Dualistic Algebra for Qualitative Analysis

A central point in qualitative models is that of calculation . The problem is to represent, compare, and combine, approximate or symbolic values with the major constraint of consistency with some numerical counterpart . Aware of the fact that many qualitative characterizations are in the form of antagonistic concepts (such as low vs. high, weak vs. strong, bad vs . good, etc.) more or less explicitly separated by a reference (we call the norm), we exploit this idea to build a dualistic algebra based on a linear order and an homomorphism with the set of integers Z, which accounts for the ranks of qualitative values (viewed as labels expressing magnitudes) ordered with respect to the norm. We define operations on those values, globally consistent with classical arithmetic (+, -, x, /, power, root) as well as fitting common sense in reasoning . The major goal of qualitative reasoning (QR) is to provide engineers with computational theories to allow them to draw inferences about systems, using information of various origins and different natures . Due to the complexity of systems, qualitative models should be able to involve variables having possibly different domains such as real numbers, numerical intervals or even purely symbolic elements . Measurements (numerical values) as well as basic observations (symbolic values) are often translated into more meaningful concepts within the framework of the system under study : 0.476 is far above normal, 150 is very low, or small is beautiful. Concepts used to characterize quantities are often antagonistic : low/high, good/bad, above/below, false/true . This dualism implies the existence of a reference representing normality within the context of reasoning. It is either explicitly stated or implicit. Actually it corresponds to the idea of being located between both antagonistic concepts which yields an order w.r.t . the norm. It may be necessary to go beyond a binary opposition and characterize a quantity at different granularity, i.e . distinguishing among good and bad concepts more subtle characterizations such as very good or very very bad. Another big point is compositionality, i.e . how to combine very high with bad, below with average and even blue with beautiful and cold! These needs of comparison and combination of heterogeneous things lead us to find some set to play the role of a "common currency". Because qualitative values are intrinsically discreet entities, we chose the integers Z. Since we do not base principally our formalism on a partition of the real line (as opposed to most of QR approaches), we will refer to quality spaces (in the sense of Hayes, 1985) to denote the set of possible values of a given qualitative variable . This paper addresses three main aspects : (i) how to build a quality space (QS) with a dualistic structure from a set of labels (section 2) ; (ii) how to compare values of (single or distinct) variables expressed on different QS at varying granularity (section 3) ; (iii) how to combine values through operations, consistent with both conventional arithmetic and common sense expectations (section 4) . Then, we provide two examples of qualitative analysis (section 5) . Finally we discuss (too briefly) this so called dualistic algebra (DuAl) formalism in the light of quantity spaces theory (section 6) .