OpenMath: compositionality achieved at last

As a language for exchanging computer-"understandable" representations of mathematical formulas and concepts, OpenMath has a fairly standard syntactic structure for most of the mathematical notions that have so far been formalized in its "Content Dictionaries." The syntax for most operators is closely related to the well-known and decades-old LISP prefix notation for function application.For syntactic representations of a particular large class of mathematical operators, however, OpenMath takes a very unusual approach when compared with any of the existing Computer Algebra systems, say. This class contains integration and differentiation operators, sums and products, and other generalized quantifiers.In our paper we argue that OpenMath's novel approach to representing generalized quantifiers is superior to the classic representations. In particular, we show that this unusual feature of OpenMath has been designed to adhere to the Compositionality Principle[10], a design principle that classic representations, including older versions of OpenMath, have been violating[12].Having thus shown the importance of compositionality for the design of modern OpenMath, we then proceed to show that the Compositionality Principle is a fundamental research instrument in the Formal Semantics branch of linguistics[9], and argue that, like the use of this particular guiding principle for improvements in the design of a representation for generalized quantifiers, the study of the underlying structure of natural language as discovered by various branches of linguistics can provide many more suggestions for further improvements to OpenMath and to its sibling, MathML. We give some examples to substantiate our claim.