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Purpose: We propose a formal framework for the modeling and segmentation of minimally-invasive surgical tasks using a unified set of motion primitives (MPs) to enable more objective labeling and the aggregation of different datasets. Methods: We model dry-lab surgical tasks as finite state machines, representing how the execution of MPs as the basic surgical actions results in the change of surgical context, which characterizes the physical interactions among tools and objects in the surgical environment. We develop methods for labeling surgical context based on video data and for automatic translation of context to MP labels. We then use our framework to create the COntext and Motion Primitive Aggregate Surgical Set (COMPASS), including six dry-lab surgical tasks from three publicly-available datasets (JIGSAWS, DESK, and ROSMA), with kinematic and video data and context and MP labels. Results: Our context labeling method achieves near-perfect agreement between consensus labels from crowd-sourcing and expert surgeons. Segmentation of tasks to MPs results in the creation of the COMPASS dataset that nearly triples the amount of data for modeling and analysis and enables the generation of separate transcripts for the left and right tools. Conclusion: The proposed framework results in high quality labeling of surgical data based on context and fine-grained MPs. Modeling surgical tasks with MPs enables the aggregation of different datasets and the separate analysis of left and right hands for bimanual coordination assessment. Our formal framework and aggregate dataset can support the development of explainable and multi-granularity models for improved surgical process analysis, skill assessment, error detection, and autonomy.