This Article 
 Bibliographic References 
 Add to: 
An Extended Petri Net Model for Normal Logic Programs
February 1995 (vol. 7 no. 1)
pp. 150-162

Abstract—This paper presents an application of the concepts of siphons (deadlocks) and inhibitor arcs in Petri net theory to logic programs with negations. More specifically, an extended Petri net is used to model function-free normal logic programs. In this model, because of the presence of inhibitor arcs, the arbitrary applications of firing rule may cause a contradictory situation. We suggest two directions to avoid contradictions: greedy and secure applications of firing rule. We choose the secure application in this paper and show that this is a direct translation of the well-founded semantics in the net model. Furthermore, we show that the greatest unfounded set corresponds to the greatest siphon in Petri net theory when we delete the transitions disabled by the secure application of firing rule, and that the property of siphon simplifies the computation of well-founded semantics for logic programs. We also propose the reduced-Petri-net method by which we can reduce an extended Petri net to a Petri net without inhibitor arcs and compute the well-founded model by iterative applications of this transformation using conventional application of firing rule.

[1] K.R. Apt and M.H. van Emden, "Contributions to the Theory of Logic Programming," J. ACM, vol. 29, no. 3, pp. 841-862, July 1982.
[2] C. Baral and V.S. Subrahmanian,“Dualities between alternative semantic for logic programming andnonmonotonic reasoning,” Proc. First Int’l Workshop of Logic Programming and Non-Monotonic Reasoning,Cambridge, Mass.: MIT Press, pp. 69-86, 1991.
[3] M. Gelfond and V. Lifschitz,“The stable model semantics for logic programming,” Proc. Fifth Int’l Conf. and Symp. on Logic Programming,New York: IEEE, pp. 1070-1080, 1988.
[4] M. Gelfond and H. Przymusinska,“Definitions in epistemic specifications,” Proc. First Int’l Workshop of Logic Programming and Non-Monotonic Reasoning,Cambridge, Mass.: MIT Press, pp. 245-259, 1991.
[5] V. Lifschitz,“Logical foundationsof deductive databases,” Information Processing Letters, vol. 89, pp. 315-321, 1989.
[6] C. Lin and T. Murata,“Applications of Petri nets to non-monotonic logic,” Proc. Joint Technical Conf. Circuits-Systems, Computer, and Comm., pp. 530-535,Cheju, Korea, Dec.10-11, 1990.
[7] J.W. Lloyd, Foundations of Logic Programming, Springer Series in Symbolic Computation, second ed. New York: Springer-Verlag, 1987.
[8] W. Marek and V.S. Subrahmanian,“The relationship between logic programs and non-monotonic reasoning,” Proc. Sixth Int’l Conf. Logic Programming, pp. 600-620, 1989.
[9] T. Murata, “Petri Nets: Properties, Analysis and Application,” Proc. IEEE, vol. 77, no. 4, 1989.
[10] T. Murata, V.S. Subrahmanian, and T. Wakayama, "A Petri Net Model for Reasoning in the Presence of Inconsistency," IEEE Trans. Knowledge and Data Eng., Vol. 3, No. 3, Sept. 1991, pp. 281-292.
[11] T. Murata and J. Yim,“Petri net method for real-time control of rule-based systems,” KSI Proc. First Int’l Conf. Software Eng. and Knowledge Eng.,Skokie, Ill., June 1989.
[12] T. Murata and D. Zhang, "A Predicate-Transition Net Model for Parallel Interpretation of Logic Programs," IEEE Trans. Software Eng., Vol. 14, No. 4, Apr. 1988, pp. 481-497.
[13] J.L. Peterson, Petri Net Theory and the Modeling of Systems.Englewood Cliffs, N.J.: Prentice Hall, 1981.
[14] T.C. Przymusinski, "Every Logic Program has a Natural Stratification and an Iterated Least Fixed Point Model," ACM SIGACT-SIGMOD-SIGART Symp. Principles of Database Systems, pp. 11-21, 1989.
[15] T.C. Przymusinski, "On the Declarative Semantics of Deductive Databases and Logic Programs," J. Minker, ed., Foundations of Deductive Databases and Logic Programming, pp. 193-216. Morgan Kaufmann, 1988.
[16] T.C. Przymusinski,“Three-valued formalizations of non-monotonic reasoning and logic programming,” Proc. First Int’l Conf. Knowledge Representation and Reasoning, pp. 341-348, 1989.
[17] R. Reiter,“A logic for default reasoning,” Artificial Intelligence, vol. 13, pp. 81-132, Apr. 1980.
[18] T. Shimura,J. Lobo,, and T. Murata,“A Petri net semantics for logic programs with negation,” Proc. Fourth Int’l Conf. Software Eng. and Knowledge Eng.,Capri, Italy, June, 1992.
[19] M.H. van Emden and R.A. Kowalski, “The Semantics of Predicate Logic as a Programming Language,” J. ACM, vol. 23, no. 4, pp. 733-742, Oct. 1976.
[20] A. van Gelder, "The Alternating Fixpoint of Logic Programs with Negation," Proc. ACM Symp. Principles of Database Systems, pp. 1-10, 1989.
[21] A. van Gelder,K. Ross, and J.S. Schlipf,"The well-founded semantics for general logic programs," J. ACM, vol. 38, no. 3, pp. 620-650, July 1991.
[22] D. Zhang and T. Murata,“Fixpoint semantics for predicate/transition netmodel for Horn clause logic programs,” to appear in Advances in Theory of Computation and Computational Mathematics, vol. 1, Norwood, N.J: Ablex.
[23] J. Lobo, J. Minker, and A. Rajasekar, Foundations of Disjunctive Logic Programming. Cambridge, Mass.: MIT Press, 1992.

Index Terms:
Logic programming, inhibitor arcs, negations, normal logic programs, Petri nets, siphons (deadlocks), well-founded semantics.
Teruhiro Shimura, Jorge Lobo, Tadao Murata, "An Extended Petri Net Model for Normal Logic Programs," IEEE Transactions on Knowledge and Data Engineering, vol. 7, no. 1, pp. 150-162, Feb. 1995, doi:10.1109/69.368512
Usage of this product signifies your acceptance of the Terms of Use.