Research Project: Algebra of frames
Lattice structure of families of frames
Algebraic analysis of independence of frames
A major pillar of evidential reasoning is the formalization of the idea of structured collection of representations of the external world, encoded by the notion of 'family of frames'.
We lay the foundations for a rigorous algebraic analysis of the conflict problem, by studying the algebraic structure of the families of compatible frames as mathematical objects obeying a small number of axioms, originally proposed by Shafer.
We distinguish finite from general families of frames, describe the monoidal properties of compatible collections of both frames and refinings, and introduce the internal operation of 'maximal coarsening', which in turns induces in a family of frames the structures of Birkhoff, upper semimodular and lower semimodular lattice.
We outline a proposal for dealing with possibly conflicting belief functions defined on different compatible frames in an algebraic setting, based on building a new collection of combinable BFs via a 'pseudo Gram-Schmidt' algorithm. To investigate this possibility, we investigate the relation between Shafer's definition of independence of frames and various extensions of matroidal independence to compatible frames as elements of a semimodular lattice, in order to draw some conclusions on a conjectured algebraic solution to the conflict problem.
Relevant papers:
  •  Fabio Cuzzolin
    An algebraic study of the notion of independence of frames
    in Mathematics of Uncertainty Modeling in the Analysis of Engineering and Science Problems Prof. S. Chakraverty (Editor), IGI Publishing, January 2014
  •  Fabio Cuzzolin
    Algebraic structure of the families of compatible frames of discernment
    Annals of Mathematics and Artificial Intelligence
    Vol. 45, No. 1-2, pp. 241-274, 2005
    Algebraic structure of the families of compatible frames of discernment
  • Fabio Cuzzolin
    On the relationship between the notions of independence in matroids, lattices, and Boolean algebras
    21th British Combinatorial Conference (BCC'07)
    University of Reading, UK, July 8-13, 2007
    Abstract Presentation
  •  Fabio Cuzzolin
    A lattice-theoretic interpretation of independence on frames
    in "Interval / Probabilistic Uncertainty and Non-classical Logics",
    Advances in Soft Computing, Vol. 46
    Huynh, V.-N.; Nakamori, Y.; Ono, H.; Lawry, J.; Kreinovich, V.; Nguyen, H.T. (Eds.)
    Springer Berlin / Heidelberg, 2008
      PDF slides
  • Fabio Cuzzolin and Ruggero Frezza
    Lattice structure of the families of compatible frames of discernment
    Proceedings of the 2nd International Symposium on Imprecise Probabilities and Their Applications (ISIPTA2001)
    Cornell University, Ithaca, NY, June 26-29, 2001
  • Fabio Cuzzolin
    Families of compatible frames of discernment as semimodular lattices
    International Conference of the Royal Statistical Society (RSS2000)
    Reading, UK, September 12-15, 2000