Logicographic Symbols: A New Feature in Theorema Bruno Buchberger In: Symbolic Computation - New Horizons (Proceedings of the 4th International Mathematica Symposium, Tokyo Denki University, Chiba Campus, Japan, June 25-27, 2001), Tokyo Denki University Press, 2001, pp. 23-30. (ISBN 4-501-73020-X C3041. Copyright: Tokyo Denki University Press.) ABSTRACT: We introduce the concept of logicographic symbols and describe a first implementation (by the author's PhD student Koji Nakagawa) in the Theorema system. Logicographic symbols are two-dimensional symbols (e.g. graphics or photos) to be used as function or predicate constants in predicate logic that may be designed freely by the user so that the intuitive meaning of the symbols can be expressed by their shape. Logicographic symbols have slots at arbitrary position for the parameters. Logically and internally in Theorema, logicographic symbols are just ordinary constants so that text using logicographic symbols is both formal in the sense of predicate logic and intuitive in the sense of conveying meaning by the graphical information contained in the external appearance of the symbols. Terms and formulae containing logicographic symbols and identifier constants can be nested to arbitrary depth. In contrast to ordinary notations of predicate logic that use parantheses for indicating the nesting of expressions, we introduce two-dimensional colored boxes for supporting easy readability of nested expressions. Also, logicographic symbols may have internal structure: For logico- graphic ymbols composed from smaller constituents one may declare whether the composed meaning is the conjunction or disjunction of the meaning of the constituent symbols. Also, parameters of logico- graphic symbols may be left out with the intention of quantifying (universally or existentially) over the parameter omitted. These systematic variation rules for logicographic symbols make it possible to introduce a huge number of variants of mathematical concepts with a few basic definitions only. We demonstrate the usage and practical advantage of logicographic symbols by the example of the merge sort algorithm and variants of the notion of function and relation in set theory.