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In this paper the Ashenhurst-Curtis theory of complex disjunctive decompositions is extended to the realm of incompletely specified Boolean functions. A compatibility relation on the column vectors of the decomposition chart is introduced, which is applied to identify all possible simple disjunctive decompositions for each input partition. The assignments of the DON'T CARE (φ) conditions that are required to realize these simple decompositions are described by a vector listing the constraints on these φ's by new Boolean variables caled constrained DON'T CAREs. A compatibility relation is introduced on these vectors, caled constrained Boolean vectors, which is applied to form complete decompositions. A Complete decomposition is one for which al possible simple decompositions have been combined into a complex decomposition. Throughout the procedure, the freedom of choice implied by the φ's is maintained as far as is alllowed by the choices that have been made to achieve the decompositions.