Representation of Mathematical Concepts for Inferencing and for Presentation Purposes

Knowledge bases that represent some domain expertise for use in a formal system typically serve one of two major purposes: (1) inferencing in some problem-solving context, or (2) interfacing the system by natural language. Interfacing may be for accessing the system's functionality or for presenting its results. Techniques for addressing the natural language interfacing purpose effectively are generally less understood, especially when connections to the problem-solving perspective have to be taken into account more deeply. Addressing this issue in a principled way for the domain of mathematics, we are currently developing an integrated knowledge base as part of the system ΩMEGA, which serves the purposes of reasoning and presentation in a coordinated way at an ambitious level of capabilities. We formulate and compare representation demands from both perspectives, sketch the current state of development of the mathematical knowledge base of ΩMEGA and the linguistic knowledge base of the attached proof explanation system P.rex, and outline our plans for extending their capabilities. Building representation facilities that can satisfy demands originating from heterogeneous purposes is a crucial prerequisite for increasing the still mediocre quality in illustrating the results of formal inference systems by natural language presentations.