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This paper considers a generalization of various communication models based on the P system paradigm where two objects synchronously move across components. More precisely, the model uses blocks of four cells such that pairs of objects from two input cells travel together to target output cells. It is shown that the model introduced, based on interactions between blocks, is complete, being able to generate all recursively enumerable sets of natural numbers. It is also proven that completeness is achievable by using a minimal interaction between blocks, i.e. every pair of cells is the input or output for at most one block. It is also shown that the concepts introduced in this paper to define the model may be simulated by more particular communication primitives, including symport, antiport and uniport rules. This enables us to automatically translate a system using interaction rules in any of minimal symport, minimal antiport or conditional uniport P systems.