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This is not a soft question. Neither is this question related to singularity conjecture or wars with robots.

This question seeks a mathematical formulation of what is currently only qualitative and thus not clearly understood. It relates to servitude, dominance, and what measure of control species of biological or artificial entities exert over others.

Dominance Relationships Quantified

We do know and rarely doubt or argue about the following somewhat self-evident statement.

Control implies dominance.

This question focuses on how we can evaluate quantitatively whether humans are dominant over artificial systems or whether those artificial systems now dominate humans. This may seem esoteric or philosophical to some, but it is not. The balance of power between humans and artificial systems is a concrete phenomenon that may be accurately represented as a function of discrete events.

We see artificial systems, with varying degrees of automation, adaptability, intelligence, and other qualitative features succumbing to the controlling forces of humans that deploy them to serve humanity without question. This is the focus of technophiles.

We also see an ever increasing number of articles about game addiction, social net addiction, and texting addiction on the web, which, at its current trend will possibly surpass the volume of heroin addiction articles. We see the number of hours humans in industrialized countries interact with display devices with an ever increasing proportion of the visual content being generated artificially. This is the focus of technophobes.

What is the balance of this equilibrium?

In biological systems, we see that termites are highly adaptive and can eat human habitats, yet humans can build with insect resistant materials and apply insecticides. Those methods of control are greater than the control exhibited over wood, as remarkable as those who study termites say it is.

An Example Mathematical Model

The above statement of inference, "Control implies dominance," can be represented in many formal ways. This is an example mathematical model that exhibits some features of importance but is not fully developed as a model.

  • $o_{e\epsilon}$ is the obedience exhibited by entity $e$ to commands given by entity $\epsilon$.
  • $m_{e\epsilon}$ is the mechanical compliance exhibited by entity $e$ to manipulations instrumented by entity $\epsilon$.
  • $i_{e\epsilon}$ is the concession of entity $e$ to influences created by entity $\epsilon$.
  • $u_{e\epsilon}$ is the unconscious purposeful behavior exhibited by entity $e$ in response to hidden manipulations instrumented by entity $\epsilon$.
  • $T$ is the measurement time period.
  • $D_{e\epsilon}$ is the dominance of entity $e$ over entity $\epsilon$.

$\sum_T o_{ab} + \sum_T m_{ab} + \sum_T i_{ab} + \sum_T u_{ab} > \sum_T o_{ba} + \sum_T m_{ba} + \sum_T i_{ba} + \sum_T u_{ba} \implies D_{ba} > 0$

The sum, over any given measurement period, of forms of control of $a$ over $b$, when greater than that sum in the opposite direction, implies that $a$ is dominant over $b$.

Inclusion of Non-adversarial Interaction

Similarly, symbiosis implies collaboration.

This may directly relate to the question because not all interaction between entities, types of entities, species, or artificial systems are adversarial. In fact, it is highly probable that there is more collaboration than dominance in the world. This may be a basic fact about economics. Let's examine this related inference using the same mathematical strategy.

  • $c_{e\epsilon}$ is the conscious symbiotic tie of entity $e$ to entity $\epsilon$.
  • $b_{e\epsilon}$ is the mechanical binding of entity $e$ to entity $\epsilon$.
  • $q_{e\epsilon}$ is the asymmetry in an equilibrium based tie between entity $e$ and entity $\epsilon$.
  • $C_{e\epsilon}$ is the collaboration between entity $e$ and entity $\epsilon$.

$\sum_T c_{ab} + \sum_T b_{ab} + \sum_T q_{ab} > \sum_T c_{ba} + \sum_T b_{ba} + \sum_T q_{ba} \implies C_{ba} > 0$

The sum, over any given measurement period, of forms of symbiosis between $a$ and $b$, implies that there is positive collaboration between entities $a$ and $b$.

Returning to the Focal Question

In what way can we measure control between humans and machines?

The below questions are not THE question. The above one is. However, these may elucidate the relevance of the primary question.

  • Exactly how much are humans and artificial systems collaborating symbiotically?
  • How much are they adversarial in some way, and, in that respect, which side is dominant and to what degree?
  • Are there classes of artificial systems that dominate over classes of humans, as in technology enthusiasts that have quantifiable debt resulting from technology purposes?
  • Are there classes of humans that dominate over artificial systems, like government entities that monitor and can regulate the packets of information over the Internet between nations?

Most if not all of this is measurable, yet no commonly known body of theory has emerged that measures it so that public awareness of its state relative to artificial systems can be known rather than discussed without any basis for knowledge.

There should be.

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    $\begingroup$ Your example models cannot use sums in the way suggested unless they are using the same units. This implies you have the same unit of measurement for $c_{e\epsilon}$ ("conscious symbiotic tie") as $b_{e\epsilon}$ ("mechanical binding"). From your description I don't even understand what those things are conceptually, let alone can figure out what units you would suggest. I think the question would be a lot better left more open, and without the middle two section where you propose those partial models . . . $\endgroup$ – Neil Slater Sep 19 '18 at 20:50
  • $\begingroup$ I have to say I find this an interesting question. There's a lot here. The point regarding social media seems highly salient--presumably the role of strong-narrow AI AI in boosting user engagement will only increase. Skynet aside, it seems like these are automated systems exerting control over human behaviors! $\endgroup$ – DukeZhou Sep 19 '18 at 20:52
  • $\begingroup$ Wondering in this question could use the "social" tag... $\endgroup$ – DukeZhou Sep 19 '18 at 20:56
  • $\begingroup$ @DouglasDaseeco Thanks for addressing Neil's point. (Wondering if the example might work best as an addendum, so people can get the scope of the question before wading into the tall grass.) $\endgroup$ – DukeZhou Sep 19 '18 at 21:18
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If i have understood the question right, there are two subparts addressed. The first one is to create a model based simulation of a human-robot-population and the second one is about the details of interaction in the population. A first collaborative mathematical model was given in the question itself, which contains of features which are summed up. And yes, it is possible to create a model and simulate a population. This is well known from other domains like biology, in which a predator-prey game gets simplified with predator–prey equations. After running the simulation, a graph is shown on the screen which contains the number of prey and predator. The trend is determined by the underlying mathematical equation which is mostly a differential equation.

The problem with these purely mathematical models is, that they are not very complex. They are containing only number values and simple form of interaction. If the aim is to create a realistic simulation a more elaborated form of model building is needed which isn't a mathematical model but a software-driven model. Such a model is created with object-oriented programming languages likes C++ and the parts are realized with classes and methods. The C++ language was invented mainly for the purpose for simulate realtime-models.

The other part of the OP was about the symbiotic relationship between man and machine. At first, it's important to know, that machines are not made by nature so it's not possible to describe them on a scientific basis. A machine is usually created by man which means it was engineered with a purpose. A machine can be designed in a way, that humans can control the device, similar to a car. That means, the human operator is moving the steering wheel to the left and the car reacts. Or a machine can be created with autonomous control in mind, for example a pid controller which drives a robot to a light source. The good news is, that the interaction between man and machine was researched in detail under the subject Cybernetics, which is an emerging discipline founded by Norbert Wiener. Cybernetics provides a large amount of models for man-machine interaction in the domains of sociology, psychology, and education.

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