## Zeno Squeezing of Cellular Automata (2010)

Venue: | INT. JOURN. OF UNCONVENTIONAL COMPUTING, VOL. 6, PP. 399–416 |

### BibTeX

@MISC{Schaller10zenosqueezing,

author = {Martin Schaller and Karl Svozil},

title = { Zeno Squeezing of Cellular Automata},

year = {2010}

}

### OpenURL

### Abstract

### Citations

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Citation Context ...on. This hypercomputer simulates a Turing machine and is capable of performing infinitely many steps of the Turing machine in finite time. We assume the following Turing machine model as described in =-=[HU79]-=-. Formally, a Turing machine is a tuple M=(Q,Σ,Γ,δ,q0,B,F), where Q is the finite set of states, Γ is the finite set of tape symbols, Σ⊂Γ is the set of input symbols, q0∈ Q is the start state, B∈ Γ/Σ ... |

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Citation Context ...lfsimilar cellular automaton that involves an infinite number of steps might lead to indeterminism. The carry-over of the self-similar cellular automaton model to the theory of Petri nets (see, e.g., =-=[Mur89]-=-) yields self-similar Petri nets. They are equivalent ∗ Algorithmics, Parkring 10, 1010 Vienna, Austria. EMail: martin schaller@acm.org �Institut für Theoretische Physik, University of Technology Vien... |

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Citation Context ...and Zeno-like processes, is discussed elsewhere (see, e.g., [Svo98]). Originally conceived as a means to demonstrate self-reproduction capabilities in a universal computing environment by von Neumann =-=[vN66]-=-, the idea of perceiving the physical universe as cellular automaton goes back to Zuse [Zus67] and was developed further by other researchers [Fre90, TM90, Wol02]. Cellular automata based on other tes... |

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Citation Context ...e form ...◻◻q1> q2< q3> q4< ...qn−1> qn< ◻◻..., where all qi are in{0,1}. It is easy to see that the evolution of A is deterministic and that A runs the one-dimensional billard ball model of Margolus =-=[Mar84]-=-. Furthermore, the construction shows that a self-similar cellular automaton can simulate any 3-site one-dimensional cellular automaton. 3 Construction of a Hypercomputer 3.1 Specification In what fol... |

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Citation Context ...ilar cellular automata are closely related to cellular automata, a class of dynamical systems characterized by discreteness in space, time, state values, determinism, and local interaction (see e.g., =-=[Gut91]-=-). A cellular automaton is an infinite lattice of finite automata, each linked with its neighboring automata, whose underlying space-time structure results from a uniform tessellation of space and tim... |

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Citation Context ...veral aspects that make both self-similar cellular automata as well as self-similar Petri nets interesting. Both are extending Zeno’s original paradox, leading to a new class of supertasks (see e.g., =-=[EN96]-=-). Another notable aspect is the hypercomputing capabilities of both models, both of them are capable of working as right-accelerated Turing machines [SS09], which is a subclass of accelerated Turing ... |

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Citation Context ...spect is the hypercomputing capabilities of both models, both of them are capable of working as right-accelerated Turing machines [SS09], which is a subclass of accelerated Turing machines (see e.g., =-=[Ord06]-=-). Both computing models result from a composition of very basic building blocks — either finite automata or Petri net transitions — and thus demonstrate that it is at least conceptually possible to b... |

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Citation Context ... cellular automaton goes back to Zuse [Zus67] and was developed further by other researchers [Fre90, TM90, Wol02]. Cellular automata based on other tessellations than the uniform grid were studied in =-=[MM99]-=-. Hypercomputing is a fast growing field (see, e.g., [Ord06]), despite criticicism related to the methodology and the classification of what should be considered a valid computing process [Dav04, Dav0... |

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Citation Context ...hodology and the classification of what should be considered a valid computing process [Dav04, Dav06, Pot06]. In particular, Newtonian mechanics facilitates the construction of hypercomputers. Davies =-=[Dav01]-=- describes in some detail how to build an accelerated Turing machine within a Newtonian universe. For other approaches within Newtonian mechanics, see [BT06, Svo07]. As already mentioned, both self-si... |

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Citation Context ...ing to a new class of supertasks (see e.g., [EN96]). Another notable aspect is the hypercomputing capabilities of both models, both of them are capable of working as right-accelerated Turing machines =-=[SS09]-=-, which is a subclass of accelerated Turing machines (see e.g., [Ord06]). Both computing models result from a composition of very basic building blocks — either finite automata or Petri net transition... |

1 |
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Citation Context ...ing that the evolution of A is indeterministic and independent of its initial configuration. Classifying the evolution as indeterministic raises subtle questions that relate to Thomson’s lamp paradox =-=[Tho54]-=-. We take the point of view that each cell is at any time in a given state of the state set, even if the initial configuration and the update rule do not uniquely determine the state. That the evoluti... |

1 |
English translation [Zus70
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Citation Context ... a means to demonstrate self-reproduction capabilities in a universal computing environment by von Neumann [vN66], the idea of perceiving the physical universe as cellular automaton goes back to Zuse =-=[Zus67]-=- and was developed further by other researchers [Fre90, TM90, Wol02]. Cellular automata based on other tessellations than the uniform grid were studied in [MM99]. Hypercomputing is a fast growing fiel... |