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The tile assembly model is intrinsically universal
 In Proceedings of the 53rd Symposium on Foundations of Computer Science
, 2012
"... We prove that the abstract Tile Assembly Model (aTAM) of nanoscale selfassembly is intrinsically universal. This means that there is a single tile assembly system U that, with proper initialization, simulates any tile assembly system T. The simulation is “intrinsic ” in the sense that the selfasse ..."
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We prove that the abstract Tile Assembly Model (aTAM) of nanoscale selfassembly is intrinsically universal. This means that there is a single tile assembly system U that, with proper initialization, simulates any tile assembly system T. The simulation is “intrinsic ” in the sense that the selfassembly process carried out by U is exactly that carried out by T, with each tile of T represented by an m×m “supertile ” of U. Our construction works for the full aTAM at any temperature, and it faithfully simulates the deterministic or nondeterministic behavior of each T. Our construction succeeds by solving an analog of the cell differentiation problem in developmental biology: Each supertile of U, starting with those in the seed assembly, carries the “genome ” of the simulated system T. At each location of a potential supertile in the selfassembly of U, a decision is made whether and how to express this genome, i.e., whether to generate a supertile and, if so, which tile of T it will represent. This decision must be achieved using asynchronous communication under incomplete information, but it achieves the correct
Letting Alice and Bob choose which problem to solve: Implications to the study of cellular automata ✩
"... In previous works we found necessary conditions for a cellular automaton (CA) in order to be intrinsically universal (a CA is said to be intrinsically universal if it can simulate any other). The idea was to introduce different canonical communication problems, all of them parameterized by a CA. The ..."
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In previous works we found necessary conditions for a cellular automaton (CA) in order to be intrinsically universal (a CA is said to be intrinsically universal if it can simulate any other). The idea was to introduce different canonical communication problems, all of them parameterized by a CA. The necessary condition was the following: if Ψ is an intrinsically universal CA then the communication complexity of all the canonical problems, when parameterized by Ψ, must be maximal. In this paper, instead of introducing a new canonical problem, we study the setting where they can all be used simultaneously. Roughly speaking, when Alice and Bob –the two parties of the communication complexity model – receive their inputs they may choose online which canonical problem to solve. We give results showing that such freedom makes this new problem, that we call Ovrl, a very strong filter for ruling out CAs from being intrinsically universal. More precisely, there are some CAs having high complexity in all the canonical problems but have much lower complexity in Ovrl. Key words: automata. communication complexity, intrinsic universality, cellular 1.
Communications in cellular automata 1
"... The goal of this paper is to show why the framework of communication complexity seems suitable for the study of cellular automata. Researchers have tackled different algorithmic problems ranging from the complexity of predicting to the decidability of different dynamical properties of cellular autom ..."
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The goal of this paper is to show why the framework of communication complexity seems suitable for the study of cellular automata. Researchers have tackled different algorithmic problems ranging from the complexity of predicting to the decidability of different dynamical properties of cellular automata. But the difference here is that we look for communication protocols arising in the dynamics itself. Our work is guided by the following idea: if we are able to give a protocol describing a cellular automaton, then we can understand its behavior. Key words: automata, communication complexity, classification, universality.
Traced communication complexity of cellular automata ⋆
"... We study cellular automata with respect to a new communication complexity problem: each of two players know half of some finite word, and must be able to tell whether the state of the central cell will follow a given evolution, by communicating as little as possible between each other. We present so ..."
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We study cellular automata with respect to a new communication complexity problem: each of two players know half of some finite word, and must be able to tell whether the state of the central cell will follow a given evolution, by communicating as little as possible between each other. We present some links with classical dynamical concepts, especially equicontinuity, expansivity, entropy and give the asymptotic communication complexity of most elementary cellular automata. Key words: cellular automata, communication complexity
Communication Complexity and Intrinsic Universality in Cellular Automata ∗
"... Let F be a cellular automaton (CA). This paper establishes necessary conditions for F in order to be intrinsically universal. The central idea is to consider the communication complexity of various “canonical problems ” related to the dynamics of F. We show that the intrinsic universality of F impli ..."
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Let F be a cellular automaton (CA). This paper establishes necessary conditions for F in order to be intrinsically universal. The central idea is to consider the communication complexity of various “canonical problems ” related to the dynamics of F. We show that the intrinsic universality of F implies high communication complexity for each of the canonical problems. This result allows us to rule out many CAs from being intrinsically universal: The linear CAs, the expansive CAs, the reversible CAs and the elementary CAs 218, 33 and 94. The notion of intrinsic universality is related to a process by which we change the scale of spacetime diagrams. Therefore, in this work we are answering pure dynamical question by using a computational theory. This communication complexity theory, on the other hand, provides a finer tool than the one given by classical computational complexity analysis. In fact, we prove that for two of the canonical problems there exists a CA for which the computational complexity is maximal (Pcomplete, or Π0 1complete) while the corresponding communication complexity is rather low. We also show the orthogonality of the problems. More precisely, for any pair of problems there exists a CA with low communication complexity for one but high communication complexity for the other. Key words: cellular automata, communication complexity, intrinsic universality. 1.
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"... journal homepage: www.elsevier.com/locate/tcs Communication complexity in numberconserving and monotone cellular automata ..."
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journal homepage: www.elsevier.com/locate/tcs Communication complexity in numberconserving and monotone cellular automata
Automata 2009 Traced communication complexity of cellular automata 4
"... We study cellular automata with respect to a new communication complexity problem: each of two players know half of some finite word, and must be able to tell whether the central cell will ever reach a particular state. We present some links with classical dynamical concepts, and give the asymptotic ..."
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We study cellular automata with respect to a new communication complexity problem: each of two players know half of some finite word, and must be able to tell whether the central cell will ever reach a particular state. We present some links with classical dynamical concepts, and give the asymptotic communication complexity of many elementary cellular automata.
Structural DNA nanotechnology, pioneered by Seeman in the 1980s [42], exploits the infor...
"... We prove that the abstract Tile Assembly Model (aTAM) of nanoscale selfassembly is intrinsically universal. This means that there is a single tile assembly system U that, with proper initialization, simulates any tile assembly system T. The simulation is “intrinsic ” in the sense that the selfasse ..."
Abstract
 Add to MetaCart
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We prove that the abstract Tile Assembly Model (aTAM) of nanoscale selfassembly is intrinsically universal. This means that there is a single tile assembly system U that, with proper initialization, simulates any tile assembly system T. The simulation is “intrinsic ” in the sense that the selfassembly process carried out by U is exactly that carried out by T, with each tile of T represented by an m×m “supertile ” of U. Our construction works for the full aTAM at any temperature, and it faithfully simulates the deterministic or nondeterministic behavior of each T. Our construction succeeds by solving an analog of the cell differentiation problem in developmental biology: Each supertile of U, starting with those in the seed assembly, carries the “genome ” of the simulated system T. At each location of a potential supertile in the selfassembly of U, a decision is made whether and how to express this genome, i.e., whether to generate a supertile and, if so, which tile of T it will represent. This decision must be achieved using asynchronous communication under incomplete information, but it achieves the correct global outcome(s).