This study proposes a new approach that recovers the system from deadlock states to its former live states, and reaches the same number of states as the original uncontrolled model by adding monitors (and control arcs) with no new problematic siphons. We further propose a lossless approach by coloring some arcs to avoid the material loss.

It has been a hot research topic to compare the effectiveness of new control policies by testing against a wellknown S 3 PR model. So far, only the control policy by Piroddi et al. may reach the optimal number of states among all approaches for a well-known benchmark using a siphon-based approach. The resulting model is a generalized Petri net since some control arcs are weighted, which complicates the system. The only improvement that can be made is to reduce the number of control arcs (by 3), and the number of weighted control arcs (by 9) as we report in this paper. Also the token count is reduced. This is achieved by replacing two monitors with weighted arcs by two new monitors without weighted arcs. INA (Integrated Net Analyzer) analysis indicates that the resulting controlled model is live and reaches the same 21581 states by Piroddi et al. We develop a formal theory for explaining the cause of state losses and providing the foundation for the above improvement model.

This paper proposes to combine the elementary siphons controlled policy (ESCP) by Li et al. and our recovery methods (Chao DY et al.) into an integrated approach. An example is demonstrated to show that it reaches more states and uses fewer monitors. It is more efficient, since there is no need to solve a large number of inequalities in marking/transition–separation instances (MTSIs) as required in the two-stage method proposed by Li et al. to reduce the number of MTSI and the crucial-MTSI (CMTSI) method by Huang and Pan, which further reduces the number of MTSIs by reducing some MTSIs to a CMTSI. We further propose a lossless approach by colouring some arcs. This not only avoids material loss but also tackles the livelock problem and achieves higher throughput, since no states are avoided.

When programmers try to reuse a software system developed by other programmers, the difficulty of understanding the system limits reuses [1]. It is not easy to measure software understandability because understanding is an internal process of humans. This paper proposes "integral of understandability" as a model for measuring software understandability which can extract first the best value of software understandability from factors with higher weight. In other words, we gave an integrated aspect of measuring software understandability through the literature on the subject.