Its application involves a wide range of area including expert systems[3][4][5], information fusion[6], pattern classfication[7][8][9], risk evaluation [10,11] [12], image recognition [13], classification[14,15] and data mining [16] etc. The original DS theory requires deterministic belie degrees and belief structures. However, in practical situations, evidence coming from multiple sources may be influenced by unexpected extraneous factors. The lack of information, linguistic ambiguity or vagueness and cognitive bias all contribute to the uncertain evidence obtained in practical situations. For example, during risk assessment, expert may be unable to provide a precise assessment if he/she is not 100% sure.

Symmetries are common in many constraint problems. They can be broken statically or dynamically. The focus of this paper is the symmetry breaking during search (SBDS) method that adds conditional symmetry breaking constraints upon each backtracking during search. To trade completeness for efficiency, partial SBDS (ParSBDS) is proposed by posting only a subset of symmetries. We propose an adaptation method recursive SBDS (ReSBDS) of ParSBDS which extends ParSBDS to break more symmetry compositions. We observe that the symmetry breaking constraints added for each symmetry at a search node are nogoods and increasing. A global constraint (incNGs), which is logically equivalent to a set of increasing nogoods, is derived. To further trade pruning power for efficiency, we propose weak-nogood consistency (WNC) for nogoods and a lazy propagator for SBDS (and its variants) using watched literal technology. We further define generalized weak-incNGs consistency (GWIC) for a conjunction of increasing nogoods, and give a lazy propagator for incNGs.

The generation and GAC enforcement of a large number of weak nogoods in Symmetry Breaking During Search (SBDS) is costly and often not worthwhile in terms of prunings. In this paper, we propose weak-nogood consistency (WNC) for nogoods and a lazy propagator for SBDS (and its variants) using watched literal technology. We give formal results on the strength and relatively low space and time complexities of the lazy propagator. Nogoods collected for each symmetry are increasing. We further define generalized weak-incNGs consistency (GWIC) for a conjunction of increasing nogoods, and give a lazy propagator for the incNGs global constraint. We prove GWIC on a conjunction is equivalent to WNC on individual nogoods, and give the space and time complexities. Various lazy versions of SBDS and its variants are implemented. We give experimentation to demonstrate the efficiency of the lazy versions as compared to state of the art symmetry breaking methods.