbay � @coli.uni-sb.de Abstract Discourse in formal domains, such as mathemat- ics, is characterized by a mixture of telegraphic nat- ural language and embedded (semi-)formal sym- bolic mathematical expressions. We present lan- guage phenomena observed in a corpus of dialogs with a simulated tutorial system for proving theo- rems as evidence for the need for deep syntactic and semantic analysis. We propose an approach to input understanding in this setting. Our goal is a uniform analysis of inputs of different degree of verbaliza- tion: ranging from symbolic alone to fully worded mathematical expressions. 1 Introduction Our goal is to develop a language understanding module for a flexible dialog system tutoring math- ematical problem solving, in particular, theorem proving (Benzm¨uller et al., 2003a).1 As empirical findings in the area of intelligent tutoring show, flex- ible natural language dialog supports active learn- ing (Moore, 1993). However, little is known about the use of natural language in dialog setting in for- mal domains, such as mathematics, due to the lack of empirical data. To fill this gap, we collected a corpus of dialogs with a simulated tutorial dialog system for teaching proofs in naive set theory. An investigation of the corpus reveals various phenomena that present challenges for such input understanding techniques as shallow syntactic anal- ysis combined with keyword spotting, or statistical methods, e.g., Latent Semantic Analysis, which are commonly employed in (tutorial) dialog systems. The prominent characteristics of the language in our corpus include: (i) tight interleaving of natural and symbolic language, (ii) varying degree of natural language verbalization of the formal mathematical 1This work is carried out within the DIALOG project: a col- laboration between the Computer Science and Computational Linguistics departments of the Saarland University, within the Collaborative Research Center on Resource-Adaptive Cognitive Processes, SFB 378 (www.coli.uni-sb.de/ sfb378). content, and (iii) informal and/or imprecise refer- ence to mathematical concepts and relations. These phenomena motivate the need for deep syntactic and semantic analysis in order to ensure correct mapping of the surface input to the under- lying proof representation. An additional method- ological desideratum is to provide a uniform treat- ment of the different degrees of verbalization of the mathematical content. By designing one grammar which allows a uniform treatment of the linguistic content on a par with the mathematical content, one can aim at achieving a consistent analysis void of example-based heuristics. We present such an ap- proach to analysis here. The paper is organized as follows: In Section 2, we summarize relevant existing approaches to in- put analysis in (tutorial) dialog systems on the one hand and analysis of mathematical discourse on the other. Their shortcomings with respect to our set- ting become clear in Section 3 where we show ex- amples of language phenomena from our dialogs. In Section 4, we propose an analysis methodology that allows us to …

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