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Towards Concept Analysis in Categories: Limit Inferior as Algebra, Limit Superior as Coalgebra

Authors Toshiki Kataoka, Dusko Pavlovic

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Toshiki Kataoka
Dusko Pavlovic

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Toshiki Kataoka and Dusko Pavlovic. Towards Concept Analysis in Categories: Limit Inferior as Algebra, Limit Superior as Coalgebra. In 6th Conference on Algebra and Coalgebra in Computer Science (CALCO 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 35, pp. 130-155, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015) https://doi.org/10.4230/LIPIcs.CALCO.2015.130

Abstract

While computer programs and logical theories begin by declaring the concepts of interest, be it as data types or as predicates, network computation does not allow such global declarations, and requires concept mining and concept analysis to extract shared semantics for different network nodes. Powerful semantic analysis systems have been the drivers of nearly all paradigm shifts on the web. In categorical terms, most of them can be described as bicompletions of enriched matrices, generalizing the Dedekind-MacNeille-style completions from posets to suitably enriched categories. Yet it has been well known for more than 40 years that ordinary categories themselves in general do not permit such completions. Armed with this new semantical view of Dedekind-MacNeille completions, and of matrix bicompletions, we take another look at this ancient mystery. It turns out that simple categorical versions of the limit superior and limit inferior operations characterize a general notion of Dedekind-MacNeille completion, that seems to be appropriate for ordinary categories, and boils down to the more familiar enriched versions when the limits inferior and superior coincide. This explains away the apparent gap among the completions of ordinary categories, and broadens the path towards categorical concept mining and analysis, opened in previous work.

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Keywords
  • concept analysis
  • semantic indexing
  • category
  • completion
  • algebra

