Labeled Null Semantics for Incomplete Decision Tables: Certain, Possible, and Conditional Decision Reasoning

Decision tables are widely used for representing and executing decision logic in business rules systems, regulatory compliance, and automated workflows. However, real-world decision tables often contain incomplete condition values. Existing rough set approaches typically treat missing values uniformly and cannot distinguish between occurrences of the same unknown value and occurrences of different unknown values. This paper introduces Labeled Null Decision Tables (LN-DTs), a framework that extends decision table semantics with labeled nulls from incomplete database theory. Missing values are represented by labeled nulls, where identical labels denote the same unknown value and distinct labels may denote different values. We define a possible-world semantics in which each possible world corresponds to a complete valuation of nulls. Based on this semantics, we define certain, possible, and conditional decisions. We develop symbolic algorithms that avoid explicit enumeration of possible worlds, analyze their computational complexity, and establish precise relationships with rough set approximations. We show that LN-DT possible decisions are always contained in the rough upper approximation, that the rough lower approximation is sound for LN-DT certain decisions under completeness and well-formedness, and that labeled null identities and constraints can yield strict precision gains by eliminating tolerant rules that cannot match in any admissible possible world. To provide evidence beyond small illustrative examples, we also include a medium-scale synthetic benchmark suite containing 105 instances with up to 500 rules and 32 labeled nulls. The benchmarks vary the number of rules, the number of labeled nulls, and the constraint type, and report runtime, precision, and verification metrics. Across all benchmark instances, the symbolic evaluator produces known results with no solver timeouts, the LN-DT possible decision set is contained in the rough upper approximation in 100% of cases, and the hybrid evaluator agrees with exact symbolic evaluation in 100% of cases. The average rough upper decision set size is 1.905, while the average LN-DT possible decision set size is 1.019, corresponding to an average upper-bound reduction of approximately 46.5%. These results provide medium-scale evidence that LN-DT is practically feasible and that labeled-null constraints can yield measurable precision gains over tolerance-based rough set approximation.