Exact Support Identification of Boolean Guards from Sector Counts and Weight-Limited Queries

Many models of regulation and control describe a component by a Boolean rule on its inputs, and the first question about such a rule is which inputs matter. This paper studies how well the set of essential inputs, the support, can be recovered when the number of accepting input words in each Hamming-weight layer, the sector profile, is known in advance, and how many additional addressed queries are needed when the profile does not settle the answer. An inclusion–exclusion formula gives, for every profile, the support sizes that occur and the number of functions realizing each, and monotone examples show that profiles can collide on supports of different size. In 3,181 node functions from 81 published Boolean network models, 6.3% of the functions with at least three inputs have an ambiguous support size. Of these, 32 are rules with an inessential input, and the other 39 are 1.2 to 1.4 times the number expected for random truth tables with the same arity and number of positive rows. For a family of threshold profiles, the exact adaptive minimax number of weight-capped queries is (𝑚-1 𝑗-1 ) + ⌈(𝑚 - 1)/𝑗⌉ and it is infinite when the cap is below 𝑗. For the complete family of rules on n inputs with sector polynomial t²+2t³+t⁴, the exact nonadaptive query complexity is proved and an adaptive strategy gives an unbounded additive saving; for n=5 exhaustive search gives 22 nonadaptive and 12 adaptive queries.