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References

  1. Yossi Azar, Amos Fiat, Anna Karlin, Frank McSherry, and Jared Saia. Spectral analysis of data. In Proceedings of the thirty-third annual ACM Symposium on Theory of Computing, STOC'01, pages 619-626, New York, NY, USA, 2001. ACM. Google Scholar
  2. Bernhard Banaschewski and Gunter Bruns. Categorical characterization of the MacNeille completion. Archiv der Mathematik, 18(4):369-377, September 1967. Google Scholar
  3. Michael Barr and Charles Wells. Toposes, Triples, and Theories. Number 278 in Grundlehren der mathematischen Wissenschaften. Springer-Verlag, 1985. Google Scholar
  4. Claudio Carpineto and Giovanni Romano. Concept Data Analysis: Theory and Applications. John Wiley & Sons, 2004. Google Scholar
  5. Richard O. Duda, Peter E. Hart, and David G. Stork. Pattern Classification. Wiley-Interscience, 2000. Google Scholar
  6. Bernhard Ganter, Gerd Stumme, and Rudolf Wille, editors. Formal Concept Analysis, Foundations and Applications, volume 3626 of Lecture Notes in Computer Science. Springer, 2005. Google Scholar
  7. Bernhard Ganter and Rudolf Wille. Formal Concept Analysis: Mathematical Foundations. Springer, Berlin/Heidelberg, 1999. Google Scholar
  8. Peter Gärdenfors. The Geometry of Meaning: Semantics Based on Conceptual Spaces. MIT Press, 2014. Google Scholar
  9. Alexander Grothendieck. Revêtements étales et groupe fondamental (SGA 1), volume 224 of Lecture notes in mathematics. Springer-Verlag, 1971. Google Scholar
  10. John R. Isbell. Small subcategories and completeness. Mathematical Systems Theory, 2(1):27-50, 1968. Google Scholar
  11. Nicolas Jardine and Robin Sibson. Mathematical Taxonomy. John Wiley & Sons, Ltd, 1971. Google Scholar
  12. Peter Johnstone. Stone Spaces. Number 3 in Cambridge Studies in Advanced Mathematics. Cambridge University Press, 1982. Google Scholar
  13. Ian T. Jolliffe. Principal Component Analysis. Springer Series in Statistics. Springer, 2002. Google Scholar
  14. Gregory Max Kelly. Basic Concepts of Enriched Category Theory. Number 64 in London Mathematical Society Lecture Notes. Cambridge University Press, 1982. Google Scholar
  15. Joachim. Lambek. Completions of categories : seminar lectures given 1966 in Zurich. Number 24 in Springer Lecture Notes in Mathematics. Springer-Verlag, 1966. Google Scholar
  16. Joachim Lambek and Philip Scott. Introduction to Higher Order Categorical Logic. Number 7 in Cambridge Studies in Advanced Mathematics. Cambridge University Press, 1986. Google Scholar
  17. Thomas K. Landauer, Danielle S. Mcnamara, Simon Dennis, and Walter Kintsch, editors. Handbook of Latent Semantic Analysis. Lawrence Erlbaum Associates, 2007. Google Scholar
  18. F. William Lawvere. Metric spaces, generalised logic, and closed categories. Rendiconti del Seminario Matematico e Fisico di Milano, 43:135-166, 1973. Google Scholar
  19. Saunders Mac Lane and Ieke Moerdijk. Sheaves in Geometry and Logic: A First Introduction to Topos Theory. Universitext. Springer-Verlag, New York, 1992. Google Scholar
  20. Saunders MacLane. Categories for the Working Mathematician. Number 5 in Graduate Texts in Mathematics. Springer-Verlag, 1971. Google Scholar
  21. Holbrook Mann MacNeille. Extensions of partially ordered sets. Proc. Nat. Acad. Sci. USA, 22(1):45-50, 1936. Google Scholar
  22. Oded Maimon and Lior Rokach, editors. Data Mining and Knowledge Discovery Handbook, 2nd ed. Springer, 2010. Google Scholar
  23. Dusko Pavlovic. Network as a computer: ranking paths to find flows. In Alexander Razborov and Anatol Slissenko, editors, Proceedings of CSR 2008, volume 5010 of Lecture Notes in Computer Science, pages 384-397. Springer Verlag, 2008. arxiv.org:0802.1306. Google Scholar
  24. Dusko Pavlovic. On quantum statistics in data analysis. In Peter Bruza, editor, Quantum Interaction 2008. AAAI, 2008. arxiv.org:0802.1296. Google Scholar
  25. Dusko Pavlovic. Quantifying and qualifying trust: Spectral decomposition of trust networks. In Pierpaolo Degano, Sandro Etalle, and Joshua Guttman, editors, Proceedings of FAST 2010, volume 6561 of Lecture Notes in Computer Science, pages 1-17. Springer Verlag, 2011. arxiv.org:1011.5696. Google Scholar
  26. Dusko Pavlovic. Relating toy models of quantum computation: comprehension, complementarity and dagger autonomous categories. E. Notes in Theor. Comp. Sci., 270(2):121-139, 2011. arxiv.org:1006.1011. Google Scholar
  27. Dusko Pavlovic. Quantitative Concept Analysis. In Florent Domenach, Dmitry I. Ignatov, and Jonas Poelmans, editors, Proceedings of ICFCA 2012, volume 7278 of Lecture Notes in Artificial Intelligence, pages 260-277. Springer Verlag, 2012. arXiv:1204.5802. Google Scholar
  28. Dusko Pavlovic. Bicompletions of distance matrices. In Bob Coecke, Luke Ong, and Prakash Panangaden, editors, Computation, Logic, Games and Quantum Foundations. The Many Facets of Samson Abramsky, volume 7860 of Lecture Notes in Computer Science, pages 291-310. Springer Verlag, 2013. Google Scholar
  29. Dusko Pavlovic and Samson Abramsky. Specifying interaction categories. In E. Moggi and G. Rosolini, editors, Category Theory and Computer Science 1997, volume 1290 of Lecture Notes in Computer Science, pages 147-158. Springer Verlag, 1997. Google Scholar
  30. Duško Pavlović and Martín Escardó. Calculus in coinductive form. In V. Pratt, editor, Proceedings. Thirteenth Annual IEEE Symposium on Logic in Computer Science, pages 408-417. IEEE Computer Society, 1998. Google Scholar
  31. F. Ricci, L. Rokach, B. Shapira, and P.B. Kantor. Recommender Systems Handbook. Springer, 2010. Google Scholar
  32. Ashok N. Srivastava and Mehran Sahami. Text Mining: Classification, Clustering, and Applications. Data Mining and Knowledge Discovery Series. CRC Press, 2009. Google Scholar
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