Automated manufacturing system: virtual-nets or non-virtual-nets?

Proving liveness for automated manufacturing systems such as systems of simple sequential processes with resources (S³PR) has been mind boggling. This is because deadlock-freeness does not imply liveness in general. In an earlier paper, it was shown that the virtual first-order structure (VFOS) is the key structure to make some transitions not live. A virtual-net (non-virtual-net) or V-net (NV-net) is a net with (without) VFOS. It was proposed to enlarge the class of NV-nets to NV⁺-nets to include S³PR by imposing conditions upon VFOS. Showing that S³PR belongs to neither NV-nets nor V-nets, it was proposed that a new large class of resource allocation systems called S⁺PR and proved that it belonged to NV⁺-nets. There is no need to prove the liveness of an NV⁺-net without emptiable siphons, if it is an ordinary Petri net as in traditional techniques is shown. The absence of the above conditioned VFOS serves as one regulation structural mechanism ensuring a siphon to be controlled other than that based on the trap or invariant concept.