Abstract
In the context of research efforts on causal sets as discrete models of physical spacetime, and on their derivation from simple, deterministic, sequential models of computation, we consider boolean nets, a transition system that generalises cellular automata, and investigate the family of causal sets that derive from their computations, in search for interesting emergent properties. The choice of boolean nets is motivated by the fact that they naturally support compositions via a LOTOS-inspired parametric parallel operator, with possible interesting effects on the emergent structure of the derived causal sets.
More generally, we critically reconsider the whole issue of algorithmic causet construction and expose the limitations suffered by these structures w.r.t. to the requirements of Lorentz invariance that even discrete models of physical spacetime, as recently shown, can and should satisfy. We conclude by hinting at novel ways to add momentum to the bold research programme that attempts to identify the natural with the computational universe.
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Notes
- 1.
Of course the range of Ed’s activities is broader, as suggested by the Festschrift title ‘ModelEd, TestEd, TrustEd’. Indeed, the addition of ‘randomisEd’ wouldn’t be completely inappropriate, in light of an episode which involved a small group of ‘LOTOS-eaters’ during a relaxing late-evening walk in a forgotten European city. On that occasion Prof. Brinksma, dissatisfied with the manipulations performed on the Rubik Magic Rings puzzle by the author – dismissed as insufficiently random – gave a public, truly brilliant demonstration of his unexpected randomisation skills.
- 2.
Note that, unless decorated with appropriate edge priority assignments, the graph is not sufficient for correctly identifying the order of function arguments: this is disambiguated in F.
- 3.
In four dimensional, Minkowski spacetime \(M^{1+3}\), with time dimension t and spatial dimensions x, y, z, the squared Lorentz distance between events \(e_1(t_1, x_1, y_1, z_1)\) and \(e_2(t_2, x_2, y_2, z_2)\) is \(L^2(e_1,e_2) = +(t_2-t_1)^2 - (x_2-x_1)^2 - (y_2-y_1)^2 - (z_2-z_1)^2\).
- 4.
For k = 3, for example, we typically set \(\alpha (0,0,0) = 0\), \(\alpha (0,0,1) = 1, \dots \), \(\alpha (1,1,1) = 7\), with \(L = \{0 \dots 7\}\) ordered in the natural way. In the sequel we shall also create a different labelling function for each different bool net bit by considering different rotations of the range tuple \((0 \dots 7)\).
- 5.
Recall that we always consider the causet in its transitively reduced form, or Hasse diagram, whose edges are often called ‘links’.
- 6.
In the graphical rendering of causets, we may render differently (black/gray/dashed) the edges that point to a node, depending on whether that node corresponds to a transition from P, from Q, or from both. This is the criterion adopted for Figs. 4 and 5. As an alternative, we may directly paint the causets node differently, as done for the subsequent figures.
- 7.
Several techniques are available for measuring the dimension of a graph [23]. Unfortunately their estimates may disagree! For the mentioned example we have used the ‘node shell growth rate’ technique, which provided a dimension 3 estimate but only relative to the node shells centered a the causet root.
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Acknowledgements
The author expresses his warmest gratitude to Larissa Albantakis for many stimulating exchanges and discussions on notions of causality, Effective Information, Integrated Information. Lack of time has prevented us from completing our joint investigation of the possible applications of these recently proposed informational measures to the synchronous/asynchronous, deterministic/nondeterministic, unstructured/composite boolean networks considered here. This will be the subject of a forthcoming paper.
This work has been partially funded by FQXi Mini-Grant number: FQXi-MGA-1702.
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Bolognesi, T. (2017). LOTOS-Like Composition of Boolean Nets and Causal Set Construction. In: Katoen, JP., Langerak, R., Rensink, A. (eds) ModelEd, TestEd, TrustEd. Lecture Notes in Computer Science(), vol 10500. Springer, Cham. https://doi.org/10.1007/978-3-319-68270-9_2
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