Petri nets are a kind of formal language that are widely applied in concurrent systems associated with resource allocation due to their abilities of the natural description on resource allocation and the precise characterization on deadlock. Weighted System of Simple Sequential Processes with Resources (WS3PR) is an important subclass of Petri nets that can model many resource allocation systems in which 1) multiple processes may run in parallel and 2) each execution step of each process may use multiple units from a single resource type but cannot use multiple resource types. We first prove that the liveness problem of WS3PR is co-NP-hard on the basis of the partition problem. Furthermore, we present a necessary and sufficient condition for the liveness of WS3PR based on two new concepts called Structurally Circular Wait (SCW) and Blocking Marking (BM), i.e., a WS3PR is live iff each SCW has no BM. A sufficient condition is also proposed to guarantee that an SCW has no BM. Additionally, we show some advantages of using SCW to analyze the deadlock problem compared to other siphon-based ones, and discuss the relation between SCW and siphon. These results are valuable to the further research on the deadlock prevention or avoidance for WS3PR.

In the classical computation theory, the language of a system features the computational behavior of the system but it does not distinguish the determinism and nondeterminism of actions. However, Milner found that the determinism and nondeterminism affect the interactional behavior of interactive systems and thus the notion of language does not features the interactional behavior. Therefore, Milner proposed the notion of (weak) bisimulation to solve this problem. With the development of internet, more and more interactive systems occur in the world, such as electronic trading system. Security is one of the most important topics for these systems. We find that different security policies can also affect the interactional behavior of a system, which exactly is the reason why a good policy can strengthen the security. In other words, two interactive systems with different security policies are not of an equivalent behavior although their functions (or business processes) are identical. However, the classic (weak) bisimulation theory draws an opposite conclusion that their behaviors are equivalent. The notion of (weak) bisimulation is not suitable for these security-oriented interactive systems since it does not consider a security policy. This paper proposes the concept of secure bisimulation in order to solve the above problem.