Improvement on structure of weakly 2-dependent siphons

Abstract

Li and Zhou propose simpler Petri net controllers based on the concept of elementary siphons (generally much smaller than the set of all strict minimum siphons (SMSs) in large Petri nets) to minimise the addition of control places. SMSs can be divided into two groups: elementary and dependant; characteristic T-vectors of the latter are linear combinations of that of the former. A T-vector η is associated with each siphon S such that η(i) is the number of tokens gained in or lost from S by firing transition ti once. A dependent siphon S0 strongly depends on elementary siphons S1, S2, … , Sk if η0 = a1η1 + a2η2 + ⋅⋅⋅ + akηk with all ai (i = 1, 2, 3, … , k) positive. S0 is a weakly dependent siphon if some ai is negative. The T-vectors (resp. number) for elementary siphons are mutually independent (linear to the size of the net). In an earlier paper, we show that there exists a third siphon S3 such that ηβ = η1 + η2 - η3. This equation (called η relationship) plays an important role for optimal control of weakly dependent siphons. However, it assumes that all above S span between exactly two processes. For a well-known benchmark, however, most dependent siphons span more than two processes. This paper improves by removing this restriction and shows that ηβ = η1 + η2 - η3 holds as long as S1∩S2 is another emptiable siphon.