## T.: Dynamics and computation in functional shifts

Venue: | Nonlineality |

Citations: | 5 - 3 self |

### BibTeX

@ARTICLE{Namikawa_t.:dynamics,

author = {Jun Namikawa and Takashi Hashimoto},

title = {T.: Dynamics and computation in functional shifts},

journal = {Nonlineality},

year = {},

volume = {17},

pages = {2004}

}

### Years of Citing Articles

### OpenURL

### Abstract

Submitted to Nonlinearity We introduce a new type of shift dynamics as an extended model of symbolic dynamics, and investigate the characteristics of shift spaces from the viewpoints of both dynamics and computation. This shift dynamics is called a functional shift that is defined by a set of bi-infinite sequences of some functions on a set of symbols. To analyze the complexity of functional shifts, we measure them in terms of topological entropy, and locate their languages in the Chomsky hierarchy. Through this study, we argue that considering functional shifts from the viewpoints of both dynamics and computation give us opposite results about the complexity of systems. We also describe a new class of shift spaces whose languages are not recursively enumerable. 1

### Citations

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Citation Context ... some results are discussed to bring dynamics and computation with the generative operation into focus. 2 Definition Since we will study shift dynamics, we first give some definitions of shift spaces =-=[9]-=-. Let A be a nonempty finite set of symbols called an alphabet. The full A-shift (simply the full shift) is the collection of all bi-infinite sequences of symbols from A. Here such a sequence is denot... |

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Citation Context .... Analyzing such dynamics is a challenge for the theory of dynamical systems, because a function is fixed in time in the usual theory. Several models are proposed to study dynamic change of functions =-=[5, 6, 7, 8]-=-. Sato and Ikegami [5] introduce switching map systems in which maps to govern evolution of the systems are dynamically switched with change of time. For another example, Kataoka and Kaneko [6, 7] inv... |

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Citation Context ...ves. In this paper, we study functional shifts from the viewpoints of both dynamics and computation. In recent years, several studies have focused on the relationship between dynamics and computation =-=[1, 2, 3, 4]-=-. The central idea in these studies is to regard the time evolution of dynamics as a computational process. From correspondence between the unpredictability of dynamical systems and the halting proble... |

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Citation Context .... Analyzing such dynamics is a challenge for the theory of dynamical systems, because a function is fixed in time in the usual theory. Several models are proposed to study dynamic change of functions =-=[5, 6, 7, 8]-=-. Sato and Ikegami [5] introduce switching map systems in which maps to govern evolution of the systems are dynamically switched with change of time. For another example, Kataoka and Kaneko [6, 7] inv... |

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Citation Context ...ves. In this paper, we study functional shifts from the viewpoints of both dynamics and computation. In recent years, several studies have focused on the relationship between dynamics and computation =-=[1, 2, 3, 4]-=-. The central idea in these studies is to regard the time evolution of dynamics as a computational process. From correspondence between the unpredictability of dynamical systems and the halting proble... |

4 | Nonlinear computation with switching map systems
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(Show Context)
Citation Context .... Analyzing such dynamics is a challenge for the theory of dynamical systems, because a function is fixed in time in the usual theory. Several models are proposed to study dynamic change of functions =-=[5, 6, 7, 8]-=-. Sato and Ikegami [5] introduce switching map systems in which maps to govern evolution of the systems are dynamically switched with change of time. For another example, Kataoka and Kaneko [6, 7] inv... |

3 |
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Citation Context ...selves [8, 9]. To describe self-modifying systems, we must take the self-referential nature of a dynamical system into account. We can represent this characteristic using functional shifts as follows =-=[17]-=-. Let us consider that a conjugacy from a functional shift to a generated shift given by the functional shift is a ‘self-reference code’ between functions 17sand states. Considering that each sequence... |