Homotopies in Multiway (Nondeterministic) Rewriting Systems as n-Fold Categories

Arsiwalla, Xerxes D.; Gorard, Jonathan; Elshatlawy, Hatem

doi:10.25088/complexsystems.31.3.261

Cited by 11 publications

(11 citation statements)

References 13 publications

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“…This means that we have a good model for exploring the larger-scale effect of growth on the information content of PIPs and testing it against cosmological models and some aspects of the mycelial comparison. However, it also means that specifically non-apical lateral branching, and anastamosis are not observed 46 . Nevertheless, the spanning-tree structures and non-trivial connections are observed.…”

Section: Resultsmentioning

confidence: 99%

“…In subsection 4.1 we introduce the growth only model. In Subsection 4.2 we show some graphical implementations of the growth-only model with spring electrical embedding and discuss correspondences with mycelium networks 47 . In subsection 4.3, building on the results from [61], we show evidence of Hubble horizons, resulting from the exponential spatial expansion and information propagation of the SERD network.…”

Section: Resultsmentioning

confidence: 99%

“…These behaviours will be observed once reductions are also included in the base code of the SERD network. 46 This will be the topic of a follow-up paper 47 Albeit somewhat dampened by the exclusion of SE reductions 48 Another way to define this is that pr → 0. 49 This code is simpler than one which includes reductions since it does not require deleting elements and IGs, updating SE neighbours in PSBs and superposing information on IGs.…”

Section: Structural Similarities In Emergent Topology Of Serd Network...mentioning

confidence: 99%

“…More recent work by Gorard and others has demonstrated that such discrete networks, with specific substitution rules, are not only numerous and compatible with physical laws [42, 43, 44, 45, 46, 47], but have a whole host of applications relating to the symbolic structure of mathematics, logic and algorithms, implying underlying structural connections between of what could be the foundations the structure of space-time and relations in abstract areas of thought [48]. There is some demonstration of the compatibility with special and general relativity and a structure for understanding quantum mechanical properties.…”

Section: Introductionmentioning

confidence: 99%

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Exploring Discrete Space-Time Models for Information Transfer: Analogies from Mycelial Networks to the Cosmic Web

Wood,

Sorakivi,

Ayres

et al. 2023

Preprint

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Fungal mycelium networks are large scale biological networks along which nutrients, metabolites flow. Recently, we discovered a rich spectrum of electrical activity in mycelium networks, including action-potential spikes and trains of spikes. Reasoning by analogy with animals and plants, where travelling patterns of electrical activity perform integrative and communicative mechanisms, we speculated that waves of electrical activity transfer information in mycelium networks. Using a new discrete space-time model with emergent radial spanning-tree topology, hypothetically comparable mycelial morphology and physically comparable information transfer, we provide physical arguments for the use of such a model, and by considering growing mycelium network by analogy with growing network of matter in the cosmic web, we develop mathematical models and theoretical concepts to characterise the parameters of the information transfer.

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Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

Section: Structural Similarities In Emergent Topology Of Serd Network...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Exploring Discrete Space-Time Models for Information Transfer: Analogies from Mycelial Networks to the Cosmic Web

Wood,

Sorakivi,

Ayres

et al. 2023

Preprint

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“…a massive scalar field obeying the discrete Klein-Gordon equation) for the stress-energy tensor. Recent work investigating global homotopic aspects of Wolfram model systems [95][96] provides one plausible direction by which it might conceivably be possible to formulate a more global topological theorem regarding geodesic incompleteness in generic Wolfram model systems, in a similar style to Penrose's original 1964 singularity theorem [7].…”

Section: Discussionmentioning

confidence: 99%

Non-Vacuum Solutions, Gravitational Collapse and Discrete Singularity Theorems in Wolfram Model Systems

