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.

The reliability of digital computing systems is a significant problem. A scheme using multiply redundant computing devices with a majority voting device to determine the output has been shown to improve the reliability of that output. In this paper, a devices called a dissent detector is suggested as an addition to the system. The function of this device is that of an alarm to indicate the quality of the information (likelihood that the information is correct). Dissent is defined as a family of logic functions that compare the outputs of the voters to the output of the majority gate and report the disagreement or degree of disagreement that takes place. The operation of a dissent-detector that reports any disagreement in a triplicated majority voting system is demonstrated by example. The example illustrates the importance of the dissent information to the conditional reliability of the output of the majority gate. The results show that, with very little additional equipment, the conditional reliability of a majority voting system can be greatly enhanced. In fact, the conditional reliability of the example system exceeds the reliability of any of the system components, including the majority gate and the dissent detector.