Computational Aspects of the Mobius Transform

In this paper we associate with every (directed) graph G a transform called the MObius transf orm of the graph G. The Mobius transform of the graph (pO,�) is of major significance for Dempster-Shafer theory of evidence. However, because it is computationally very heavy, the Mobius transform together with Dempster's rule of combination is a major obstacle to the use of Dempster­ Shafer theory for handling uncertainty in expert systems. The major contribution of this paper is the discovery of the 'fast Mobius transforms' of (PO.�). These 'fast Mt>bius transforms' are the fastest algorithms for computing the Ml>bius transform of (pO,�). As an easy but useful application, we provide, via the conunonality function, an algorithm for computing Dempster's rule of combination which is much faster than the usual one. 0. Introduction and motivations Let 0 be a fmite non empty set and p 0 be its power set equipped with the inclusion relation. In the Dempster­ Shafer theory of evidence the standard reference of which is [Shafer 76], see also [Smets 88] a basic belief assignment (bba) on 0 is any function m : p0-+[0 1] such that L m(X) = 1 XepO m(X) is the mass of X. Most often it is also required that m(0)=0. Anyway, any basic belief assignment m determines its belief function belm: p 0 -+ [0 11 defmed by 'VAe po: belm(A) = L m(X) Xs;A,X.c(J belm(A) is the belief of A induced by the bba m. If, more generally, we consider any function m: pO-+ R, the previous formula defmes the functional R�'0-+ R110: m-+ belm • "The following text pmenu research mul u of t!'e B�gi� National incentive-prognm for fundamental research m artific�al intelli gence initiated by lhe BeJaian Swe, Prime Minister's Office, Science Policy Proaramming. The scientific responsibility is assumed by lhe authors." where R p 0 denotes, as usual, the set of functions from p 0 to the set of real numbers. The notation belm , although rather explicit, does not do justice to the most important protagonist of the formula, that is the binary relation {(X,Y)e pOxpO I X�0. X�Y}. Thus the above functional, which will be called the Mobius t r a ns form, is induced by the relation {(X,Y)e poxpo I X�0. X�Y}, or less exactly by the finite boo lean lattice (pQ,�). The ftrst section of the paper begins with the defmition of the Mobius transform induced by an arbitrary graph. The Mobius transform defines in an obvious way a map between two categories. The just defined map is not a functor, but by generalizing the Ml>bius transform induced by a graph to the M�bius transform induced by a weighted graph, the map becomes a functor. Such a generalization sheds some light on the preceding situation by providing a recursive formula for computing the M()bius transform. The fundamental fact is that recursion is neither on the set 0 nor on the power set pO but on the inclusion relation. In both situations, graphs and weighted graphs, we defme what we call M-algorithms: since a graph determines a functional, a sequence of graphs determines the composite of the functionals induced by each graph of the sequence. A natural problem is then to decompose a graph into subgraphs in order to get various algorithms computing the MObius transform induced by the graph. In the second section, as an application of that decomposition, we provide 'fast' M-algorithms for computing the MObius transform of (pQ,�). In the third section we defme the computational complexity of M-algorithms and in the fourth section we show that the previously defmed 'fast' M-algorithms are the fastest among all M-algorithms computing the same Mobius transform. In the fJ.fth part, as an application, we compute Dempster's rule of combination in a much faster way than the usual one. Lydia Kronsjo points out, in her boo k [Kronsj() 85 p.20], that efficient algorithms for solving the problems of arithmetic complexity are frequently based on a technique known as recursion. She mentions that during the 1960's three very surprising algorithms were discovered: for the multiplication of two integers, for computing the discrete Fourier transform, and for the product of two matrices. As a matter of fact all these efficient algorithms are based on recursive formulas. The present paper is in keeping with this observation. Due to lack of space, no proof will be given in this paper. All theorems are proved in [Kennes 90]. 1. The Mobius functor 1.1 How graphs operate on functions LetS and T be finite sets. A subset G of the cartesian productS x Tis called a (directed) graph from S to T. We write indifferently G: S � T or G � S x T. Sometimes we will say arrows of G instead of ordere d pairs of G. When no confusion is possible we use the same symbol to denote a binary relation and the set of ordered pairs it determines on a particular set. Explicitly, if R is a binary relation, the graph {(s,t)e SxS I sRt) it determines on the set S will also be denoted by R. Throughout this paper all graphs are finite. SET denotes the category of sets. FGRAPH denotes (confusingly!) the category of which the objects are the finite sets and the arr ows are the graphs G: S � T together with the usual composition of graphs. Any arr ow of the category FGRAPH, i.e. any graph, determines a MObius transform. More explicitly: Definition 1. The graph G: S � T determines the fwlctional: defined by V'te T : fG(t) = I, f(s) = I, f(s) IGt MO-I(t) We call � the Mobius tr�orm of G. (We recall that: I, f(s) = 0. )