GenI Process

The GenI Process ([dʒiːnɑɪ] for Generic Intelligence), German GenI process , describes a time-discrete stochastic process in the state space of the finite subsets of a countable set E, together with a mapping of the power set to E in an n-dimensional complex vector space . In principle, it can be classified as a Markov chain of the first order, but with variable transition probabilities (parallels to the Galton-Watson process can be seen ). ${\displaystyle X:[0;1]\times \mathbb {N} _{0}\mapsto 2^{E}}$ ${\displaystyle \rho :2^{E}\rightarrow \mathbb {C} ^{n}}$ ${\displaystyle P(X_{n+1}\vline X_{n})}$

background

The GenI random process leads abrupt changes in the complex vector space back to the random behavior of independent individuals within a swarm-like construct. The swarm has an overlapping state that controls individual activities via a target variable (stimulation). The amplitudes are also referred to as ideas (cf. "Generalized Quantum Modeling"). After a finite number of steps, the swarm assumes one of the eigen-states with the well-defined probability . The individuals follow defined rules and are allowed to make mistakes, based on the processes in simulated schools of fish. The GenI algorithm starts a chaotic decision-making process as a competition of ideas, as it happens , for example, in a team that has to choose from a limited number of solutions for a given task. In the course of the process, a selection mechanism leads to the fact that ideas die out one after the other, until finally exactly one survives that represents the solution to the task. ${\displaystyle \sum _{j=1}^{n}\beta _{j}e_{j}\in \mathbb {C} ^{n}}$ ${\displaystyle \beta _{j}e_{j}}$ ${\displaystyle \gamma e_{j}}$ ${\displaystyle {\frac {\vert {\beta _{j}}\vert ^{2}}{\sum _{k}{\vert \beta _{k}\vert }^{2}}}}$

The special properties of the GenI process also make it interesting for the interpretation of physical processes.

definition

Terms

Let E be a countable set and the set of finite subsets of E. Let the canonical basis in and . ${\displaystyle {\tilde {E}}:=\{S\subset E:\vert S\vert <\infty \}\subset 2^{E}}$ ${\displaystyle B=(e_{1},\ldots ,e_{n})}$ ${\displaystyle \mathbb {C} ^{n}}$ ${\displaystyle {\tilde {B}}:=\{i^{k}e_{j}:k=0\ldots 3,j=1\ldots n\}}$

The GenI process: The swarm constantly picks up zero rings from its environment and "burns" them randomly. The GenI process determines a gradient to reduce the excitation. The state of the swarm eventually converges to one of the options given.

A given mapping maps each element of E onto a basis vector multiplied by a complex unit , so that . For a swarm denotes its state with complex amplitudes . ${\displaystyle \rho :E\mapsto {\tilde {B}}}$ ${\displaystyle \{1;i;-1;-i\}}$ ${\displaystyle \forall e\in {\tilde {B}}:\vert \rho ^{-1}(\{e\})\vert =\infty }$ ${\displaystyle S\in {\tilde {E}}}$ ${\displaystyle \rho (S):=\sum _{s\in S}\rho (s)=\sum _{j}\beta _{j}e_{j}}$ ${\displaystyle \beta _{j}}$

A pair with is a zero pair . A tuple is called a null ring generated by if . ${\displaystyle s,t\in E}$ ${\displaystyle \rho (s)+\rho (t)=0}$ ${\displaystyle (s_{0},s_{1},s_{2},s_{3})}$ ${\displaystyle s_{0}}$ ${\displaystyle \exists j:\rho (s_{k})=i^{k}e_{j}}$

A set is called a null set if . A maximal null set is called the entropy of S and an entropy-free residual swarm . ${\displaystyle N\in {\tilde {E}}}$ ${\displaystyle \rho (N)=0}$ ${\displaystyle N\subset S}$ ${\displaystyle S\setminus N}$

The term denotes the excitation of the swarm in index j. ${\displaystyle \epsilon _{j}(S):=2{\frac {\vert \beta _{j}\vert {\sqrt {\sum _{k\neq j}\vert \beta _{k}\vert ^{2}}}}{\sum _{k}\vert \beta _{k}\vert ^{2}}}\in [0;1]}$

algorithm

Let be a sequence of swarms (as an instance of ) with the respective decomposition into a maximum zero swarm and the entropy-free residual swarm , the corresponding states and the excitations. ${\displaystyle S^{(l)}=S_{D}^{(l)}+N_{S}^{(l)}}$ ${\displaystyle X(\omega ,l)}$ ${\displaystyle N_{S}^{(l)}}$ ${\displaystyle S_{D}^{(l)}}$ ${\displaystyle \rho (S^{(l)})=\sum _{k=1}^{l}\beta _{k}^{(l)}e_{k}}$ ${\displaystyle \epsilon _{j}^{(l)}:=\epsilon _{j}(S^{(l)})}$

