Advertisement

Towards a Self-Reflective, Context-Aware Semantic Representation of Mathematical Specifications

  • Peter Schodl
  • Arnold Neumaier
  • Kevin Kofler
  • Ferenc Domes
  • Hermann Schichl
Chapter
Part of the Applied Optimization book series (APOP, volume 104)

Abstract

We discuss a framework for the representation and processing of mathematics developed within and for the MoSMathproject. The MoSMathproject aims to create a software system that is able to translate optimization problems from an almost natural language to the algebraic modeling language AMPL. As part of a greater vision (the FMathL project), this framework is designed both to serve the optimization-oriented MoSMathproject, and to provide a basis for the much more general FMathL project. We introduce the semantic memory, a data structure to represent semantic information, a type system to define and assign types to data, and the semantic virtual machine (SVM), a low level, Turing-complete programming system that processes data represented in the semantic memory. Two features that set our approach apart from other frameworks are the possibility to reflect every major part of the system within the system itself, and the emphasis on the context-awareness of mathematics. Arguments are given why this framework appears to be well suited for the representation and processing of arbitrary mathematics. It is discussed which mathematical content the framework is currently able to represent and interface.

Keywords

Turing Machine Semantic Memory Mathematical Content Parse Tree Semantic Mapping 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

Support by the Austrian Science Fund (FWF) under contract number P20631 is gratefully acknowledged.

