Bulking II: Classifications of cellular automata

Delorme, Marianne; Mazoyer, J.; Ollinger, Nicolas; Theyssier, Guillaume

doi:10.1016/j.tcs.2011.02.024

Cited by 50 publications

(67 citation statements)

References 34 publications

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“…Does there exist a tile set U for the abstract Tile Assembly Model, such that for any (adversarially chosen) seed assembly σ , at temperature 2, this tile assembly system simulates some tile assembly system T ? Moreover, U should be able to simulate all such members T of some non-trivial class S. U is a tile set that can do one thing and nothing else: simulate tile assembly systems from the class S. This question about U is inspired by the factor simulation question in CA [29], although it differs in the details.…”

Section: Discussionmentioning

confidence: 99%

Intrinsic universality and the computational power of self-assembly

Woods

2015

Phil. Trans. R. Soc. A.

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Molecular self-assembly, the formation of large structures by small pieces of matter sticking together according to simple local interactions, is a ubiquitous phenomenon. A challenging engineering goal is to design a few molecules so that large numbers of them can self-assemble into desired complicated target objects. Indeed, we would like to understand the ultimate capabilities and limitations of this bottom-up fabrication process. We look to theoretical models of algorithmic self-assembly, where small square tiles stick together according to simple local rules in order to carry out a crystal growth process. In this survey, we focus on the use of simulation between such models to classify and separate their computational and expressive powers. Roughly speaking, one model simulates another if they grow the same structures, via the same dynamical growth processes. Our journey begins with the result that there is a single intrinsically universal tile set that, with appropriate initialization and spatial scaling, simulates any instance of Winfree's abstract Tile Assembly Model. This universal tile set exhibits something stronger than Turing universality: it captures the geometry and dynamics of any simulated system in a very direct way. From there we find that there is no such tile set in the more restrictive non-cooperative model, proving it weaker than the full Tile Assembly Model. In the two-handed model, where large structures can bind together in one step, we encounter an infinite set of infinite hierarchies of strictly increasing simulation power. Towards the end of our trip, we find one tile to rule them all: a single rotatable flipable polygonal tile that simulates any tile assembly system. We find another tile that aperiodically tiles the plane (but with small gaps). These and other recent results show that simulation is giving rise to a kind of computational complexity theory for self-assembly. It seems this could be the beginning of a much longer journey, so directions for future work are suggested.

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Section: Discussionmentioning

confidence: 99%

Intrinsic universality and the computational power of self-assembly

Woods

2015

Phil. Trans. R. Soc. A.

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“…Moreover, a detailed classification of cellular automata based on their properties has been an open problem for a long time [24,25,26]. This wide variety of behaviors and characteristics of these systems make them particularly relevant for the modeling of some natural phenomena (when many identical elements are in short-range interaction).…”

Section: Proposed Approachmentioning

confidence: 99%

New Symmetrical Cryptosystems Based on a Dynamic Optimization Model : two-Dimensional non-Uniform Cellular Automata

Souici¹,

Seridi²,

Akdag³

2015

IJCIS

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Section: Introductionmentioning

confidence: 99%

“…This paper is concerned with the choice of the main ingredients to define an interesting bulking. The second paper, Bulking II: Classifications of Cellular Automata [4], studies the structure of the main three varieties of bulking.A cellular automaton is a discrete dynamical system consisting of a network of cells fulfilling the following properties: each cell acts as a finite state machine; the network is regular; interactions are local, uniform and synchronous. A spacetime diagram is the geometrical representation of an orbit obtained by piling up the successive configurations.…”

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confidence: 99%

Bulking I: An abstract theory of bulking

Delorme¹,

Mazoyer²,

Ollinger³

et al. 2011

Theoretical Computer Science

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a b s t r a c tThis paper is the first part of a series of two papers dealing with bulking: a quasi-order on cellular automata comparing space-time diagrams up to some rescaling. Bulking is a generalization of grouping taking into account universality phenomena, giving rise to a maximal equivalence class. In the present paper, we discuss the proper components of grouping and study the most general extensions. We identify the most general space-time transforms and give an axiomatization of bulking quasi-order. Finally, we study some properties of intrinsically universal cellular automata obtained by comparing grouping to bulking.Bulking is introduced as a tool to structure cellular automata, considered as the sets of their orbits. To achieve this goal, sets of orbits are considered up to spatio-temporal transforms. Such quotients are then compared according to algebraic relations to obtain quasi-orders on the set of cellular automata, in a way similar to reductions in the case of recursive functions. It turns out that the obtained equivalence classes tend to capture relevant properties: in particular, the greatest element, when it exists, corresponds to a notion of intrinsic universality. This paper is concerned with the choice of the main ingredients to define an interesting bulking. The second paper, Bulking II: Classifications of Cellular Automata [4], studies the structure of the main three varieties of bulking.A cellular automaton is a discrete dynamical system consisting of a network of cells fulfilling the following properties: each cell acts as a finite state machine; the network is regular; interactions are local, uniform and synchronous. A spacetime diagram is the geometrical representation of an orbit obtained by piling up the successive configurations. The present paper aims at structuring the sets of space-time diagrams generated by cellular automata.To deal with the richness of these objects, some families of space-time diagrams are identified through spatio-temporal transforms preserving the notion of cellular automata. In particular, one type of transform, commonly used in algorithmic constructions in the cellular automata literature, is to be taken into account: cells grouping.A very common grouping transform appears early in algorithmic constructions on cellular automata as a tool to simplify the description of the algorithm. A typical use of this tool appear in the work of Fischer [6]. To recognize prime numbers in real time, a first construction is given to recognize the primality of n at time 3n, the cells are then packed in 3 × 3 blocks defining a new cellular automaton achieving real-time recognition, as depicted in Fig. 1.Rescaling also appears when comparing neighborhoods, in particular the relation between first neighbors and the oneway neighborhood. To simulate, up to a translation, a first neighbors cellular automaton by a one-way cellular automaton, Choffrut and Čulik II [3] propose to add to the set of states every pair of original states and compute a transition in two time ✩ The re...

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Bulking II: Classifications of cellular automata

Cited by 50 publications

References 34 publications

Intrinsic universality and the computational power of self-assembly

Intrinsic universality and the computational power of self-assembly

New Symmetrical Cryptosystems Based on a Dynamic Optimization Model : two-Dimensional non-Uniform Cellular Automata

Bulking I: An abstract theory of bulking

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