Recent results on the complexity of problems related to Petri nets

In this paper, we examine the complexity of the boundedness, containment, equivalence, and reachability problems for certain subclasses of Petri nets (PNs) (equivalently vector addition systems (VASs), vector addition systems with states (VASSs), or vector replacement systems (VRSs)). Specifically, we consider the complexity of the boundedness problem for general VASSs, fixed dimensional VASSs, and conflict-free VRSs. We consider the complexity of the remaining problems for bounded VASSs, 2-dimensional VASSs, and conflict-free VRSs. Instances in each of these classes are known to have effectively computable semilinear reachability sets (SLSs). In each case, our results are derived by showing how to obtain succinct and sometimes special representations of the associated SLSs. The results discussed here constitute a summary of results obtained elsewhere by the authors. No proofs appear in this document, although we do strive to outline the general strategies involved. Readily available sources for the detailed proofs are indicated.

[1]  Horst Müller,et al.  Weak Petri Net Computers for Ackermann Functions , 1985, J. Inf. Process. Cybern..

[2]  Edward L. Robertson,et al.  Properties of Conflict-Free and Persistent Petri Nets , 1978, JACM.

[3]  Larry J. Stockmeyer,et al.  The Polynomial-Time Hierarchy , 1976, Theor. Comput. Sci..

[4]  Michael Yoeli,et al.  Vector Addition Systems and Regular Languages , 1980, J. Comput. Syst. Sci..

[5]  Albert R. Meyer,et al.  The Complexity of the Finite Containment Problem for Petri Nets , 1981, JACM.

[6]  Marvin Minsky,et al.  Computation : finite and infinite machines , 2016 .

[7]  Kenneth McAloon,et al.  Petri Nets and Large Finite Sets , 1984, Theor. Comput. Sci..

[8]  Peter Clote,et al.  On the Finite Containment Problem for Petri Nets , 1986, Theor. Comput. Sci..

[9]  Walter J. Savitch,et al.  Relationships Between Nondeterministic and Deterministic Tape Complexities , 1970, J. Comput. Syst. Sci..

[10]  Richard J. Lipton,et al.  Exponential space complete problems for Petri nets and commutative semigroups (Preliminary Report) , 1976, STOC '76.

[11]  Richard M. Karp,et al.  Parallel Program Schemata , 1969, J. Comput. Syst. Sci..

[12]  Neil D. Jones,et al.  Complete problems for deterministic polynomial time , 1974, Symposium on the Theory of Computing.

[13]  Horst Müller.  Decidability of Reachability in Persistent Vector Replacement Systems , 1980, MFCS.

[14]  Ernst W. Mayr.  An Algorithm for the General Petri Net Reachability Problem , 1984, SIAM J. Comput..

[15]  Neil D. Jones,et al.  Complexity of Some Problems in Petri Nets , 1977, Theor. Comput. Sci..

[16]  John E. Hopcroft,et al.  On the Reachability Problem for 5-Dimensional Vector Addition Systems , 1976, Theor. Comput. Sci..

[17]  James Lyle Peterson,et al.  Petri net theory and the modeling of systems , 1981 .

[18]  Dung T. Huynh,et al.  The complexity of the equivalence problem for commutative semigroups and symmetric vector addition systems , 1985, STOC '85.

[19]  A. Meyer,et al.  The complexity of the word problems for commutative semigroups and polynomial ideals , 1982 .

[20]  Hsu-Chun Yen,et al.  An O(n^(1.5)) Algorithm to Decide Boundedness for Conflict-Free Vector Replacement Systems , 1987, Inf. Process. Lett..

[21]  Hsu-Chun Yen,et al.  A Multiparameter Analysis of the Boundedness Problem for Vector Addition Systems , 1985, J. Comput. Syst. Sci..

[22]  Thiet-Dung Huynh.  The Complexity of Semilinear Sets , 1982, J. Inf. Process. Cybern..

[23]  Stefano Crespi-Reghizzi,et al.  A Decidability Theorem for a Class of Vector-Addition Systems , 1975, Inf. Process. Lett..

[24]  Jan Grabowski,et al.  The Decidability of Persistence for Vector Addition Systems , 1980, Information Processing Letters.

[25]  Hsu-Chun Yen,et al.  Some Complexity Bounds for Problems Concerning Finite and 2-Dimensional Vector Addition Systems with States , 1986, Theor. Comput. Sci..

[26]  Rüdiger Valk,et al.  Petri Nets and Regular Languages , 1981, J. Comput. Syst. Sci..

[27]  Michel Hack,et al.  The Equality Problem for Vector Addition Systems is Undecidable , 1976, Theor. Comput. Sci..

[28]  Dung T. Huynh.  A Simple Proof for the Sump2 Upper Bound of the Inequivalence Problem for Semilinear Sets , 1986, J. Inf. Process. Cybern..

[29]  I. Borosh,et al.  Bounds on positive integral solutions of linear Diophantine equations , 1976 .

[30]  Charles Rackoff,et al.  The Covering and Boundedness Problems for Vector Addition Systems , 1978, Theor. Comput. Sci..