Geodesic
Homology Groups of a Pipeline Petri Net
Modelirovanie i analiz informacionnyh sistem, Tome 20 (2013) no. 2, pp. 92-103 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Petri net is said to be elementary if every place can contain no more than one token. In this paper, it is studied topological properties of the elementary Petri net for a pipeline consisting of n functional devices. If the work of the functional devices is considered continuous, we can come to some topological space of “intermediate” states. In the paper, it is calculated the homology groups of this topological space. By induction on n, using the Addition Sequence for homology groups of semicubical sets, it is proved that in dimension 0 and 1 the integer homology groups of these nets are equal to the group of integers, and in the remaining dimensions are zero. Directed homology groups are studied. A connection of these groups with deadlocks and newsletters is found. This helps to prove that all directed homology groups of the pipeline elementary Petri nets are zeroth.
Keywords: trace monoid, elementary Petri net, pipeline, semicubical set, homology of small categories.
Mots-clés : asynchronous transition system 
@article{MAIS_2013_20_2_a6,
     author = {A. A. Husainov and E. S. Bushmeleva and T. A. Trishina},
     title = {Homology {Groups} of a {Pipeline} {Petri} {Net}},
     journal = {Modelirovanie i analiz informacionnyh sistem},
     pages = {92--103},
     year = {2013},
     volume = {20},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MAIS_2013_20_2_a6/}
}
TY  - JOUR
AU  - A. A. Husainov
AU  - E. S. Bushmeleva
AU  - T. A. Trishina
TI  - Homology Groups of a Pipeline Petri Net
JO  - Modelirovanie i analiz informacionnyh sistem
PY  - 2013
SP  - 92
EP  - 103
VL  - 20
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/MAIS_2013_20_2_a6/
LA  - ru
ID  - MAIS_2013_20_2_a6
ER  - 
%0 Journal Article
%A A. A. Husainov
%A E. S. Bushmeleva
%A T. A. Trishina
%T Homology Groups of a Pipeline Petri Net
%J Modelirovanie i analiz informacionnyh sistem
%D 2013
%P 92-103
%V 20
%N 2
%U http://geodesic.mathdoc.fr/item/MAIS_2013_20_2_a6/
%G ru
%F MAIS_2013_20_2_a6
A. A. Husainov; E. S. Bushmeleva; T. A. Trishina. Homology Groups of a Pipeline Petri Net. Modelirovanie i analiz informacionnyh sistem, Tome 20 (2013) no. 2, pp. 92-103. http://geodesic.mathdoc.fr/item/MAIS_2013_20_2_a6/

[1] A. A. Husainov, “On the homology of small categories and asynchronous transition systems”, Homology Homotopy Appl., 6:1 (2004), 439–471 http://www.rmi.acnet.ge/hha | MR | Zbl

[2] E. Goubault, The Geometry of Concurrency, Thesis Doct. Phylosophy (Mathematics), Ecole Normale Supérieure, 1995

[3] P. Gaucher, “About the globular homology of higher dimensional automata”, Topol. Geom. Differ., 43:2 (2002), 107–156 | MR | Zbl

[4] E. Goubault, E. Haucourt, S. Krishnan, “Covering space theory for directed topology”, Theory Appl. Categ., 22:9 (2009), 252–268 | MR | Zbl

[5] A. A. Husainov, “The Homology of Partial Monoid Actions and Petri Nets”, Appl. Categor. Struct., 2012 | DOI

[6] A. A. Husainov, E. S. Bushmeleva, “Gomologii asinhronnyh sistem”, Aktualnye problemy matematiki, fiziki, informatiki v vuze i shkole, Materialy Mezhdunarodnoj nauchno-prakticheskoj konferencii (23 marta 2012, Komsomolsk-na-Amure), Izd-vo AmGPGU, Komsomolsk-na-Amure, 2012, 24–31 (in Russian)

[7] G. Winskel, M. Nielsen, “Models for Concurrency”, Handbook of Logic in Computer Science, v. IV, eds. Abramsky, Gabbay, Maibaum, Oxford University Press, 1995, 1–148 | MR

[8] M. Nielsen, G. Winskel, “Petri nets and bisimulation”, Theoretical Computer Science, 153:1–2 (1996), 211–244 | DOI | MR | Zbl

[9] A. A. Khusainov, “Homology groups of semicubical sets”, Sib. Math. J., 49:1 (2008), 593–604 | DOI | MR | Zbl