Research Theme: Algebra and Independence of
Frames |

Lattice structure of compatible frames of discernmentAlgebraic analysis of independence of frames

The introduction of belief functions as the most suitable mathematical descriptions of empirical evidence or subjective states of belief, and the related mechanisms for their combination in a belief revision process is the most important contribution of the theory of evidence. Another 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 b.f.s via a 'pseudo Gram-Schmidt' algorithm. To investigate this possibility, we analyse the algebraic structure of families of frames and show that they form upper and lower semimodular lattices, depending on which order relation we pick

We then take a step forward and 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. We study relationships and differences between the different forms of lattice-theoretical independence, and understand whether classical independence of frames can be reduced to one of them.

As a result, we pose the notion of independence of frames in a wider context by highlighting its relation with classical independence in modern algebra. Although independence of frames turns out not to be a cryptomorphic form of matroidal independence, it does exhibit correlations with several extensions of matroidal independence to lattices, stressing the need for a more general, comprehensive definition of this widespread and important notion.