References

  1. 1.
    Andrews, P.: A Universal Automated Information System for Science and Technology. In: First Workshop on Challenges and Novel Applications for Automated Reasoning, pp. 13–18 (2003)Google Scholar
  2. 2.
    Beasley, J.: OR-Library: Distributing test problems by electronic mail. Journal of the Operational Research Society 41(11), 1069–1072 (1990)Google Scholar
  3. 3.
    Boyer, R., et al.: The QED Manifesto. Automated Deduction–CADE 12, 238–251 (1994)Google Scholar
  4. 4.
    Clark, J., Murata, M., et al.: Relax NG specification – Committee Specification 3 December 2001. Web document (2001). http://www.oasis-open.org/committees/relaxng/spec-20011203.html
  5. 5.
    Covington, M.: A fundamental algorithm for dependency parsing. In: Proceedings of the 39th annual ACM southeast conference, pp. 95–102. Citeseer (2001)Google Scholar
  6. 6.
    Cramer, M., Fisseni, B., Koepke, P., Kühlwein, D., Schröder, B., Veldman, J.: The Naproche Project Controlled Natural Language Proof Checking of Mathematical Texts. Controlled Natural Language pp. 170–186 (2010)Google Scholar
  7. 7.
    Fourer, R., Gay, D., Kernighan, B.: A modeling language for mathematical programming. Management Science 36(5), 519–554 (1990)CrossRefGoogle Scholar
  8. 8.
    Ganter, B., Wille, R.: Formale Begriffsanalyse: Mathematische Grundlagen. Springer-Verlag Berlin Heidelberg New York (1996)Google Scholar
  9. 9.
    Humayoun, M., Raffalli, C.: MathNat – Mathematical Text in a Controlled Natural Language. Special issue: Natural Language Processing and its Applications p. 293 (2010)Google Scholar
  10. 10.
    Jefferson, S., Friedman, D.: A simple reflective interpreter. LISP and symbolic computation 9(2), 181–202 (1996)CrossRefGoogle Scholar
  11. 11.
    Jurafsky, D., Martin, J., Kehler, A., van der Linden, K., Ward, N.: Speech and language processing: An introduction to natural language processing, computational linguistics, and speech recognition, vol. 163. MIT Press (2000)Google Scholar
  12. 12.
    Klarlund, N., Moeller, A., I., S.M.: Meta-DSD. Web document (1999). http://www.brics.dk/DSD/metadsd.html
  13. 13.
    Kofler, K.: A Dynamic Generalized Parser for Common Mathematical Language. PhD thesis (In preparation)Google Scholar
  14. 14.
    Kofler, K., Schodl, P., Neumaier, A.: Limitations in Content MathML. Technical report (2009). http://www.mat.univie.ac.at/~neum/FMathL.html%5C#Related
  15. 15.
    Kofler, K., Schodl, P., Neumaier, A.: Limitations in OpenMath. Technical report (2009). http://www.mat.univie.ac.at/~neum/FMathL.html%5C#Related
  16. 16.
    Lee, D., Chu, W.: Comparative analysis of six XML schema languages. ACM SIGMOD Record 29(3), 76–87 (2000)CrossRefGoogle Scholar
  17. 17.
    Lee, T., Hendler, J., Lassila, O., et al.: The semantic web. Scientific American 284(5), 34–43 (2001)Google Scholar
  18. 18.
    Manola, F., Miller, E., et al.: RDF Primer. Web document (2004). http://www.w3.org/TR/2004/REC-rdf-primer-20040210/
  19. 19.
    McCarthy, J.: A micro-manual for LISP – not the whole truth. ACM SIGPLAN Notices 13(8), 215–216 (1978)CrossRefGoogle Scholar
  20. 20.
    Miller, B.: LaTeXML the manual. Web document (2011). http://dlmf.nist.gov/LaTeXML/manual.pdf
  21. 21.
    Neumaier, A.: Analysis und lineare Algebra. Lecture notes (2008). http://www.mat.univie.ac.at/~neum/FMathL.html%5C#ALA
  22. 22.
    Neumaier, A.: The FMathL mathematical framework. Draft version (2009). http://www.mat.univie.ac.at/~neum/FMathL.html%5C#foundations
  23. 23.
    Neumaier, A., Marginean, F.A.: Models for context logic. Draft version (2010). http://www.mat.univie.ac.at/~neum/FMathL.html%5C#Contextlogic
  24. 24.
    Neumaier, A., Schodl, P.: A Framework for Representing and Processing Arbitrary Mathematics. Proceedings of the International Conference on Knowledge Engineering and Ontology Development pp. 476–479 (2010). An ealier version is available at http://www.mat.univie.ac.at/~schodl/pdfs/IC3K_10.pdf
  25. 25.
    Neumaier, A., Schodl, P.: A semantic virtual machine. Draft version (2011). http://www.mat.univie.ac.at/~neum/FMathL.html%5C#SVM
  26. 26.
    Ranta, A.: Grammatical framework. Journal of Functional Programming 14(02), 145–189 (2004)CrossRefGoogle Scholar
  27. 27.
    Schodl, P.: Foundations for a self-reflective, context-aware semantic representation of mathematical specifications. PhD thesis (2011)Google Scholar
  28. 28.
    Schodl, P., Neumaier, A.: An experimental grammar for German mathematical text. Manuscript (2009). http://www.mat.univie.ac.at/~neum/FMathL.html%5C#ALA
  29. 29.
    Schodl, P., Neumaier, A.: A typesheet for optimization problems in the semantic memory. Web document (2011). Available at http://www.mat.univie.ac.at/~neum/FMathL.html%5C#TypeSystems
  30. 30.
    Schodl, P., Neumaier, A.: A typesheet for types in the semantic memory. Web document (2011). Available at http://www.mat.univie.ac.at/~neum/FMathL.html%5C#TypeSystem
  31. 31.
    Schodl, P., Neumaier, A.: Representing expressions in the semantic memory. Draft version (2011). http://www.mat.univie.ac.at/~neum/FMathL.html%5C#TypeSystem
  32. 32.
    Schodl, P., Neumaier, A.: The FMathL type system. Draft version (2011). http://www.mat.univie.ac.at/~neum/FMathL.html%5C#TypeSystem
  33. 33.
    Shapiro, S.: An introduction to SNePS 3. Conceptual Structures: Logical, Linguistic, and Computational Issues pp. 510–524 (2000)Google Scholar
  34. 34.
    Sutcliffe, G., Suttner, C.: The TPTP problem library. Journal of Automated Reasoning 21(2), 177–203 (1998)CrossRefGoogle Scholar
  35. 35.
    Trybulec, A., Blair, H.: Computer assisted reasoning with Mizar. In: Proceedings of the 9th International Joint Conference on Artificial Intelligence, pp. 26–28. Citeseer (1985)Google Scholar
  36. 36.
    Walsh, T.: A Grand Challenge for Computing Research: a mathematical assistant. In: First Workshop on Challenges and Novel Applications for Automated Reasoning, pp. 33–34 (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Peter Schodl
    • 1
  • Arnold Neumaier
    • 1
  • Kevin Kofler
    • 1
  • Ferenc Domes
    • 1
  • Hermann Schichl
    • 1
  1. 1.Fakultät für MathematikUniversität WienWienAustria

Personalised recommendations