Advertisement

Labeled homology of higher-dimensional automata

  • Thomas KahlEmail author
Article

Abstract

We construct labeling homomorphisms on the cubical homology of higher-dimensional automata and show that they are natural with respect to cubical dimaps and compatible with the tensor product of HDAs. We also indicate two possible applications of labeled homology in concurrency theory.

Keywords

Higher-dimensional automata Labeled homology Cubical homology Cubical dimap 

Mathematics Subject Classification

55N35 55U15 68Q85 68Q45 68Q60 

Notes

References

  1. Bourbaki, N.: Algebra I. Addison-Wesley, Reading (1974)zbMATHGoogle Scholar
  2. Dijkstra, E.: Hierarchical ordering of sequential processes. Acta Inform. 1(2), 115–138 (1971)MathSciNetGoogle Scholar
  3. Dubut, J., Goubault, E., Goubault-Larrecq, J.: Directed homology theories and Eilenberg-Steenrod axioms. Appl. Categ. Struct. 25(5), 775–807 (2017)MathSciNetzbMATHGoogle Scholar
  4. Fahrenberg, U.: Directed homology. Electron. Notes Theor. Comput. Sci. 100, 111–125 (2004)MathSciNetzbMATHGoogle Scholar
  5. Fajstrup, L., Raußen, M., Goubault, E.: Algebraic topology and concurrency. Theor. Comput. Sci. 357, 241–278 (2006)MathSciNetzbMATHGoogle Scholar
  6. Fajstrup, L., Goubault, E., Haucourt, E., Mimram, S., Raussen, M.: Directed Algebraic Topology and Concurrency. Springer, Berlin (2016)zbMATHGoogle Scholar
  7. Gaucher, P.: Homological properties of non-deterministic branchings and mergings in higher dimensional automata. Homol. Homotopy Appl. 7(1), 51–76 (2005)MathSciNetzbMATHGoogle Scholar
  8. Glabbeek, R.V.: On the expressiveness of higher dimensional automata. Theor. Comput. Sci. 356(3), 265–290 (2006)MathSciNetzbMATHGoogle Scholar
  9. Goubault, E., Jensen, T.: Homology of higher-dimensional automata. In: Proceedings of CONCUR ’92, Springer, Lecture Notes in Computer Science, vol. 630, pp. 254–268 (1992)Google Scholar
  10. Grandis, M.: Directed Algebraic Topology: Models of Non-reversible Worlds, New Mathematical Monographs, vol. 13. Cambridge University Press, Cambridge (2009)zbMATHGoogle Scholar
  11. Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2001)zbMATHGoogle Scholar
  12. Hilton, P., Wylie, S.: Homology Theory: An Introduction to Algebraic Topology. Cambridge University Press, Cambridge (1967)zbMATHGoogle Scholar
  13. Kahl, T.: The homology graph of a precubical set. Homol. Homotopy Appl. 16(1), 119–138 (2014a)MathSciNetzbMATHGoogle Scholar
  14. Kahl, T.: Weak morphisms of higher dimensional automata. Theor. Comput. Sci. 536, 42–61 (2014b)MathSciNetzbMATHGoogle Scholar
  15. Kahl, T.: Topological abstraction of higher-dimensional automata. Theor. Comput. Sci. 631, 97–117 (2016)MathSciNetzbMATHGoogle Scholar
  16. Kahl, T.: Higher-dimensional automata modeling shared-variable systems, pp. 1–21 (2018a). arXiv:1801.08451 [csLO]
  17. Kahl, T.: pg2hda [Computer software]. (2018b). http://w3.math.uminho.pt/~kahl/. Accessed 2 Oct 2018
  18. Krishnan, S.: Flow-cut dualities for sheaves on graphs, pp. 1–29 (2014). arXiv:1409.6712 [mathAT]
  19. Massey, W.: Singular Homology Theory, Graduate Texts in Mathematics, vol. 70. Springer, Berlin (1980)Google Scholar
  20. Peterson, G.: Myths about the mutual exclusion problem. Inf. Process. Lett. 12(3), 115–116 (1981)zbMATHGoogle Scholar
  21. Pilarczyk, P.: CHomP [Computer software]. (2002–2018). http://chomp.rutgers.edu/Projects/Computational_Homology/OriginalCHomP/software/. Accessed 12 Feb 2018
  22. Pratt, V.: Modeling concurrency with geometry. In: POPL ’91, Proceedings of the 18th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages. ACM New York, NY, USA, pp. 311–322 (1991)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Centro de MatemáticaUniversidade do MinhoBragaPortugal

Personalised recommendations