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Branching Space-Times and Parallel Processing

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New Challenges to Philosophy of Science

Part of the book series: The Philosophy of Science in a European Perspective ((PSEP,volume 4))


There is a remarkable similarity between some mathematical objects used in the Branching Space-Times framework and those appearing in computer science in the fields of event structures for concurrent processing and Chu spaces. This paper introduces the similarities and formulates a few open questions for further research, hoping that both BST theorists and computer scientists can benefit from the project.

This paper stems from a joint project with Thomas Müller (Universiteit Utrecht), who told me of the idea, triggered by a remark by Hu Liu, of connecting the Branching Space-Times theory to the approaches to parallel processing found in computer science.

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  1. 1.

    For the discussion of point events see the founding paper for the BST theory: Nuel Belnap, 1992, “Branching Space-Time”, in: Synthese 92, 3, pp. 385-434.

  2. 2.

    Ibid., p. 385.

  3. 3.

    See e.g. Nuel Belnap and László Szabó, 1992, “Branching Space-Time Analysis of the GHZ Theorem”, in: Foundations of Physics 26, 8, pp. 989-1002; and Tomasz Placek, 2010, “On Propensity-Frequentist Models for Stochastic Phenomena with Applications to Bell’s Theorem”, in: Tadeusz Czarnecki, Katarzyna Kijania-Placek, Olga Poller and Jan Woleński (Eds.), The Analytical Way, London: College Publications, pp. 105-140.

  4. 4.

    See e.g. Nuel Belnap, 2011, “Prolegomenon to Norms in Branching Space-Times”, in: Journal of Applied Logic 9, pp. 83-94.

  5. 5.

    See Thomas Müller, 2005, “Probability Theory and Causation: A Branching Space-Times Analysis”, in: British Journal for the Philosophy of Science 56, 3, pp. 487-520.

  6. 6.

    See Thomas Müller, 2002, “Branching Space-Time, Modal Logic and the Counterfactual Conditional”, in: Tomasz Placek and Jeremy Butterfield (Eds.), Non-locality and Modality, Dordrecht: Kluwer, pp. 273-291 and Leszek Wroński and Tomasz Placek, 2009, “On Minkowskian Branching Structures”, in: Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 40, 3, pp. 251-258.

  7. 7.

    This is an abbreviation dating back to the 1992 paper by Belnap (op. cit.), denoting “Our World”.

  8. 8.

    See Thomas Müller, 2010, “Towards a Theory of Limited Indeterminism in Branching Space-Times”, in: Journal of Philosophical Logic 39, pp. 395-423.

  9. 9.

    Building the formal machinery needed for this required many pages in ibid.

  10. 10.

    This opinion is based on conversations with other people accustomed with the BST framework.

  11. 11.

    We are very grateful to the Reviewer for suggesting this.

  12. 12.

    For details, see e.g. Tomasz Placek and Nuel Belnap, 2010, “Indeterminism is a Modal Notion: Branching Spacetimes and Earman’s Pruning”, in: Synthese, DOI 10.1007/s11229-010-9846-8.

  13. 13.

    For a description of the notion in the general context of modal logic, see e.g. Patrick Blackburn, Maarten de Rijke and Yde Venema, 2001, Modal Logic, Cambridge: Cambridge University Press.

  14. 14.

    For an overview see Vaughan Pratt, 2003, “Transition and Cancellation in Concurrency and Branching Time”, in: Mathematical Structures in Computer Science 13, 4, pp. 485-529.

  15. 15.

    Daniele Varacca, Hagen Völzer, and Glynn Winskel, 2006, “Probabilistic Event Structures and Domains”, in: Theoretical Computer Science 358, 2-3, pp. 173-199.

  16. 16.

    As a side-note, the following is a problem stated only once the connection between BST and ES has been noticed, but which relates to the current lack of deeper understanding of some fundamental aspects of BST: what are the sufficient and necessary conditions for a set of transitions to be the set of all transitions for some BST history?

  17. 17.

    Vaughan Pratt, 1995, “Chu Spaces and Their Interpretation as Concurrent Objects”, in: Computer Science Today 1000, pp. 392-405 (2005 version from the Author’s homepage, ).

  18. 18.

    Ibid., p. 3

  19. 19.

    The mathematical value of Chu spaces seems to stem from the so called “Chu transforms”, not introduced in this paper.

  20. 20.


  21. 21.

    Perhaps the Reader will find the following quote illuminating: “The rows present the physical, concrete, conjunctive, or yang aspects of the space, while the columns present the mental, coconcrete, disjunctive, or yin aspects” (ibid., p. 4).

  22. 22.

    See Müller, “Towards a Theory of Limited Indeterminism in Branching Space-Times”, op. cit.

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Correspondence to Leszek Wroński .

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Wroński, L. (2013). Branching Space-Times and Parallel Processing. In: Andersen, H., Dieks, D., Gonzalez, W., Uebel, T., Wheeler, G. (eds) New Challenges to Philosophy of Science. The Philosophy of Science in a European Perspective, vol 4. Springer, Dordrecht.

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