NPN calculi: a family of three strict Q-algebras

NPN (negative-positive-neutral) calculi, a family of three mathematical structures are introduced for qualitative reasoning. NPN crisp logic extends the usual 4-valued model {+, -, 0, ?) to a 6-valued model {1, 0, +1, (-1, 0), (0, +1), (-1, +1)} and adds one more level of specification to the usual model. NPN fuzzy logic extends the NPN model to the space {∀(x,y)|x, yε[-1, 1] and x⩽y} and adds infinite levels of specifications to the usual model. NPN algebra generalizes the NPN model to include algebraic operations on NPN variables defined in the space of [-∞, ∞]. Based on the three models, NPN relations and NPN matrices are proposed. It is proved that the three models provide three strict qualitative algebras after the usual 4-valued model. Properties of different models are discussed. Potential applications of NPN calculi in qualitative reasoning are outlined