Topological Interpretation of Interactive Computation

  • Emanuela MerelliEmail author
  • Anita Wasilewska
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11500)


It is a great pleasure to write this tribute in honor of Scott A. Smolka on his 65th birthday. We revisit Goldin, Smolka hypothesis that persistent Turing machine (PTM) can capture the intuitive notion of sequential interaction computation. We propose a topological setting to model the abstract concept of environment. We use it to define a notion of a topological Turing machine (TTM) as a universal model for interactive computation and possible model for concurrent computation.


Persistent Turing machine Topological environment Topological Turing machine 



E. M. thanks Mario Rasetti for bringing her to conceive a new way of thinking about computer science and for numerous and lively discussions on topics related to this article; and Samson Abramsky with his group for insightful conversations on the topological interpretation of contextuality and contextual semantics. E. M. and A. W. thank the anonymous referees for suggesting many significant improvements.

Funding statements

We acknowledge the financial support of the Future and Emerging Technologies (FET) programme within the Seventh Framework Programme (FP7) for Research of the European Commission, under the FP7 FET-Proactive Call 8 - DyMCS, Grant Agreement TOPDRIM, number FP7-ICT-318121.


  1. 1.
    Goldin, D.Q., Smolka, S.A., Attie, P.C., Sondereggera, E.L.: Turing machines, transition systems, and interaction. In: Information and Computation 194, 2004. - ENTCS Vol. 52, No. 1, Elsevier (2001)Google Scholar
  2. 2.
    Goldin, D.Q.: Persistent turing machines as a model of interactive computation. In: Schewe, K.-D., Thalheim, B. (eds.) FoIKS 2000. LNCS, vol. 1762, pp. 116–135. Springer, Heidelberg (2000). Scholar
  3. 3.
    Goldin, D.Q., Smolka, S.A., Wegner, P.: Interacting Computation: The New Paradigm. Springer, Heidelberg (2006). Scholar
  4. 4.
    Wegner, P.: Why Intera is More P Than Algorit. CACM, vol. 40, No. 5. ACM (1997)Google Scholar
  5. 5.
    Wegner, P.: Interactive foundations of computing. TCS, vol. 192. Elsevier (1998)Google Scholar
  6. 6.
    Gandy, R.O.: Church’s thesis and principles for mechanisms. In: Barwise, J., Keisler, H.J., Kunen, K. (eds.) The Kleene Symposium. North-Holland Publishing Company (1980)Google Scholar
  7. 7.
    Wigderson, A.: Mathematics and Computation. IAS, Draft (March 2018)Google Scholar
  8. 8.
    Garrone, S., Marzuoli, A., Rasetti, M.: Spin networks, quantum automata and link invariants. J. Phys. Conf. Ser. 33, 95 (2006)CrossRefGoogle Scholar
  9. 9.
    Merelli, E., Pettini, M., Rasetti, M.: Topology driven modeling: the IS metaphor. Nat. Comput. 14(3), 421–430 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Carlsson, G.: Topology and data. Bull. Am. Math. Soc. 46(2), 255–308 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Rasetti, M., Merelli, E.: Topological field theory of data: mining data beyond complex networks. In: Contucci, P. (ed.) Advances in Disordered Systems. Random Processes and Some Applications. Cambridge University Press, Cambridge (2016)Google Scholar
  12. 12.
    Steenrod, N.: The topology of Fiber Bundles. Princeton Mathematical Series. Princeton University Press, Princeton (1951) CrossRefGoogle Scholar
  13. 13.
    Landin, P.J.: A Program Machine Symmetric Automata Theory. In: Meltzer and Michie (ed.) Machine Intelligence, Vol. 5. Edinburgh University Press (1969)Google Scholar
  14. 14.
    Abramsky, S.: An algebraic characterisation of concurrent composition. ArXiv:1406.1965v1 (2014)
  15. 15.
    Turing, A.M.: Lecture to the London Mathematical Society, 20 February 1947. Quoted in Carpenter, B.E., Doran, R.W. (eds.), A. M. Turing’s Ace Report of 1946 (1946)Google Scholar
  16. 16.
    Lewis, H., Papadimitriou, C.H.: Elements of the Theory of Computation, 2nd edn. Prentice Hall, Upper Saddle River (1998)Google Scholar
  17. 17.
    Abramsky, S.: Contextuality: at the borders of paradox. In: Landry, E. (ed.) Categories for the Working Philosophers (2017)Google Scholar
  18. 18.
    Abramsky, S.: Contextual semantics: from quantum mechanics to logic, databases, constraints, and complexity. ArXiv:1406.7386v1 (2014)
  19. 19.
    Abramsky, S.: What are the Fundamental Structures of Concurrency? We still don’t know! Electronic Notes in Theoretical Computer Science vol. 162 (2006)Google Scholar
  20. 20.

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of CamerinoCamerinoItaly
  2. 2.Department of Computer ScienceStony Brook UniversityStony BrookUSA

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