ABSTRACT
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.
- W. Belkhir, N. Ratier, D. D. Nguyen, N. B. T. Nguyen, M. Lenczner, and F. Zamkotsian. 2017. A tool for aided multi-scale model derivation and its application to the simulation of a micro mirror array. In 2017 18th International Conference on Thermal, Mechanical and Multi-Physics Simulation and Experiments in Microelectronics and Microsystems (EuroSimE). IEEE, 1–8.Google Scholar
- W. Belkhir, N. Ratier, D. D. Nguyen, B. Yang, M. Lenczner, F. Zamkotsian, and H. Cirstea. 2015. Towards an automatic tool for multi-scale model derivation illustrated with a micro-mirror array. In 17th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC). IEEE, 47–54.Google Scholar
- R. Brachman, E. Ciccarelli, N. Greenfeld, and M. Yonke. 1978. KL-ONE reference manual. Technical Report. BBN report.Google Scholar
- Ronald J. Brachman and Hector J. Levesque. 2004. Knowledge Representation and Reasoning. Morgan Kaufmann.Google Scholar
- MD Canonica, F Zamkotsian, P Lanzoni, W Noell, and N De Rooij. 2013. The two-dimensional array of 2048 tilting micromirrors for astronomical spectroscopy. Journal of Micromechanics and Microengineering 23, 5 (2013), 055009.Google ScholarCross Ref
- B. Glimm, I. Horrocks, B. Motik, G. Stoilos, and Z. Wang. 2014. HermiT: an OWL 2 reasoner. Journal of automated reasoning 53 (2014), 245–269.Google ScholarDigital Library
- C. Lange. 2013. Ontologies and languages for representing mathematical knowledge on the semantic web. Semantic Web 4, 2 (2013), 119–158.Google ScholarDigital Library
- C. Lutz. 2003. Description Logics with Concrete Domains – A Survey. In Advances in Modal Logics Volume 4. King’s College Publications.Google Scholar
- B. Motik, B. Cuenca Grau, I. Horrocks, Z. Wu, A. Fokoue, and C. Lutz. 2009. OWL 2 web ontology language profiles. W3C recommendation.Google Scholar
- D. D. Nguyen, W. Belkhir, N. Ratier, B. Yang, M. Lenczner, F. Zamkotsian, and H. Cirstea. 2015. A multi-scale model of a micro-mirror array and an automatic model derivation tool. In 16th Int. Conf. on Thermal, Mechanical and Multi-Physics Simulation and Experiments in Microelectronics and Microsystems. IEEE, 1–9.Google Scholar
- C. K. Riesbeck and R. C. Schank. 1989. Inside Case-Based Reasoning. Lawrence Erlbaum Associates, Inc., Hillsdale, New Jersey.Google ScholarDigital Library
Index Terms
- Combining representation formalisms for reasoning upon mathematical knowledge
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