Specifying the defective parts for modification of reactive system specifications

A reactive system is a system that maintains interactions with its environment. Therefore it is necessary for realizability[1][2] of its specification that the reactive system can keep reacting correctly to any kind of its environment behavior. Temporal logics have been studied as a language for describing reactive system specifications[3][4] and a realizability judgment method for temporal specifications and a program construction method from realizable specifications is found in [5]. As for un-realizable and satisfiable specifications, three interesting properties of them are introduced in [6], e.g. strong satisfiability, stepwise satisfiability and stepwise strong satisfiability. They and satisfiability are necessary conditions for realizability. The judgment methods are also presented in [6] and with that methods, un-realizable specifications can be divided into several classes based on the characteristics of their defects. For example, if a specification of reactive system is satisfiable and not strong satisfiable, then it is concluded that the system can not react properly to some behaviors of an environment with satisfying their specifications, although they have no contradictions. If a specification of some system is stepwise satisfiable and not realizable, then the systems can not behave satisfying the specifications without knowing whole future behaviors of an environment. As knowing characteristics of un-realizable specifications gives good hints for the way to modify them, the classification of specifications are effective in describing specifications. However, as specifications is getting bigger and bigger, it comes to be very hard to revise specifications without specifying defective parts, although we know characteristics of their defects. Moreover, defections are usually caused by joining several parts of specifications, ex. joining an old part and a new part, joining parts wrote by different persons, different companies, and so on. As there are so many kind of responsibilities and many situations, the decision which one part should be revised can not be decided automatically. Then, if all combinations of parts that make defections can be find automatically and properly as small as possible, we can select the very part that should be revised, can select the part is best fit for an evolution of the specification and its system. Here we introduce a new defective part finding method for strong unsatisfiable specifications. First, we introduce the definition of the cause of strong-unsatisfiability in a temporal specification that suit for the decision of the part which should be revised. Then we present the procedure that gives all causes, maximal sets of events occurrences that make the specification strong-unsatisfiable.