Invariants of Petri nets are fundamental algebraic characteristics of Petri nets, and are used in various situations, such as checking (as necessity of) liveness, boundedness, periodicity and so on. Any given Petri net N has two kinds of invariants: a P-invariant is a |P|-dimensional vector Y with Yt A = and a T-invariant is a |T|-dimensional vector X with A X = for the place-transition incidence matrix A of N. T-invariants are nonnegative integer vectors, while this is not always the case with P-invariants. This paper deals only with nonnegative integer invariants (invariants that are nonnegative vectors) and shows results common to the two invariants. For simplicity of discussion, only P-invariants are treated. The Fourier-Motzkin method is well-known for computing all minimal support integer invariants. This method, however, has a critical deficiency such that, even if a given Perti net N has any invariant, it is likely that no invariants are obtained because of an overflow in storing intermediate vectors as candidates for invariants. The subject of the paper is to give an overview and results known to us for efficiently computing minimal-support nonnegative integer invariants of a given Petri net by means of the Fourier-Motzkin method. Also included are algorithms for efficiently extracting siphon-traps of a Petri net.

A siphon-trap(ST) of a Petri net N = (P,T,E,α,β) is defined as a set S of places such that, for any transition t, there is an edge from t to a place of S if and only if there is an edge from a place of S to t. A P-invariant is a |P|-dimensional vector Y with YtA = for the place-transition incidence matrix A of N. The Fourier-Motzkin method is well-known for computing all such invariants. This method, however, has a critical deficiency such that, even if a given Perti net N has any invariant, it is likely that no invariants are output because of memory overflow in storing intermediary vectors as candidates for invariants. In this paper, we propose an algorithm STFM_N for computing minimal-support nonnegative integer invariants: it tries to decrease the number of such candidate vectors in order to overcome this deficiency, by restricting computation of invariants to siphon-traps. It is shown, through experimental results, that STFM_N has high possibility of finding, if any, more minimal-support nonnegative integer invariants than any existing algorithm.

A siphon-trap of a Petri net N is defined as a place set S with S = S, where S = { u| N has an edge from u to a vertex of S} and S = { v| N has an edge from a vertex of S to v}. A minimal siphon-trap is a siphon-trap such that any proper subset is not a siphon-trap. The following polynomial-time algorithms are proposed: (1) FDST for finding, if any, a minimal siphon-trap or even a maximal class of mutually disjoint minimal siphon-traps of a given Petri net; (2) FDSTi that repeats FDST i times in order to extract more minimal siphon-traps than FDST. (3) STFM_T (STFM_Ti, respectively) which is a combination of the Fourier-Motzkin method and FDST (FDSTi) and which has high possibility of finding, if any, at least one minimal-support nonnegative integer invariant.

We consider only P-invariants that are nonnegative integer vectors in this paper. An P-invariant of a Petri net N=(P,T,E,α,β) is a |P|-dimensional vector Y with Yt A = for the place-transition incidence matrix A of N. The support of an invariant is the set of elements having nonzero values in the vector. Since any invariant is expressed as a linear combination of minimal-support invariants (ms-invariants for short) with nonnegative rational coefficients, it is usual to try to obtain either several invariants or the set of all ms-invariants. The Fourier-Motzkin method (FM) is well-known for computing a set of invariants including all ms-invariants. It has, however, critical deficiencies such that, even if invariants exist, none of them may be computed because of memory overflow caused by storing candidate vectors for invariants and such that, even when a set of invariants are produced, many non-ms invariants may be included. We are going to propose the following two methods: (1) FM1_M2 that finds a smallest possible set of invariants including all ms-invariants; (2) STFM that necessarily produces one or more invariants if they exist. Experimental results are given to show their superiority over existing ones.