We propose an efficient method to infer combinatorial association logic networks from multiple genome-wide measurements from the same sample. We demonstrate our method on a genetical genomics dataset, in which we search for Boolean combinations of multiple genetic loci that associate with transcript levels.Our method provably finds the global solution and is very efficient with runtimes of up to four orders of magnitude faster than the exhaustive search. This enables permutation procedures for determining accurate false positive rates and allows selection of the most parsimonious model. When applied to transcript levels measured in myeloid cells from 24 genotyped recombinant inbred mouse strains, we discovered that nine gene clusters are putatively modulated by a logical combination of trait loci rather than a single locus. A literature survey supports and further elucidates one of these findings. Due to our approach, optimal solutions for multi-locus logic models and accurate estimates of the associated false discovery rates become feasible. Our algorithm, therefore, offers a valuable alternative to approaches employing complex, albeit suboptimal optimization strategies to identify complex models.The MATLAB code of the prototype implementation is available on: http://bioinformatics.tudelft.nl/ or http://bioinformatics.nki.nl/.