Recently, several algorithms have been proposed for decomposing reactive synthesis specifications into independent and simpler sub-specifications. Being inspired by one of the approaches, developed by Antonio Iannopollo (2018), who designed the so-called (DC) algorithm, we present here our solution that takes his ideas further and provides mathematical formalisation of the strategy behind DC. We rigorously define the main notions involved in the algorithm, explain the technique, and demonstrate its application on examples. The core technique of DC is based on the detection of independent variables in linear temporal logic formulae by exploiting the power and efficiency of a model checker. Although the DC algorithm is sound, it is not complete, as its author already pointed out. In this paper, we provide a counterexample that shows this fact and propose relevant changes to adapt the original DC strategy to ensure its correctness. The modification of DC and the detailed proof of its soundness and completeness are the main contributions of this work.

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