Summary
In a persistent vector replacement system (VRS) or Petri net, an enabled transition can become disabled only by firing itself. Here, an algorithm is presented which allows to decide whether an arbitrary VRS is persistent or not, and if so, to construct a semilinear representation of the set of states reachable in the system.
Similar content being viewed by others
References
Araki, T., Kasami, T.: Decidable problems on the strong connectivity of Petri net reachability sets. Theor. Comput. Sci 4, 99–119 (1977)
Dickson, L.E.: Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors. Amer. J. Math. 35, 413–422 (1913)
Ginsburg, S., Spanier, E.H.: Semigroups, Presburger formulas, and languages. Pacific J. Math. 16, 285–296 (1966)
Hack, M.: Decision problems for Petri nets and vector addition systems. M.I.T., Project MAC, MAC-TM 59 (1975)
Hack, M.: Decidability questions for Petri nets. M.I.T., LCS, TR 161 (1976)
Karp, R.M., Miller, R.E.: Parallel program schemata. J. Comput. System Sci. 3, 147–195 (1969)
Keller, R.M.: Vector replacement systems: A formalism for modelling asynchronous systems. Princeton University, CSL, TR 117 (1972)
Keller, R.M.: A fundamental theorem of asynchronous parallel computation. In: Parallel processing. (T.Y. Feng, ed.) Proceedings Sagamore Computer Conference. Lecture Notes in Computer Sciences, Vol. 24, pp. 102–112. Berlin Heidelberg New York: Springer, 1975
Landweber, L.H., Robertson, E.L.: Properties of conflict free and persistent Petri nets. J. Assoc. Comput. Mach. 25, 352–364 (1978)
Lipton, R.J., Miller, R.E., Snyder, L.: Synchronization and computing capabilities of linear asynchronous structures. Proc. 16th Ann. Symp. on FOCS. IEEE Computer Society 1975, pp. 19–28
Müller, H.: Decidability of reachability in persistent vector replacement systems. In: Mathematical Foundations of Computer Science 1980. Proceedings of the 9th Symposium in Rydzyna. (P. Dembinski, ed.) Lecture Notes in Computer Sciences, Vol. 88, pp. 426–438. Berlin Heidelberg New York: Springer (1980)
Muller, D.E., Bantky, M.S.: A theory of asynchronous circuits. Proc. Int. Symp. on Theory of Switching. Cambridge, MA: Harvard Univ. Press, p. 204–243, 1959
Oppen, D.C.: A 318-01 upper bound on the complexity of Presburger arithmetic. J. Comput. System Sci. 16, 323–332 (1978)
Parikh, R.J.: On context-free languages. J. Assoc. Comput. Mach. 13, 570–581 (1966)
Presburger, M.: Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt. Compte-Rendus du I. Congrès des Mathématiciens des pays Slavs, Warsaw, p. 92–101, 1930
Rosen, B.: Tree manipulating systems and Church-Rosser theorems. J. Assoc. Comput. Mach. 20, 160–187 (1973)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Mayr, E. Persistence of vector replacement systems is decidable. Acta Informatica 15, 309–318 (1981). https://doi.org/10.1007/BF00289268
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00289268