1. Step : Sit and start with a given crush . ${\displaystyle l\leftarrow 0}$ ${\displaystyle S^{(0)}}$
2. Step : If so , then end the process . ${\displaystyle \forall j:\epsilon _{j}^{(l)}=0}$
3. Step : Each element creates an additional zero ring in the swarm. ${\displaystyle s\in S_{D}^{(l)}}$
4. Step : Every zero pair with , is selected with a probability (and is "burned" in the next step). ${\displaystyle r,t\in N_{S}^{(l)}}$ ${\displaystyle \rho (r)=i^{k}e_{j}}$ ${\displaystyle \rho (t)=-i^{k}e_{j}}$ ${\displaystyle p=\epsilon _{j}^{(l)2}}$
5. Step : For each selected zero pair , t leaves the swarm with a probability . Otherwise t remains and r leaves the swarm. ${\displaystyle r,t\in N_{S}^{(l)}}$ ${\displaystyle {\frac {\epsilon _{j}(S\setminus \{r\})}{\epsilon _{j}(S\setminus \{r\})+\epsilon _{j}(S\setminus \{t\})}}}$
6. Step : The resulting swarm is labeled with . ${\displaystyle S^{(l+1)}}$
7. Step : Sit down and continue with step 2. ${\displaystyle l\leftarrow l+1}$

Explanation

As soon as the excitation in each index disappears, the process comes to rest naturally in step 4, apart from the hard termination condition in step 2, since no more zero pairs are "burned" and the state of the swarm therefore no longer changes. The role of stimulation here is reminiscent of the dynamics of a grain of sand in the creation of the Chladnian sound figures . On the other hand, the stimulation as a target variable in step 5 leads to a systematic distortion of the probability of an individual staying. This leads here to a tendency to reduce the excitation. The following interpretation is obvious, based on biological swarm behavior: Each individual tends to follow the rule "reduce the stimulation". It remains free to decide whether to do nothing (step 4), to follow the rule, or to disregard it (step 5).

simulation

Ideas competition within a swarm: Diagram a shows the development of the absolute amplitudes during a GenI process that takes place in an environment with four options. Due to its intrinsically chaotic behavior, it is impossible to predict the evolution of the GenI process at any point. Interestingly, the option with the lowest chance to start wins here. Diagram b shows the corresponding development of the entropy, which in the end increases dramatically for the winning idea. Figure c shows the paths of each idea in the complex plane.

The reference implementation under JAVA shows an extremely good convergence of the process. The table shows an example of the result from 1000 simulation runs (termination after more than 500 iterations or for swarm sizes> 10 million):

	${\displaystyle \vert \gamma e_{1}\vert }$	${\displaystyle \vert \gamma e_{2}\vert }$	${\displaystyle \vert \gamma e_{3}\vert }$	${\displaystyle \vert \gamma e_{4}\vert }$	${\displaystyle \vert \gamma e_{5}\vert }$	${\displaystyle \vert \gamma e_{6}\vert }$	${\displaystyle \vert \gamma e_{7}\vert }$	${\displaystyle \vert \gamma e_{8}\vert }$	${\displaystyle \vert \gamma e_{9}\vert }$	${\displaystyle \vert \gamma e_{10}\vert }$
Should	132	81	97	78	11	206	3	336	36	3
Is	135	74	99	76	15th	189	1	357	36	1
sigma	10.7	8.6	9.4	8.5	3.3	12.8	1.8	14.9	5.9	1.8
Measurements planned			1000	of which divergent		17th	convergent			983
Statistics: Chi square value: 7.85; Chi critical value at 95% confidence: 16.9
mean swarm size: 300,418					sigma: 281,543		maximum: 1,008,512		minimum: 9,695

The results support the following convergence statement (hypothesis):

Be a given swarm with , , . ${\displaystyle S^{(0)}=S}$ ${\displaystyle \rho (S)=\sum \beta _{j}a_{j}}$ ${\displaystyle b_{j}=\vert \beta _{j}\vert }$ ${\displaystyle \vert S\vert ={\sqrt {\sum b_{j}^{2}}}}$

Be a GenI process with . ${\displaystyle S:\mathbb {N} _{0}\times [0;1]\mapsto {\tilde {E}}}$ ${\displaystyle \rho (S^{(m)}(\omega ))=\sum \beta _{j}^{(m)}(\omega )a_{j}}$

Then is . ${\displaystyle P\left({\tilde {S}}^{(m)}{\underset {m\rightarrow \infty }{\longrightarrow }}\gamma a_{j}\right)=P\left(\sum _{k\neq j}b_{k}^{(m)2}{\underset {m\rightarrow \infty }{\longrightarrow }}0\right)={\frac {b_{j}^{2}}{\vert {S}\vert ^{2}}}}$

Individual evidence

1. ^ Siegfried Genreith: The Source of the Universe Intelligent decision making, quantum measurements and gravity are three different traits of a single flame-like random process. Is consciousness the true foundation of the universe? Books on Demand GmbH, Norderstedt 2017, ISBN 978-3-8482-2357-2 , p. 44 .
2. ↑ Liane Gabora, Kirsty Kitto: Toward a Quantum Theory of Humor . In: Frontiers in Physics . tape 4 , 2017, ISSN  2296-424X , doi : 10.3389 / fphy.2016.00053 ( frontiersin.org [accessed January 25, 2018]).
3. ↑ ^{a b} IAIN D. couzin, JENS KRAUSE, RICHARD JAMES, Graeme D. Ruxton, NIGEL R. FRANKS: Collective Memory and Spatial Sorting in Animal Groups . In: Journal of Theoretical Biology . tape 218 , no. 1 , September 7, 2002, p. 1–11 , doi : 10.1006 / jtbi.2002.3065 (English, elsevier.com [accessed January 24, 2018]).
4. ↑ Siegfried Genreith: GenI Reference Implementation. GitHub, July 2017, accessed January 24, 2018 .