## A Simple Discrete System with Chaotic Behavior*

### BibTeX

@MISC{Asveld_asimple,

author = {Peter R. J. Asveld},

title = {A Simple Discrete System with Chaotic Behavior*},

year = {}

}

### OpenURL

### Abstract

Abstract — We discuss the behavior of a particular discrete system, viz. Post’s system of tag with alphabet {0,1}, deletion number d = 3, and rules: 0 → 00, 1 → 1101. As initial string we consider all strings of length less than or equal to 15 as well as all ‘‘worst case’ ’ inputs of the form (100) m with 1 ≤ m ≤ 128. 1.

### Citations

97 |
The 3x + 1 problem and its generalizations
- Lagarias
- 1985
(Show Context)
Citation Context ...But contrary to their descriptional simplicity their behavior can only be referred to in terms of unpredictable, irregular and chaotic. A typical example of this kind of systems is the 3x + 1-problem =-=[4]-=-, i.e., given the function f : IN→IN defined by f (1) = 1 and for each x ≥ 2, f (x) = if x is even then return f (x /2) else return f (3x + 1) then the question is: ‘‘Is f (x) defined for each x ∈IN?’... |

93 |
Recursive unsolvability of Post’s problem of “tag” and other topics in theory of Turing machines
- Minsky
- 1961
(Show Context)
Citation Context ...lds ω t +1. Post’s systems of tag have been introduced in [8,9]; cf. also [7] for an introduction. They are quite powerful devices; in particular they are able to simulate Turing-machine computations =-=[6]-=- and thus − assuming the Church-Turing Thesis − to perform every effective computation. However, in this note we restrict our attention to a single, very simple instance of such a system. Viz. we cons... |

82 |
Formal Reduction of the General Combinatorial Decision Problem
- Post
(Show Context)
Citation Context ...ring P (σ) to the right end of ω t. We denote the string obtained in this fashion by ω′ t. � We delete the first d symbols from ω′ t; this yields ω t +1. Post’s systems of tag have been introduced in =-=[8,9]-=-; cf. also [7] for an introduction. They are quite powerful devices; in particular they are able to simulate Turing-machine computations [6] and thus − assuming the Church-Turing Thesis − to perform e... |

34 |
Absolutely Unsolvable Problems and Relatively Undecidable Propositions - Account of an Anticipation
- Post
- 1965
(Show Context)
Citation Context ...ring P (σ) to the right end of ω t. We denote the string obtained in this fashion by ω′ t. � We delete the first d symbols from ω′ t; this yields ω t +1. Post’s systems of tag have been introduced in =-=[8,9]-=-; cf. also [7] for an introduction. They are quite powerful devices; in particular they are able to simulate Turing-machine computations [6] and thus − assuming the Church-Turing Thesis − to perform e... |

7 | The Undecidable: Basic Papers on Undecidable - Davis - 1965 |

4 |
Periodicity of Post’s normal process of tag
- Watanabe
- 1963
(Show Context)
Citation Context ...and inputs ω 0 are taken from {0,1} + . The first time this system occurs in the literature is in [8]; cf. also [9] and [7]. The amount of theory developed for this particular system is restricted to =-=[10]-=-. Examples of its behavior are in Figure 1 for the input string ω 0 equal to 1000 and to 1001, respectively. ���������������� * This paper will be included in D. Kleima et al (Eds.): Liber Amicorum fo... |

2 |
On Post’s system of tag with 0 → 00 and 1 → 1101 as productions, Memoranda Informatica (in preparation
- Asveld
(Show Context)
Citation Context ...d m ≠ 110. The case m = 110 is treated separately in §5 because it has not yet been resolved. The present note is a survey of our simulations rather than a full account; the latter one will appear in =-=[2]-=-. Some preliminary results are mentioned in [1]. 2. Concepts and Notation Since the bit values in the input string at the (3k + 2) nd and at the (3k + 3) rd position (k ≥ 0) are immaterial for the ult... |

1 |
On a Post’s system of tag, Memorandum INF-88-20
- Asveld
(Show Context)
Citation Context ...ly in §5 because it has not yet been resolved. The present note is a survey of our simulations rather than a full account; the latter one will appear in [2]. Some preliminary results are mentioned in =-=[1]-=-. 2. Concepts and Notation Since the bit values in the input string at the (3k + 2) nd and at the (3k + 3) rd position (k ≥ 0) are immaterial for the ultimate behavior of T, we assume that the bit val... |

1 |
Chaos and order, Nieuw Archief voor Wiskunde
- Lauwerier
- 1984
(Show Context)
Citation Context ...1) then the question is: ‘‘Is f (x) defined for each x ∈IN?’’. As another example of this sort of systems we mention the difference equation x n +1 = ax n(1 − x n) with 0 < a ≤ 4 and 0 < x n < 1; cf. =-=[5]-=- for the behavior of this equation and for similar examples. In this note we consider Post’s system of tag, i.e., a system T of the form T = (Σ,d,P, ω 0), where Σ is an alphabet (a finite set of symbo... |