# An Informational Test for Random Finite Strings

^{*}

^{†}

^{‡}

## Abstract

**:**

## 1. Introduction

#### 1.1. Mathematical Definitions of Randomness

#### 1.2. True Randomness

#### 1.3. Pseudo-Randomness and Deterministic Chaos

#### 1.4. Empirical Randomness

## 2. Informational Indexes of Strings

**Proposition**

**1.**

**Proposition**

**2.**

**Proposition**

**3.**

**Proposition**

**4.**

**Proposition**

**5.**

**Proposition**

**6.**

## 3. A “Positive” Notion of a Random String

**Principle**

**1**(Random Log Normality Principle (RLNP)).

**Proposition**

**7.**

**Proof.**

**Proposition**

**8.**

**Proposition**

**9.**

**Proof.**

## 4. Log Bounds Randomness Test

## 5. Analysis of the Experimental Results

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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n | $\mathit{mhl}$ | $\lceil \mathit{LG}\rceil $ | Check | $\mathit{mrl}+1$ | $\lceil 2\mathit{LG}\rceil $ | Check |
---|---|---|---|---|---|---|

100,000 | 4 | 5 | ✓ | 10 | 10 | ✓ |

1,000,000 | 5 | 6 | ✓ | 13 | 12 | ✓ |

2,000,000 | 6 | 7 | ✓ | 13 | 14 | ✓ |

5,000,000 | 6 | 7 | ✓ | 13 | 14 | ✓ |

10,000,000 | 6 | 7 | ✓ | 15 | 14 | ✓ |

20,000,000 | 7 | 8 | ✓ | 15 | 16 | ✓ |

50,000,000 | 7 | 8 | ✓ | 16 | 16 | ✓ |

n | $\mathit{mhl}$ | $\lceil \mathit{LG}\rceil $ | Check | $\mathit{mrl}+1$ | $\lceil 2\mathit{LG}\rceil $ | Check |
---|---|---|---|---|---|---|

100,000 | 4 | 5 | ✓ | 10 | 10 | ✓ |

200,000 | 5 | 6 | ✓ | 12 | 12 | ✓ |

500,000 | 5 | 6 | ✓ | 12 | 12 | ✓ |

1,000,000 | 5 | 6 | ✓ | 13 | 12 | ✓ |

1,200,000 | 5 | 7 | ✗ | 13 | 14 | ✓ |

1,500,000 | 6 | 7 | ✓ | 13 | 14 | ✓ |

2,000,000 | 6 | 7 | ✓ | 13 | 14 | ✓ |

n | $\mathit{mhl}$ | $\lceil \mathit{LG}\rceil $ | Check | $\mathit{mrl}+1$ | $\lceil 2\mathit{LG}\rceil $ | Check |
---|---|---|---|---|---|---|

10,000 | 4 | 4 | ✓ | 8 | 8 | ✓ |

20,000 | 4 | 5 | ✓ | 8 | 10 | ✗ |

50,000 | 4 | 5 | ✓ | 10 | 10 | ✓ |

100,000 | 4 | 5 | ✓ | 11 | 10 | ✓ |

200,000 | 5 | 6 | ✓ | 11 | 12 | ✓ |

500,000 | 5 | 6 | ✓ | 11 | 12 | ✓ |

1,000,000 | 5 | 6 | ✓ | 12 | 12 | ✓ |

n | $\mathit{mhl}$ | $\lceil \mathit{LG}\rceil $ | Check | $\mathit{mrl}+1$ | $\lceil 2\mathit{LG}\rceil $ | Check |
---|---|---|---|---|---|---|

10 | 1 | 2 | ✓ | 2 | 3 | ✓ |

100 | 2 | 2 | ✓ | 3 | 4 | ✓ |

1000 | 3 | 3 | ✓ | 6 | 6 | ✓ |

10,000 | 4 | 4 | ✓ | 9 | 8 | ✓ |

100,000 | 5 | 5 | ✓ | 12 | 10 | ✗ |

1,000,000 | 5 | 6 | ✓ | 15 | 12 | ✓ |

10,000,000 | 6 | 7 | ✓ | 18 | 14 | ✗ |

n | $\mathit{mhl}$ | $\lceil \mathit{LG}\rceil $ | Check | $\mathit{mrl}+1$ | $\lceil 2\mathit{LG}\rceil $ | Check |
---|---|---|---|---|---|---|

100 | 2 | 2 | ✓ | 4 | 4 | ✓ |

1000 | 3 | 3 | ✓ | 6 | 6 | ✓ |

10,000 | 4 | 4 | ✓ | 8 | 8 | ✓ |

100,000 | 4 | 5 | ✓ | 10 | 10 | ✓ |

1,000,000 | 5 | 6 | ✓ | 12 | 12 | ✓ |

10,000,000 | 6 | 7 | ✓ | 14 | 14 | ✓ |

**Table 6.**Strings generated by logistic maps with seed $0.1$ and parameter $r=4$. Generated numbers are normalized in the interval $(0,1)$ and thus discretized into 10 and 1000 digits.

n | $\mathit{mhl}$ | $\lceil \mathit{LG}\rceil $ | Check | $\mathit{mrl}+1$ | $\lceil 2\mathit{LG}\rceil $ | Check |
---|---|---|---|---|---|---|

