Combining representation formalisms for reasoning upon mathematical knowledge

Knowledge in mathematics (definitions, theorems, proofs, etc.) is usually expressed in a way that combines natural language and mathematical expressions (e.g. equations). Using an ontology formalism such as OWL DL is well-suited for formalizing the natural language part, but complex mathematical expressions can be better handled by symbolic computation systems. We examine this representation issue and propose an original extension of OWL DL by call formulas, i.e., formulas from which assertions can be drawn thanks to calls to external functions. Using this formalism makes it possible to classify a mathematical problem defined by its relations to instances and classes and by some mathematical expressions: if a theorem for solving this problem is represented in the knowledge base, it can be retrieved, and thus, the problem can be solved by applying this theorem. We describe an inference algorithm and discuss its properties as well as its limitations. Indeed, the proposed extension, algorithm, and implementation represent a first step towards a combined formalism for representing mathematical knowledge, with some open issues regarding the representation of more complex problems: the resolution of multiscale, multiphysics cases in physics are foreseen.