(Continued from Volume 6)
Throughout the two previous volumes, I have been pointing out the difficulty of concurrently triggering various actions inside their shared gameplay space without letting themselves contradict one another, for which I have introduced a brand new system (i.e. network of signal transmitters) as a safe solution.
The remaining issue is to integrate this new system into the existing gameplay model which is quite distinct in the sense that it is based upon a mutually reinforcing cycle of narratives (hinted by "Absorb", "Expand", and "Secure") and their respective mechanics (hinted by "Region of Influence", "Resource", and "Obstacle"). We have already seen how the atom-to-atom signaling scheme may fit into the framework of compound objects and their emergent behaviors by looking at a specific example (i.e. angry cow), yet I have not yet established a unified conceptual foundation of everything which have been illustrated so far. In order to let such an establishment be undertaken thoroughly without any bit of vagueness, I will first revisit the basics of the gameplay model which was initially proposed and clarify some of its structural details for the purpose of the theory's integrity and further development.
In the beginning, I supposed that the most basic form of life is a single mathematical point in space (i.e. single atom) which exhibits 3 most fundamental goals throughout its lifespan: "Absorb", "Expand", and "Secure". And in order to describe the physical configuration of the gameplay universe which is responsible for triggering the realization of these 3 goals, I have also supposed the existence of 3 most fundamental entities in space - namely, "Region of Influence", "Resource", and "Obstacle".
The 3 goals are originated from the game's narratives, whereas the 3 entities are originated from the game's mechanics. The 3 goals encourage the lifeform (which refers to itself as "myself") to modify its spatial configuration with respect to the 3 entities, and the 3 entities encourage the lifeform to conceive these 3 goals. This, in the long run, creates an endless cycle of mutual reinforcement between the game's narratives and mechanics, which enables the growth of both of these two pillars of interactivity in the form of a rapid back-and-forth iteration.
Then I went on to suggest that these archetypal elements, just like atoms in chemistry, are able to bond with each other to form compound objects which possess their own intelligent behaviors based off of a fairly minimal set of additive building blocks such as atoms and their discrete movements.
And in order to solve the problem of resolving inner conflicts which may arise among atoms, I introduced the notion of signal-based communication which can potentially happen between any pair of atoms in a selective manner. This system of indirect atom-to-atom interaction provided us with the ability to construct complex objects simply by assembling a group of independent atoms, rather than by manually devising a bunch of additional rules to tell the system how they should be interrelated.
This model, however, raises a major concern due to its implication of complexity. As you may have noticed already, I have introduced quite a large number of concepts so far which can hardly be referred to as "minimal" in the sense that it is supposed to let us design a wide variety of games based upon a set of very few basic components. The sum of all the descriptions which have been made to represent the current model renders itself as quite contrary to such an objective. Apparently, the theory itself involves not just atoms as the ultimate constituents of the gameplay universe, but also their mutual connections, signals, regions in space, internal receptors, transmission filters, goals, actions, and so on.
This looks pretty complicated, isn't it? If we observe the entire model from a purely computational point of view, however, we can easily realize that it is possible to simplify the whole scene down to a much slimmer subset of terminologies. In order to illustrate the reasoning behind this blunt assertion of optimism, let me revisit the initially introduced model which represented the game's protagonist as a single atom that persistently strives to preserve and expand its own region of influence, by means of decreasing its distance from "resource" atoms, increasing its distance from "obstacle" atoms, and spatially expanding the region via movement and procreation.
This mathematical representation is not so minimal in the sense that it requires us to keep both of the two drastically different conceptual entities in mind, which are typically called "atom" and "space", respectively. The points in space among which we measure distances are termed "atoms", whereas a domain of existence within which individual atoms can claim themselves to be associated with specific locations (coordinates) is termed "space". This worldview is pretty reasonable in most circumstances because our common sense dictates that the universe must possess a pool of dimensional permutations called "space" in order to be able to have spatial entities in it.
The dualism between atom and space, however, can easily complicate the way in which the system is being formulated. And it is my abstract sort of conviction which insists that the entire universe can be depicted as a set of atoms only, without any necessity of involving the notion of space whatsoever.
Here is the reason. When we represent the global gameplay space as a hierarchical composition of local spaces and proceed to suppose that an atom belongs to one of these local spaces, what we are saying is that the atom's current topological position in space is essentially just a binding of an atom to the space to which it belongs. And since there is apparently no reason to assume that a space must be functionally distinct from an atom (aside from perceptual differences), it is highly sensible to claim that a space is an atom, too.
In other words, it is not so rash to imagine that any volume of space is just a graph made out of atoms, each of which corresponds to a continuous spatial region. Each binding between two atoms tells us which spatial region is contained in another region.
This alternative worldview works as a key to a much more elegant description of the gameplay universe, especially when it comes to events which occur among objects and their component atoms. For example, a movement of an atom from one point in space to another can be equated with the process of unbinding the atom from its original region of residence and rebinding it to a different region of residence.
