On the analysis of Petri nets and their synthesis from process languages

Czaja, Ludwik

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 37 (2003) no. 1, pp. 17-38.

Processes in Place/Transition (P/T) nets are defined inductively by a peculiar numbering of place occurrences. Along with an associative sequential composition called catenation and a neutral process, a monoid of processes is obtained. The power algebra of this monoid contains all process languages with appropriate operations on them. Hence the problems of analysis and synthesis, analogous to those in the formal languages and automata theory, arise. Here, the analysis problem is: for a given P/T net with an initial marking find the set of all processes the net may evoke. The synthesis problem is: given a process language L decide if there exists a marked net whose evolutions (represented by processes) are collected in L and, in the positive case, find such net and its initial marking. The problems are posed and given a general solution.

[1] E. Best and C. Fernandez, Non-sequential Processes, A Petri Net View. Springer, Berlin, Heidelberg, New York, Tokyo, EATCS Monogr. Theoret. Comput. Sci. 13 (1988). | MR | Zbl

[2] R. Berghammer, B. Karger and C. Ulke, Relation-Algebraic Analysis of Petri Nets with RELVIEW, in Tools and Algorithms for the Construction and Analysis of Systems, Second Int. Workshop, TACAS'96, Passau, Germany. Springer-Verlag, Lecture Notes in Comput. Sci. 1055 (1996) 49-69.

[3] E. Best and R. Devillers, Sequential and Concurrent Behaviour in Petri Net Theory. Theoret. Comput. Sci. 55 (1987) 87-136. | MR | Zbl

[4] N. Busi and G.M. Pinna, Synthesis with Inhibitor Arcs, in Proc. of CONCUR'97, Warsaw, Poland. Springer-Verlag, Lecture Notes in Comput. Sci. 1243 (1997) 151-165.

[5] L.A. Castelano, Beta Processes of C/E Systems, Advances in Petri Nets. Springer-Verlag, Lecture Notes in Comput. Sci. 222 (1985) 83-100. | MR | Zbl

[6] L. Czaja, Net-Definability of Process Languages. Fund. Inform. 37 (1999) 213-223. | MR | Zbl

[7] L. Czaja, Process Languages and Nets. Theoret. Comput. Sci. 238 (2000) 161-181. | MR | Zbl

[8] L. Czaja and M. Kudlek, Lematy Iteracyjne dla Rownosciowo Definiowalnych Jezykow Procesow (in Polish), in Proc. of symposium Theoretical Informatics, Methods of Analysis of Incomplete and Distributed Information. Technical University Bialystok (1999) 8-23.

[9] L. Czaja and M. Kudlek, Rational, Linear and Algebraic Process Languages and Iteration Lemmata. Fund. Inform. 43 (2000) 49-60. | MR | Zbl

[10] L. Czaja and M. Kudlek, $\omega$-Process Languages for Place/Transition Nets. Fund. Inform. 47 (2001) 217-229. | MR | Zbl

[11] P. Darondeau, Deriving unbounded Petri nets from formal languages. IRISA, Internal Report No. 1172 (1998). | MR | Zbl

[12] J. Desel and W. Reisig, Place/Transition Petri Nets, in Lecture Notes on Petri Nets 1: Basic Models, edited by R.W. Rozenberg. Springer-Verlag, Lecture Notes in Comput. Sci. 1491 (1998) 122-173. | Zbl

[13] V. Diekert and G. Rozenberg, The Book of Traces. World Scientific, Singapore, New Jersey, London, Hong Kong (1995). | MR

[14] J. Esparza and M. Silva, On the Analysis and Synthesis of Free Choice Systems, in Advances in Petri Nets 1990. Springer-Verlag, Lecture Notes in Comput. Sci. 483 (1991) 243-286. | MR

[15] U. Goltz and W. Reisig, The Non-sequential Behaviour of Petri Nets. Inform. and Comput. 57 (1983) 125-147. | MR | Zbl

[16] R. Gorrieri, Refinement, Atomicity and Transactions for Process Description Languages, Ph.D. Thesis TD-2/91. Università degli Studi di Pisa Dipartimento di Informatica (1991).

[17] W.E. Kotov, Petri Nets (in Russian). NAUKA, Moscow (1984). | MR | Zbl

[18] A. Krzywicki, Processes in Place/Transition Nets with weights (in Polish), M.Sc. Thesis. Institute of Informatics, Warsaw University (2001).

[19] T. Kuzak, Sequential Behaviour of Nets with Unbounded Capacity of Places (in Polish), Ph.D. Thesis. Institute of Computer Science Foundations, Polish Academy of Sciences (1987).

[20] I.A. Lomazova, On Occurrence Net Semantics for Petri Nets with Contacts, in Proc. of Fundamentals of Computation Theory, 11th International Symposium, FCT'97, Krakow, Poland. Springer-Verlag, Lecture Notes in Comput. Sci. 1279 (1997) 317-328.

[21] A. Mazurkiewicz, Trace theory, edited by W. Brauer et al., Petri Nets, Applications and Relationship to other Models of Concurrency. Springer, Berlin-Heidelberg-New York, Lecture Notes in Comput. Sci. 255 (1987) 279-324. | MR | Zbl

[22] A. Mazurkiewicz, Concurrency, Modularity and Synchronization. Mathematical Foundations of Computer Science, Porabka-Kozubnik, Poland. Springer-Verlag, Lecture Notes in Comput. Sci. 379 (1989) 577-598. | Zbl

[23] A. Mazurkiewicz, Introduction to Trace Theory, in [13], pp. 3-41. | MR

[24] J. Meseguer, U. Montanari and V. Sassone, On the Semantics of Place/Transition Petri Nets. Math. Struct. Comput. Sci. 7 (1997) 359-397. | MR | Zbl

[25] M. Nielsen and G. Winskel, Trace Structures and other Models of Concurrency, in [13], pp. 271-305. | MR

[26] E. Ochmanski, Occurrence traces - Processes of elementary net systems, in Advances in Petri Nets 88. Springer-Verlag, Lecture Notes in Comput. Sci. 340, 331-342. | Zbl

[27] E. Ochmanski, Recognizable Trace Languages, in [13], pp. 168-204.

[28] M. Pietkiewicz-Koutny, Transition Systems of Elementary Net Systems with Inhibitor Arcs, in 18th International Conference on Application and Theory of Petri Nets, Toulouse, France. Springer-Verlag, Lecture Notes in Comput. Sci. 1248 (1997) 310-327.

[29] W. Reisig, Petri Nets. An Introduction. Springer, Berlin-Heidelberg-New York, Tokyo, EATCS Monogr. Theoret. Comput. Sci. 4 (1985). | MR | Zbl

[30] G. Rozenberg, Behaviour of elementary net systems, edited by W. Brauer, Petri nets: Central models and their properties; advances in Petri nets; proceedings of an advanced course, Bad Honef. Springer-Verlag, Lecture Notes in Comput. Sci. 254 (1986) 8-19. | MR | Zbl

[31] G. Rozenberg and J. Engelfriet, Elementary Net Systems, in Lectures on Petri Nets 1: Basic Models, edited by W. Reisig and G. Rozenberg. Springer-Verlag, Lecture Notes in Comput. Sci. 1491 (1998) 12-121. | Zbl

[32] P.H. Starke, Petri-Netze. Grundlagen, Anwendungen, Theorie. VEB Deutscher Verlag der Wissenschaften, Berlin (1980) Polish translation WNT (1987). | MR | Zbl

[33] J. Winkowski, Behaviours of Concurrent Systems. Theoret. Comput. Sci. 12 (1980) 39-60. | MR | Zbl