On Fuzzy Arithmetic, Possibility Theory and Theory of Evidence Asuncion P. Cucala, Jose Villar Institute of Research in Technology Universidad Pontificia Comillas C/ Santa Cruz de Marcenado 26 28015 Madrid. Spain Abstract This paper explores the existing relationship between Possibility Theory and Theory of Evidence, when they are both applied to fuzzy arithmetic. Possibility Theory arithmetic is based on the extension principle (projection of the joint possibility distribution), while in Theory of Evidence, the consonant bodies of evidence obtained from each operand are combined into a new joint body of evidence, which can in general be non consonant. Identical behaviour is found when the joint possibility distribution is calculated using the min operator, while Possibility Theory gives more specific results when others T-norms are used. This has been considered by some authors as a Theory of Evidence drawback (Dubois & Prade 1989). This paper shows that Theory of Evidence may be a more realistic uncertainty model when input data are obtained from random experiments with imprecise outcomes. 1 INTRODUCTION There is a straight forward relationship between Possibility Theory and Theory of Evidence, when consonant bodies of evidence are involved. In this case possibility and plausibility measures coincide. Interpreting basic assignments as density functions, where the random variables are the focal elements, simulations can be performed from given possibility distributions. This paper shows how the sum of two fuzzy numbers A and B can be calculated, applying both the extension principle and the theory of evidence, and compares the results. Section 2 reviews some basic definitions. Section 3 shows how the joint possibility/plausibility distribution can be obtained. Section 4 compares the possibility/plausibility distribution of the union of two points. Section 5 compares the results of summing two fuzzy numbers using both approaches, and finally some conclusions are presented in Section 6. 2 DEFINITIONS REVIEW Plausibility and Belief measures are fuzzy measures defined by (see (Klir 1988) (Shafer 1987)): [ ] Pl P X : ( ) , → 0 1 [ ] Bel P X : ( ) , → 0 1 such that: Pl A A A Pl A Pl A A Pl A A A n i i j i j i n n ( . . . ) ( ) ( ) ( ) ( . . . ) 1 2 1 1 2 1 ∩ ∩ ≤ ∪ - ∪ ∪ < + ∑ ∑ Bel A A A Bel A Pl A A Bel A A A n i i j i j i n n ( . . . ) ( ) ( ) ( ) ( . . . ) 1 2 1 1 2 1 ∪ ∪ ≥ ∩ - ∩ ∩ < + ∑ ∑ where P X ( ) is the power set of crisp subsets of X. Plausibility/Belief measures can also be defined, given a body of evidence (F,m), as: Pl B m A i A B i ( ) ( ) = ∩ ≠ ∑ 0 Bel B m A i A B i ( ) ( ) = ⊆ ∑ where A i are the focal elements and m is the basic probability assignment (Alvarez 1994). When the body of evidence is consonant, that is, its focal elements are nested, then the plausibility (resp belief) measure is called possibility (resp necessity) measure, and the following properties hold: [ ] [ ] Pl A B max Pl A Pl B A B max A B ( ) ( ), ( ) ( ) ( ), ( ) ∪ = → ∪ = π π π [ ] [ ] Bel A B min Bel A Bel B A B min A B N N N ( ) ( ), ( ) ( ) ( ), ( ) ∩ = → ∩ = Taking into account the body of evidence, a fuzzy set can be given a probabilistic interpretation. The basic assignment is view as a probability density function whose random variable is the set of focal elements. The possibility/plausibility distribution function is defined from the plausibility measure definition by: { } ( 29 { } μ( ) ( ) ( ) x Pl x m A m A i i x A A x i i = = = ∈ ∩ ≠∅ ∑ ∑ μ( ) x can then be interpreted as a probability distribution function of the focal elements, and Montercarlo method can be used to obtain a realisation of the random variable, that is, to obtain a set A i .