Exploiting the Addressee's Inferential Capabilities in Presenting Mathematical Proofs

Proof presentation systems and, in some more general context, many natural language generation systems suffer from a crucial problem: they present too much information explicitly which the intended audience could more naturally infer from a less detailed text. Moreover, proofs in mathematical textbooks make extensive use of building chains of inferences in specialized notations, which is not sufficiently taken into account by proof presentation systems. Encouraged by these observations, we present a model for presenting mathematical proofs that (1) features the implicit conveyance of information through concise texts, (2) organizes major lines in the proof presentation around focused chains of inferences in a specialized notation, (3) can adapt its output to some of the capabilities of its audience. The methods described in this paper allow us to present proofs of moderately complex size in a quality approaching that of proofs found in mathematical textbooks. 1 Introduction Proof presentation systems and, more generally, many natural language generation systems suffer from a crucial problem: they present too much information explicitly which the intended audience could more naturally infer from a less detailed text. In contrast, mathematical proofs as typically found in textbooks express lines of reasoning in a rather condensed form by leaving out several elementary, but logically necessary inference steps. Moreover, the proofs emphasize conciseness by making extensive use of building chains of inferences in specialized notations, such as series of inequations. However, when presenting a mathematical proof to less trained people, those parts which require increased experience to be understood should be expressed in closer detail. Motivated by these observations, we have developed a model in which we try to mimic the properties of mathematical textbook proofs to a significant extent. Our model also supports more verbose presentations to meet the needs of formally less trained addressees. It 1. features the implicit conveyance of information through concise texts, 2. organizes major lines in the proof presentation around focused chains of inferences in a specialized notation, 3. can adapt its output to some of the capabilities of its audience. The paper is organized as follows: After discussing the role of inferences in the larger context of natural language generation, we briefly describe how the results obtained by a theorem prover are prepared for presentation. Then we introduce our inference model which is particularly dedicated to understanding mathematical proofs, and further motivate and describe our user model. Finally, we illustrate our results …