If we fancy that the notion of "binding" in this case represents some kind of land ownership, we may as well suppose that the act of movement is a micro-scale real estate transaction which happens inside a hypothetical system of finance. An atom which moves from place to place does so by "selling" its initial location and then subsequently "buying" its final location as a form of exchange. In general, every observable entity which constitutes our physical phenomena (e.g. a piece of matter) can be described as a financial asset, and every physical process such as a displacement can be described as a transaction which triggers a set of exchanges among such assets. This idea may be a bit too abstract, yet it bears a hint of pragmatism when we consider the possibility of developing a game which operates on top of a financial protocol (e.g. blockchain) and directly associates its in-game activities with means of monetization.
A potential source of confusion which may arise at this point is the concept of "connection" between atoms which allows us to create compound objects by means of composition. When we "connect" multiple atoms together, we are basically grouping them and treating them as a singular object. How does this differ from the concept of "binding" which I have introduced above?
The answer is that a "connect" is nothing more than a form of indirect binding between a pair of atoms. When two atoms are connected with each other, it is implicitly assumed that they are both contained within the same local space. Although this definition may seem a bit too arbitrary, it proves itself to be quite handy when it comes to explaining the nature of connection-based signal transmission between atoms (as explained in volume 6). So for the sake of internal consistency and convenience, I will stick to such a definition for now.
Since a local space is itself an atom, the state of two atoms residing in the same local space is an equivalent of two atoms being bound to the same target atom. This is what is meant by the word "connection".
As a result, a compound object (which is a connected set of atoms) can be thoroughly represented as a set of atoms which locally share their intermediary binding targets in a pairwise manner.
Signal transmission, too, is fully explicable in terms of bindings among atoms. A signal is just an atom (although it is not as tangible as others), and the space through which it is able to propagate, such as a copper wire, is also yet another atom to which both the emitter and receiver of the signal are bound. And from this unified mode of conceptualization, we can easily deduce that the act of sending a signal from one atom to another is made up of 2 discrete movements: (1) A movement which rebinds the signal from the emitter to the medium of communication (aka "connector"), and (2) Another movement which rebinds the signal from the medium of communication to the receiver.
Receptors and broadcasting mechanics can be explained in terms of atomic interactions as well. Previously I have mentioned that two atoms can achieve a state of cognitive synchronicity by reaching a "thermal equilibrium" of signals between them, as long as they always actively and rapidly share copies of external signals with each other. The purpose of introducing this mechanic was to prevent any pair of atoms from executing two mutually contradictory actions in parallel. This whole sharing process can be represented as a sequence of binding/unbinding actions performed by each received signal's copy shortly after its initial creation. The details are illustrated below.
However, this is not everything there is to the idea of unification between "space" and "atom". In volume 1 and 2, I explained the quantitative nature of the protagonist's "Absorb" and "Secure" goals in terms of its distance from a resource and its distance from an obstacle, respectively. But how do we define the word "distance" within the context of the current design model, if the word "space" is not even independently defined here? If every quantitatively definable entity which constitutes the game's inner universe is an atom and there is absolutely nothing which pertains to our conventional notion of "space", how can we even measure a distance between a pair of atoms in the first place without making vague assumptions?
In order to resolve this inner conflict of meanings, one must take a step back and contemplate upon the idea of space itself.
Why do we say that we are surrounded by something called "space"? First of all, we cannot see, smell, hear, or touch a volume of space, so we cannot define space in terms of direct sensory data. However, we know that assuming the existence of "space" comes in handy whenever we feel the necessity of distinguishing occasions in which two objects are close to each other from occasions in which two objects are far away from each other (for practical purposes such as avoiding the nearest source of danger first and then avoiding others later, and so forth). In other words, our definition of "space" begins with the idea of a distance between two atoms (since atoms are the simplest objects we can imagine). Space exists because atoms exist and they are separated by a certain amount of "room for occupancy" between them.
One might argue that distance is only one of many possible dimensional measures in space, such as direction, area, volume, and others. However, it is my (potentially erroneous) conviction that all these additional concepts are only secondary to the concept of distance. For example, we can measure the direction of a line segment solely in terms of distances because, if we imagine the coordinate system's point (0,0) and point (1,0) as a pair of objects and the line segment's two endpoints as yet another pair of objects, the ratios among the pairwise distances of all of these four objects will tell us the angle of the line with respect to the coordinate system. The concept of area and volume can be derived in similar ways as well, since they can be derived from the total number of atoms in addition to the statistical distribution of their pairwise distances.
The next question is, how to define "distance"? Since space is defined in terms of distances, the idea of distance itself must be defined as well in order to fully explain the nature of space. And in order to figure this out, we ought to first find out an objective way of measuring a distance between two atoms - for it appears to be quite axiomatic that, if we want to define something quantitative, we must be able to measure it in a quantitative manner. One quick method of measuring a distance is to use a ruler (or any other fixed-length object), but the problem is that such a way of measurement assumes the legitimacy of the observer's sensory (visual) data as well as a predefined distance between the two tips of the ruler (which creates a circular logic by forcing us to define a distance in terms of another distance).
