Deciding a class of path formulas for conflict-free petri nets

The aim of this paper is to develop a unified approach for deriving complexity results for problems concerning conflict-free Petri nets. To do so, we first define a class of formulas for paths in Petri nets. We then show that answering the satisfiability problem for conflict-free Petri nets is tantamount to solving a system of linear inequalities (which is known to be in P). Since a wide spectrum of Petri net problems (including various fairness-related problems) can be reduced to the satisfiability problem in a straightforward manner, our approach offers an umbrella under which many Petri net problems for conflict-free Petri nets can be shown to be solvable in polynomial time. As a side-product, our analysis provides evidence as to why detecting unboundedness for conflict-free Petri nets is easier (provided P ≠ NP) than for normal and sinkless Petri nets (which are two classes that properly contain conflict-free Petri nets).

[1]  L. G. H. Cijan.  A polynomial algorithm in linear programming , 1979 .

[2]  Peter Radford,et al.  Petri Net Theory and the Modeling of Systems , 1982 .

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

[4]  Javier Esparza.  A Solution to the Covering Problem for 1-Bounded Conflict-Free Petri Nets Using Linear Programming , 1992, Inf. Process. Lett..

[5]  Monika Heiner,et al.  Petri Net Based Software Validation Prospects and Limitations , 2004 .

[6]  M. W. Shields.  An Introduction to Automata Theory , 1988 .

[7]  Hideki Yamasaki.  Normal Petri Nets , 1984, Theor. Comput. Sci..

[8]  Hsu-Chun Yen,et al.  Normal and Sinkless Petri Nets , 1989, J. Comput. Syst. Sci..

[9]  Wolfgang Reisig,et al.  Petri Nets , 1985, EATCS Monographs on Theoretical Computer Science.

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

[11]  Hsu-Chun Yen.  A Polynomial Time Algorithm to Decide Pairwise Concurrency of Transitions for 1-Bounded Conflict-Free Petri Nets , 1991, Inf. Process. Lett..

[12]  Umberto Nanni,et al.  Linear Time Algorithms for Liveness and Boundedness in Conflict-free Petri Nets , 1992, LATIN.

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

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

[15]  Hsu-Chun Yen,et al.  A Taxonomy of Fairness and Temporal Logic Problems for Petri Nets , 1988, Theor. Comput. Sci..

[16]  Hsu-Chun Yen,et al.  A Unified Approach for Deciding the Existence of Certain Petri Net Paths , 1992, Inf. Comput..

[17]  Tadao Kasami,et al.  Three measures for synchronic dependence in Petri nets , 1983, Acta Informatica.

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

[19]  S. Rao Kosaraju,et al.  Decidability of reachability in vector addition systems (Preliminary Version) , 1982, STOC '82.

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

[21]  Tadao Murata,et al.  Petri nets: Properties, analysis and applications , 1989, Proc. IEEE.

[22]  守屋 悦朗,et al.  J.E.Hopcroft, J.D. Ullman 著, "Introduction to Automata Theory, Languages, and Computation", Addison-Wesley, A5変形版, X+418, \6,670, 1979 , 1980 .

[23]  Rüdiger Valk,et al.  The residue of vector sets with applications to decidability problems in Petri nets , 1985, Acta Informatica.

[24]  Rodney R. Howell,et al.  On Questions of Fairness and Temporal Logic for Conflict- Free Petri Nets , 1988, European Workshop on Applications and Theory of Petri Nets.

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

[26]  Grzegorz Rozenberg,et al.  Advances in Petri Nets 1988 , 1987, Lecture Notes in Computer Science.

[27]  Rodney R. Howell,et al.  Completeness Results for Conflict-Free Vector Replacement Systems , 1988, J. Comput. Syst. Sci..

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

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

[30]  Jeffrey D. Ullman,et al.  Introduction to Automata Theory, Languages and Computation , 1979 .

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

[32]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[33]  Tadao Murata,et al.  B-Fairness and Structural B-Fairness in Petri Net Models of Concurrent Systems , 1992, J. Comput. Syst. Sci..

[34]  M. Garey.  Johnson: computers and intractability: a guide to the theory of np- completeness (freeman , 1979 .

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