Summarizing and propagating uncertain information with triangular norms

A new approach to reasoning with uncertainty is proposed, which is organized in three layers: representation, inference, and control, which shows that for a common negation operator, the selection of a T-norm uniquely and completely describes an uncertainty calculus.

Abstract

A wide variety of numerical or symbolic approaches to reasoning with uncertainty have been proposed in the artificial intelligence (AI) literature. This article postulates a list of desiderata that any such formalism should try to satisfy. The author then proposes a new approach to reasoning with uncertainty, which is organized in three layers: representation, inference, and control. In the representation layer the structure required to capture information used in the inference layer and meta-information used in the control layer are described. In this structure, numerical slots take values on linguistic term sets with fuzzy-valued semantics. These term sets capture the input granularity usually provided by users or experts. In the inference layer a large number of uncertainty calculi based on triangular norms (T-norms), intersection operators whose truth functionality entails low computational complexity, are described. It is shown that for a common negation operator, the selection of a T-norm uniquely and completely describes an uncertainty calculus. Previous experiments have determined the existence of a small number of equivalence classes among the uncertainty calculi (as a function of the input granularity). This property drastically reduces the number of different combining rules to be considered. In the control layer the policy selection for the different calculi used in the inference layer, based on their meanings, properties, and contextual information, is specified. Conflicts and ignorance measurements are also defined. The proposed formalism is compared against the requirements of the desiderata and contrasted with existing schemes for reasoning with uncertainty.