Causal sets from simple models of computation

Causal Sets from simple models of computation Tommaso Bolognesi CNR/ISTI - Via Moruzzi 1, 56124 Pisa, Italy t.bolognesi@isti.cnr.it Abstract Causality among events is widely recognized as a most fundamental structure of spacetime, and causal sets have been proposed as discrete models of the latter in the context of quantum gravity theories, notably in the Causal Set Programme. In the rather different context of what might be called the 'Computational Universe Programme' -- one which associates the complexity of physical phenomena to the emergent features of models such as cellular automata -- a choice problem arises with respect to the variety of formal systems that, in virtue of their computational universality (Turing-completeness), qualify as equally good candidates for a computational, unified theory of physics. This paper proposes Causal Sets as the only objects of physical significance and relevance to be considered under the 'computational universe' perspective, and as the appropriate abstraction for shielding the unessential details of the many different computationally universal candidate models. At the same time, we propose a fully deterministic, radical alternative to the probabilistic techniques currently considered in the Causal Set Programme for growing discrete spacetimes. We investigate a number of computation models by grouping them into two broad classes, based on the support on which they operate; in one case this is linear, like a tape or a string of symbols; in the other, it is a two-dimensional grid or a planar graph. For each model we identify the causality relation among computation events, implement it, and conduct a possibly exhaustive exploration of the associated causal set space, while examining quantitative and qualitative features such as dimensionality, curvature, planarity, emergence of pseudo-randomness, causal set substructures and particles. 1. Introduction In the last few decades several scientists (K. Zuse, J. A. Wheeler, R. Feynman, E. Fredkin, S. Wolfram, G. 't Hooft, S. Lloyd, J. Schmidhuber, to mention a few) have contributed, in a variety of ways and degrees, to the birth and progression of a wave of ideas and conjectures that could be collectively named ‘Computational Universe Programme’. In its most extreme form, this programme suggests that all the complexity we observe in the physical universe might emerge from the iteration of a few simple transition rules, that could be implemented by a short computer program. The fact that simple programs can produce highly complex patterns, sometimes similar to those found in nature, is widely recognized today, and has been investigated and divulgated, in particular, by S. Wolfram, with his extensive behavioural analysis of cellular automata and other simple models [21]. These facts are taken by some researchers -- not without considerable skepticism by others -- as a basis, or, at least, an inspiring metaphor for trying to devise a radically new, computation-based, spectacularly simple fundamental theory of physics. Once the idea is adopted of understanding complexity in physics as emergence in computation, an obvious question arises: which model of computation? It seems reasonable to restrict to 'universal' models, in the sense of Turing-complete, since computational universality is indeed supported by our physical universe (e.g. in computers!). But universality turns out to be quite cheap, and at reach for many simple formal models; according to a conjecture by Wolfram [21], universality is indeed found in any artificial or natural system capable of pseudo-random behaviour. Fredkin [4] suggests an interesting choice criterion: "We might be able to demonstrate that an ordinary computer model of physics is sufficient, but we cannot normally show that it is necessary. The reason is that any and all models of finite nature can be replaced by equivalent computational models based on any universal computer. [...] DM (Digital Mechanics) implies that there is a computer-like model that has a bijective mapping, one to one, from states and function in the real world to states and function in the model. [...] The beauty of the 'one-to-one mapping onto' restriction is that it delivers us from the apparent tyranny of computation universality. It is unlikely that there will be more than one such correct model." An illustration of the above concept is provided by Fredkin himself: a flight simulator running on a PC is not a good model of an airplane since many of the data items manipulated by the computer are not related to the airplane, but, say, to the PC operating system, and, conversely, many structural and behavioral aspects of the real airplane are abstracted away in the simulation. Based on these criteria, Fredkin is lead to focus on a precise family of computation models, namely cellular automata; in particular, he investigates second-order, reversible, universal automata (RUCA), and a conservative model c

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