Defining distance as something that is proportional to the amount of time it takes for an object of a constant velocity to move from a fixed reference point to another fixed reference point seems to be a more scientific method, yet it is still bound to circular reasoning because the temporal dimension itself can be considered just yet another axis in space (which, along with the X, Y, and Z spatial axes, constitutes one unified continuum called "spacetime") as far as the theory of relativity goes, and hence requires the observer to measure the "temporal distance" between the object's two different positions at two different moments in time.
If one desires to measure a distance without either sensory bias or semantic circularity, he/she should seek a fully distance-independent causal relation between successive observations as a primary proof of the existence of distance instead. One relatively objective criterion for finding such a relation is to suppose that distance is proportional to the amount of energy it takes for a unit mass to travel from a fixed reference point to another fixed reference point. This is probably still not a fundamentally valid definition because the notion of "fixed reference point" itself indicates a spatial entity which presumes the distance between itself, the observer, and other points in space (which, again, induces circular logic), but I will stop delving into the ontological ultimacy of meanings at this point because this article is about game design and is not a treatise on metaphysics.
If we define the distance between atom B and C as the amount of energy it takes to move atom A from B to C, and further define that:
(1) Two atoms are "neighbors" to each other if at least one of them is bound to the other,
(2) An atom can only rebind itself to a neighbor of the atom to which it is currently bound, and
(3) An atom consumes the exact same quantity of energy (aka "cost") every time it rebinds itself,
we can then declare that the distance between atom B and C is the number of times A is required to rebind itself until it changes its binding target from B to C. This means that the so-called "distance" in space, in any discrete gameplay environment operated by a digital computer, can be defined as the minimum cost (in terms of energy) of traversing a path from one atom to the other inside a graph of atoms.
Now, what's the purpose of all this? Why not just place atoms inside a plain Euclidean space and move them freely, without esoteric notions such as "binding"? Here is the most significant reason why I chose to undertake the process of explaining all spatial concepts in terms of atoms and their bindings, despite its apparent obscurity.
At the very beginning of this series of articles (volume 1), I presumed that the instincts of any biological organism can ultimately be classified into 3 most fundamental goals called "Absorb", "Expand", and "Secure", and proceeded to claim that the "Absorb" goal motivates the organism to decrease its distance from a nearby resource, the "Expand" goal motivates the organism to increase the area of its region of influence, and the "Secure" goal motivates the organism to increase its distance from a nearby obstacle.
The "Absorb" and "Secure" goals were explained pretty easily in terms of distances between atoms. The "Expand" goal, on the other hand, was explained in terms of yet another type of spatial quantity called "area". Since an area is a secondary property which must be derived from multiple distance relations among multiple pairs of atoms instead of just a single pair, it has been quite complicated to describe the exact way in which the organism could carry out its "Expand" goal. As a result, I explained the sequence of actions required to pursue the "Expand" goal with vague specifications such as: "Go to a random position in space that is outside of your own body".
If we exclude the notion of space altogether (the method of which is what I have been expounding so far), the structure of the "Expand" goal can be synchronized with those of the other two fundamental goals ("Absorb" and "Secure") because we will then be letting any region of influence be represented by atoms, just like resources and obstacles are being represented by atoms. A lifeform's region of influence is a set of local regions in space, and such a set is equivalent to a set of atoms. Let us start referring to each constituent atom of any region of influence as an "influence" from now on.
The trickiest part is, how to measure the area of a region of influence? As mentioned before, since the ultimate root of the definition of the word "space" within the context of the current atom-based model is an attribute called "distance", a scalar quantity that is being referred to as "area" should be able to be computed based upon the distances among the corresponding region's component atoms. In order to identify the region's area, however, one must first be able to tell the shape of the region. And this is indeed a quite perplexing task because each atomic unit of space, in our context, is a topological being that is defined in terms of its nature of connectivity with other units instead of a uniformly sized/arranged segment which sits within a continuous field of dimensional permutations.
But, let's pause here for a bit and think about the intended usage of the concept called "area" in our purpose. Why do we want to find out the area of a region? The answer is, we need a metric which tells the game's protagonist how successful it is in terms of expanding its own region of influence. Whenever a typical Euclidean space is concerned, the area of the region of influence tends to be an obvious measure of such a type of success. The thing is, we are not dealing with a Euclidean space here, and thus do not have to force our stream of consciousness to linger upon geometric constructs such as the area of a region; all we need is a metric which reveals the organism's outgoing flux of influence that is being imposed upon the rest of the world.
This problem will be investigated in the next volume. After that, we will be able to assemble the concepts introduced so far and summarize them in the form of a single unified model.
(Will be continued in Volume 8)