Counting Minimal Unique-Cause MC/DC Test Suites for Boolean Decisions

초록

Unique-Cause Modified Condition/Decision Coverage (UC MC/DC) is a widely used structural adequacy criterion for safety-critical software decisions. For a Boolean decision with N atomic conditions, the theoretically minimal suite size under the one-bit witness interpretation is N + 1, which is exponentially smaller than exhaustive Multiple Condition Coverage. Minimality, however, does not imply uniqueness: a single decision may admit many distinct optimal suites, while another may admit only one or none. This paper defines minimal-suite multiplicity, denoted & micro;(f ), as the number of distinct size-(N+1) minimal suites for a decision f , together with a feasibility-constrained variant & micro;(f; F). Two complementary counting views are developed: a graph-theoretic model based on colored witness edges of the Boolean hypercube, and a propositional model-counting encoding that counts minimal suites without enumerating all 2(N) assignments. Exact counting is shown to be #P-complete in general under feasibility constraints. In the empirical study, exact labeled-tree enumeration remained practical for N is an element of {4, 5, 6, 7} with median runtimes of 1.9 & times; 10(-4) s, 1.3 & times;10(-3) s, 1.5 & times;10(-2 )s, and 2.3 & times;10(-1 )s per instance, respectively. A supplementary solver-based study on repeated-variable B2 instances extended the scalability picture beyond this range: exact projected counting with Ganak completed all tested cases at N = 8 and N = 10, with timeouts beginning at N = 12, while ApproxMC solved selected instances up to N = 30 but with strong instance sensitivity. Across fixed N, multiplicity varied by several orders of magnitude, and feasibility constraints sharply reduced the surviving space of optimal suites. These results position multiplicity as a quantitative indicator of robustness and substitutability among optimal UC MC/DC evidence.

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