Alphabet size 10 | ||||||

10 | 1 | 2 | ✓ | 2 | 4 | ✗ |

50 | 1 | 2 | ✓ | 7 | 4 | ✗ |

100 | 1 | 2 | ✓ | 10 | 4 | ✗ |

200 | 2 | 3 | ✓ | 11 | 6 | ✗ |

500 | 2 | 3 | ✓ | 14 | 6 | ✗ |

1000 | 2 | 3 | ✓ | 17 | 6 | ✗ |

10,000 | 2 | 4 | ✗ | 21 | 8 | ✗ |

100,000 | 2 | 5 | ✗ | 32 | 10 | ✗ |

1,000,000 | 2 | 6 | ✗ | 35 | 12 | ✗ |

5,000,000 | 2 | 7 | ✗ | 39 | 14 | ✗ |

10,000,000 | 2 | 7 | ✗ | 42 | 14 | ✗ |

52,000,000 | 2 | 8 | ✗ | 61 | 16 | ✗ |

Alphabet size 1000 | ||||||

10 | 1 | 2 | ✓ | 2 | 4 | ✗ |

50 | 1 | 2 | ✓ | 3 | 4 | ✓ |

100 | 1 | 2 | ✓ | 3 | 4 | ✓ |

200 | 1 | 2 | ✓ | 5 | 4 | ✓ |

500 | 1 | 2 | ✓ | 8 | 4 | ✗ |

1000 | 1 | 2 | ✓ | 8 | 4 | ✗ |

10,000 | 1 | 2 | ✓ | 18 | 4 | ✗ |

100,000 | 2 | 2 | ✓ | 25 | 4 | ✗ |

1,000,000 | 2 | 2 | ✓ | 28 | 4 | ✗ |

5,000,000 | 2 | 3 | ✓ | 34 | 6 | ✗ |

10,000,000 | 2 | 3 | ✓ | 38 | 6 | ✗ |

52,000,000 | 2 | 3 | ✓ | 52 | 6 | ✗ |

n | $\mathit{mhl}$ | $\lceil \mathit{LG}\rceil $ | Check | $\mathit{mrl}+1$ | $\lceil 2\mathit{LG}\rceil $ | Check |
---|---|---|---|---|---|---|

100,000 | 2 | 3 | ✓ | 5 | 6 | ✓ |

500,000 | 2 | 3 | ✓ | 6 | 6 | ✓ |

1,000,000 | 3 | 3 | ✓ | 6 | 6 | ✓ |

5,000,000 | 3 | 3 | ✓ | 6 | 6 | ✓ |

10,000,000 | 3 | 3 | ✓ | 7 | 6 | ✓ |

50,000,000 | 3 | 4 | ✓ | 7 | 8 | ✓ |

n | $\mathit{mhl}$ | $\lceil \mathit{LG}\rceil $ | Check | $\mathit{mrl}+1$ | $\lceil 2\mathit{LG}\rceil $ | Check |
---|---|---|---|---|---|---|

100,000 | 3 | 4 | ✓ | 7 | 8 | ✓ |

200,000 | 3 | 4 | ✓ | 31 | 8 | ✗ |

500,000 | 3 | 4 | ✓ | 31 | 8 | ✗ |

1,000,000 | 4 | 4 | ✓ | 315 | 8 | ✗ |

1,500,000 | 4 | 4 | ✓ | 315 | 8 | ✗ |

2,000,000 | 4 | 5 | ✓ | 315 | 10 | ✗ |

2,500,000 | 4 | 5 | ✓ | 315 | 10 | ✗ |

3,000,000 | 4 | 5 | ✓ | 315 | 10 | ✗ |

n | $\mathit{mhl}$ | $\lceil \mathit{LG}\rceil $ | Check | $\mathit{mrl}+1$ | $\lceil 2\mathit{LG}\rceil $ | Check |
---|---|---|---|---|---|---|

100 | 2 | 4 | ✗ | 8 | 7 | ✓ |

1000 | 3 | 5 | ✗ | 14 | 10 | ✗ |

10,000 | 4 | 7 | ✗ | 17 | 14 | ✗ |

100,000 | 5 | 9 | ✗ | 116 | 17 | ✗ |

1,000,000 | 6 | 10 | ✗ | 381 | 20 | ✗ |

10,000,000 | 7 | 12 | ✗ | 2,721 | 24 | ✗ |

13,033,770 | 7 | 12 | ✗ | 2,721 | 24 | ✗ |

n | $\mathit{mhl}$ | $\lceil \mathit{LG}\rceil $ | Check | $\mathit{mrl}+1$ | $\lceil 2\mathit{LG}\rceil $ | Check |
---|---|---|---|---|---|---|

10,000 | 1 | 3 | ✗ | 25 | 6 | ✗ |

100,000 | 2 | 4 | ✗ | 42 | 8 | ✗ |

200,000 | 2 | 4 | ✗ | 117 | 8 | ✗ |

500,000 | 2 | 5 | ✗ | 287 | 10 | ✗ |

1,000,000 | 2 | 5 | ✗ | 287 | 10 | ✗ |

1,500,000 | 2 | 5 | ✗ | 287 | 10 | ✗ |

2,000,000 | 2 | 5 | ✗ | 287 | 10 | ✗ |

2,500,000 | 2 | 5 | ✗ | 287 | 10 | ✗ |

3,000,000 | 2 | 5 | ✗ | 286 | 10 | ✗ |

3,301,740 | 2 | 5 | ✗ | 286 | 10 | ✗ |

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**MDPI and ACS Style**

Bonnici, V.; Manca, V.
An Informational Test for Random Finite Strings. *Entropy* **2018**, *20*, 934.
https://doi.org/10.3390/e20120934

**AMA Style**

Bonnici V, Manca V.
An Informational Test for Random Finite Strings. *Entropy*. 2018; 20(12):934.
https://doi.org/10.3390/e20120934

**Chicago/Turabian Style**

Bonnici, Vincenzo, and Vincenzo Manca.
2018. "An Informational Test for Random Finite Strings" *Entropy* 20, no. 12: 934.
https://doi.org/10.3390/e20120934