# Time for Change: Implementation of Aksentijevic-Gibson Complexity in Psychology

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## Abstract

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## 1. String Complexity

#### 1.1. Complexity as Magnitude

_{2}n, where n is the number of cells).

#### 1.2. Complexity as Structure

Suppose we divide the space into little volume elements. If we have black and white molecules, how many ways could we distribute them among the volume elements so that white is on one side and black is on the other? On the other hand, how many ways could we distribute them with no restriction on which goes where? Clearly, there are many more ways to arrange them in the latter case. We measure "disorder" by the number of ways that the insides can be arranged, so that from the outside it looks the same… The number of ways in the separated case is less, so the entropy is less, or the “disorder” is less.[23]; p. 1

#### 1.3. Complexity as Change

#### 1.4. Computing and Testing AG Complexity

#### 1.5. Examined Studies

#### 1.6. Results

#### 1.6.1. General Findings

#### 1.6.2. Usefulness of Complexity Profiles

## 2. Complexity of Visual Form

#### 2.1. Computing 2D AG Complexity

- -
- create a zero matrix ${M}_{LxL}$
- -
- for $i=1\text{}\mathrm{to}\text{}L-1$: If $\left({b}_{i}\ne {b}_{i+1}\right)$, then set ${M}_{i2}=1,$
- -
- for $j=3\text{}\mathrm{to}\text{}L,\text{}i=1\text{}\mathrm{to}\text{}L-j+1,\text{}k=2$ to $j-1:r=i+j-k,$ If ${M}_{ik}\ne {M}_{rk},$ then set ${M}_{ij}=1$ and continue to the next value of $i$,
- -
- calculate ${p}_{j}={{\displaystyle \sum}}_{i=1}^{L-j+1}{M}_{ij}$, $j=2$ to $L$ and obtain $P=\left({p}_{2},{p}_{3},\dots ,{p}_{L}\right)$ which is a change profile of $B$,
- -
- the $AG$ complexity of $B$ is $AG\left(B\right)={{\displaystyle \sum}}_{j=2}^{L}{p}_{j}{w}_{j}$, where ${w}_{j}=\frac{1}{L-j+1}.$

- $d=m+n-1$
- $S=\frac{R}{m}+\frac{C}{n}+\frac{D}{d}$
- $X=d-1+\frac{2\left(m-1\right)\left(n-1\right)}{d}$
- $L=\frac{4mn}{3d+1}$
- $N=\frac{S}{X}$
- $U=\left(L-1\right)N$

#### 2.2. Applying 2D AG

#### 2.2.1. Subjective Complexity/Goodness

#### 2.2.2. Geometric Transformations

#### 2.2.3. Form and Complexity

#### 2.2.4. Proximity and Similarity

#### 2.2.5. Local Field Interactions

#### 2.2.6. Global Field Interactions

#### 2.2.7. AG and Transition to Disorder

## 3. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Statistical uncertainty (entropy) as a measure of magnitude/size. Panels under (

**a**) show that entropy (6 bits) does not depend on the location of the “submarine”. Panel (

**b**) shows that the only factor affecting entropy (in this case 8 bits or binary decisions) is the size of the space in which the “submarine” is located.

**Figure 3.**Two views of the structure of the output of the periodic algorithm P16. (

**a**) The front and side of the profile relief map showing inter-level periodicities (low-level structure). (

**b**) The back side of the map highlights intra-level periodicities (high-level structure).

**Figure 4.**Profile relief maps for the outputs of (

**a**) the deterministic-stochastic (GM) and (

**b**) stochastic (B-0.5) algorithms.

**Figure 7.**2D AG and spatial transformations for intact and perturbed patterns. VR = vertical rotation, HT = horizontal translation, VR + HR = vertical rotation plus horizontal rotation, VR + VT = vertical rotation plus horizontal translation.

**Figure 8.**2D AG complexity of geometric figures of equal area possessing four, two, one, and zero axes of symmetry.

**Figure 9.**Grouping of squares and triangles and the associated 2D AG complexities. There are three stimulus configurations (square, triangle, both) and three levels of proximity (far, near, compact) giving nine conditions (SF, SN, SC, TF, TN, TC, BF, BN, BC).

**Figure 10.**Local spatial distortion created by the two grouping configurations. The vertex creates a steep local gradient that precludes formation of a joint basin between the figures. (

**A**): The rhomboids vs. bow ties display. (

**B**): Local spatial distortions caused by different aspects of the triangle.

**Figure 12.**Complexity of two-triangle configurations in a 340 × 340-pixel field. Original distance units were retained because the relationship between the figures and the field was maintained.