Gorard¹

2023

Preprint

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The celebrated geodesic congruence equation of Raychaudhuri, together with the resulting singularity theorems of Penrose and Hawking that it enabled, yield a highly general set of conditions under which a spacetime (or, more generically, a pseudo-Riemannian manifold) is expected to become geodesically incomplete. However, the proofs of these theorems traditionally depend upon a collection of assumptions about the continuum spacetime (and, in the physical case, the stress-energy distribution defined over it), including its global structure, its energy conditions, the existence of trapped null surfaces and the various volume/intersection properties of geodesic congruences, that are inherently difficult to translate to the case of discrete spacetime formalisms, such as causal set theory or the Wolfram model. Some, such as the discrete analog of Raychaudhuri's equation for the volumes of geodesic congruences, are subtle to formulate due to intrinsic differences in the behavior of discrete vs. continuous geodesics; for others, such as the definition of a trapped null surface, no appropriate translation is known for the general discrete case due to a lack of a priori coordinate information. It is therefore a non-trivial question to ask whether (and to what extent) there exist equivalently general conditions under which one expects discrete spacetimes to become geodesically incomplete, and how these conditions might differ from those in the continuum. This article builds upon previous work, in which the conformal and covariant Z4 (CCZ4) formulation of the Cauchy problem for the Einstein field equations, with constraint-violation damping, was defined in terms of Wolfram model evolution over discrete (spatial) hypergraphs for the case of vacuum spacetimes, and proceeds to consider a minimal extension to the non-vacuum case by introducing a massive scalar field distribution, defined in either spherical or axial symmetry. Under appropriate assumptions, this scalar field distribution admits a physical interpretation as a collapsing (and, in the axially-symmetric case, uniformly rotating) dust, and we are able to show, through a combination of rigorous mathematical analysis and explicit numerical simulation, that the resulting discrete spacetimes converge asymptotically to either non-rotating Schwarzschild black hole solutions or maximally-rotating (extremal) Kerr black hole solutions, respectively. Although the assumptions used in obtaining these preliminary results are very strong, they nevertheless offer hope that a more general, perhaps ultimately ``Penrose-like'', singularity theorem may be provable in the discrete spacetime case too.

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Pregeometric Spaces from Wolfram Model Rewriting Systems as Homotopy Types

Arsiwalla,

Gorard

2024

Int J Theor Phys

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How do spaces in physics emerge from pregeometric discrete building blocks governed by computational rules? To address this question we investigate non-deterministic rewriting systems (so-called multiway systems) of the Wolfram model. We express these rewriting systems as homotopy types. Using this new formulation of the model, we motivate how spatial structures are functorially inherited from pregeometric type-theoretic constructions. We show how higher homotopies are constructed from rewriting rules. These correspond to morphisms of an n-fold category. Subsequently, the $$n \rightarrow \infty $$ n → ∞ limit of the Wolfram model rulial multiway system is identified as an $$\infty $$ ∞ -groupoid, with the latter being relevant in light of Grothendieck’s homotopy hypothesis. We then go on to show how this construction extends to the classifying space of rulial multiway systems, which forms a multiverse of multiway systems and carries the formal structure of an $${\left( \infty , 1\right) }$$ ∞ , 1 -topos. This correspondence to higher categorical structures potentially offers a new way to understand how the kinds of spaces relevant to physics result from pregeometric combinatorial models. The key issue we have addressed in this work is to relate abstract non-deterministic rewriting systems to higher homotopy spaces. A consequence of constructing spaces and geometry synthetically is that it removes the need to make ad hoc assumptions about geometric attributes of a model such as an a priori background or any pre-assigned geometric data. Instead, geometry is inherited functorially from higher structures. This can be particularly useful for formally justifying different choices of underlying spacetime discretization schemes adopted by various models of quantum gravity. We conclude with comments on how our framework of higher category-theoretic combinatorial constructions closely corroborates with other approaches investigating higher categorical structures relevant to the foundations of physics.

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Homotopies in Multiway (Nondeterministic) Rewriting Systems as n-Fold Categories

Cited by 11 publications

References 13 publications

Exploring Discrete Space-Time Models for Information Transfer: Analogies from Mycelial Networks to the Cosmic Web

Exploring Discrete Space-Time Models for Information Transfer: Analogies from Mycelial Networks to the Cosmic Web

Non-Vacuum Solutions, Gravitational Collapse and Discrete Singularity Theorems in Wolfram Model Systems

Pregeometric Spaces from Wolfram Model Rewriting Systems as Homotopy Types

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