Bipolar Fuzzy Minimum-Weight Satisfiability under Continuous t-Norms: A Clause-Witness Branch-and-Bound Framework
Minimum-weight satisfiability asks for a satisfying Boolean assignment of minimum cost, while many fuzzy decision and knowledge systems operate with graded positive and negative relations rather than crisp literals. This paper introduces a bipolar fuzzy minimum-weight satisfiability problem in which positive and negative clause--variable relations, target clause levels, and variable states are represented in the unit interval. The clause requirements are expressed by bipolar fuzzy relational equalities defined with an arbitrary continuous t-norm. Classical minimum-weight CNF satisfiability is recovered as a crisp special case, independently of the selected t-norm, and an optimal Boolean solution is shown to exist in that specialization even though the proposed formulation uses continuous variables. For the graded problem, clause satisfaction is characterized through admissible variable domains and effective clause-witness sets. This structure leads to an exact clause-witness branch-and-bound method with preprocessing, propagation, lower-point closure, and a disjoint-witness packing bound. Computational experiments verify the solver against explicit witness enumeration, mixed-integer baselines, and external SATLIB formulas; on a focused graded ablation set, the packing bound reduces the median number of explored nodes by about 45 percent relative to the single-row witness bound.