**Figure 14.**Transition to disorder. 2D AG complexity over the course of the simulation. Snapshots of the dissipative process have been taken every 20 steps.

**Figure 15.**AG complexity of four representative ECA rules (rule number in parentheses; patterns obtained from https://en.wikipedia.org/wiki/Elementary_cellular_automaton).

Study | N (Patterns) | Mode | Presentation | Length | Measure | Correlation with AG | Correlation with KC ^{a} |
---|---|---|---|---|---|---|---|

Galanter & Smith (1958) | 6 | S | Seq | 2–5 | Prediction accuracy | 0.94 ** | 0.77 |

Glanzer & Clark (1962) | 256 | V | Sim | 8 | Reproduction accuracy | −0.39 *** (−0.71 ***) | −0.18 * |

Alexander & Carey (1967) | 35 | V | Sim | 7 | Overall goodness | −0.73 *** | 0.05 |

Exp. 1 Search | −0.41 * | −0.15 | |||||

Exp. 2 Sorting | −0.61 *** | 0.01 | |||||

Exp. 3a Time | −0.58 *** | 0.19 | |||||

Exp. 3b Confusion | −0.68 *** | −0.06 | |||||

Exp. 4 Description | −0.68 *** | −0.06 | |||||

Griffiths & Tenenbaum (2003) | 127 | V | Seq | 8 | Perceived randomness | 0.66 *** | 0.31 *** |

Vitz (1968) | 26 | V | Seq | 1–8 | Judged complexity | 0.87 *** | 0.77 *** |

Psotka (1975) | 35 | V | Seq | 8 | Judged complexity | 0.68 *** | −0.02 |

Judged symmetry | −0.80 *** | −0.30 | |||||

Judged syntely | −0.26 | 0.06 | |||||

Garner & Gottwald (1967) | 10 | V | Seq | 5 | Trials to criterion | 0.75 * | −0.58 |

Number of errors | 063^{†} | −0.69 * | |||||

Royer & Garner (1966) | 19 | A | Seq | 8 | Response uncertainty | 0.71 ** | 0.36 |

Response delay | 0.69 ** | 0.40 ^{†} | |||||

Error rate | 0.65 ** | 0.44 ^{†} | |||||

138 | Freq. SP | 0.09 | 0.02 | ||||

Response delay | 0.49 *** | 0.20 * | |||||

Error rate | 0.52 *** | 0.26 ** | |||||

Falk & Kondold (1997) | 40 | V | Sim | 21 | Apparent randomness | 0.72 *** | 0.77 |

Memorization difficulty | 0.79 *** | −0.18 * | |||||

Copying difficulty | 0.80 *** | 0.05 | |||||

Memorization time | 0.86 *** | −0.15 | |||||

De Fleurian et al. (2016) | 48 | A | Seq | 49 | Correct ending | −0.32 * | 0.01 |

Ease | −0.75 *** | 0.19 |

^{†}p < 0.1A = Auditory, V = visual, S = speech; SP = starting point

^{a}KC = CTM for strings under 13 bits; BDM for strings over 13 bits.

String | Algorithm | KC |
---|---|---|

10111011101110101011101110111010101110111011101010 | P16 | 0.00178 |

10110110101101010101101101010101101111010110101111 | GM | 0.0415 |

01010110100111011011111010110100101101111001100110 | B-0.5 | 0.0574 |

Study | N (Patterns) | Dimensions | Measure | Correlation with 2D AG |
---|---|---|---|---|

Chipman (1977) | 45 | 6 × 6 | Subjective complexity | 0.74 *** |

Howe (1980) | 60 | 5 × 5 | Subjective goodness | 0.72 *** |

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**MDPI and ACS Style**

Aksentijevic, A.; Mihailovic, A.; T. Mihailovic, D.
Time for Change: Implementation of Aksentijevic-Gibson Complexity in Psychology. *Symmetry* **2020**, *12*, 948.
https://doi.org/10.3390/sym12060948

**AMA Style**

Aksentijevic A, Mihailovic A, T. Mihailovic D.
Time for Change: Implementation of Aksentijevic-Gibson Complexity in Psychology. *Symmetry*. 2020; 12(6):948.
https://doi.org/10.3390/sym12060948

**Chicago/Turabian Style**

Aksentijevic, Aleksandar, Anja Mihailovic, and Dragutin T. Mihailovic.
2020. "Time for Change: Implementation of Aksentijevic-Gibson Complexity in Psychology" *Symmetry* 12, no. 6: 948.
https://doi.org/10.3390/sym12060948