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With the rise of new DNA part libraries and technologies for assembling DNA, synthetic biologists are increasingly constructing and screening combinatorial libraries to optimize their biological designs. As combinatorial libraries are used to generate data on design performance, new rules for composing biological designs will emerge. Most formal frameworks for combinatorial design, however, do not yet support formal comparison of design composition, which is needed to facilitate automated analysis and machine learning in massive biological design spaces. To address this need, we introduce a combinatorial design framework called GOLDBAR. Compared with existing frameworks, GOLDBAR enables synthetic biologists to intersect and merge the rules for entire classes of biological designs to extract common design motifs and infer new ones. Here, we demonstrate the application of GOLDBAR to refine/validate design spaces for TetR-homologue transcriptional logic circuits, verify the assembly of a partial gene cluster, and infer novel gene clusters for the biosynthesis of rebeccamycin. We also discuss how GOLDBAR could be used to facilitate grammar-based machine learning in synthetic biology.