The Quantification Of Complexity

If there was one store a kilometer from where you live and another store four kilometers from where you live, you would naturally describe the furthest store as being four times the distance from where you live as the closer store. If you were comparing the weight of two objects, you might say that one object is 2.13 times as heavy as the other. If you wanted to tell someone how long it took you to complete a task, you may possibly say that it took one hour and ten minutes.

These are examples of quantification, or using numbers to express a quantity, which is what numbers are for.

Now, let me ask you a question. If you had two machines, one more complex than the other, and someone asked you how much more complex the one was than the other, what would you answer? You would most likely say that "The one machine is quite a bit more complex than the other." You might use other terms such as "a lot more complex" or "somewhat more complex".

Have you ever wondered why human beings use numbers, quantification, for expression of all kinds of things, but when we want to express or compare complexity, we invariably use subjective terms instead of numbers? Just why is it that we do not express something as primal to life as complexity in numbers and units as we do distance, time, temperature or, weight? I have really put quite a bit of thought into this.

Complexity is a conservative quantity in being simply the amount of information in something, which is why I believe that we could and should quantify it, that is express it as a number instead of in the vague and subjective terms that we tend to use now. The quantification of complexity would be a great advantage in all that human beings do, particularly in science and engineering. It seems to me that the increasing use of complex documents like computer programs, blueprints and, product specifications in recent decades has prepared us to begin the expression of complexity with precise numbers.

We live in a world and universe that consists of matter in space. This matter can form many, many different patterns. This is what brings complexity into existence. The people of long ago were very much affected by complexity, just as we are today. But they did not have the knowledge to quantify it and got in the habit of expressing it in subjective terms, which we still do today.

The quantification of complexity does not require and kind of measuring device. It does, however, require a new way of thinking. I have developed some general rules of complexity to get us started.

THE RULES OF COMPLEXITY

Any type of measurement, unless it is a comparison, requires units. What does the number 17.33 mean? The answer is that it means absolutely nothing until it is used with a unit, such as meters or ounces.

The unit of complexity will be the "level". A level is anything that is quantified, which can be expressed in numbers. If you have a box, you can describe the box with three levels, that of it's length, width and, height. We could say that the box, basically, has a complexity of three levels.

It was the writing of my book, "The Theory of Primes" that led me to the idea of quantifying complexity. Every domain in existence can be differentiated from every other domain by it's levels. This makes it possible for complexity to be expressed by simply the number of levels that is manifested by a domain.

RELEVANCE

In the expression of complexity, a level must be expressed if there is any possible alternative to what that level is now that would affect the purpose for the measurement of the complexity. This is the great factor that we must work around in the quantification of complexity, that it's expression will be done for a variety of different reasons and that not all levels that can be expressed will be relevant to a measurement of complexity for a specific purpose.

To give an example, if a meteorologist is measuring snowfall, the complexity of the measurement does not normally need to include the patterns in the snowflakes. For another example, measurement of the temperature of an object can be expressed as one level and does not normally need to include the trajectories of the fast-moving atoms and molecules that create heat. We can describe these as "irrelevant" levels.

REDUNDANCY

The domains with the least complexity is a line, circle, cube or, sphere requiring only one number, one level, to describe it. It is important that, in measurement of complexity, there be no redundancy in the levels expressed. Redundant levels are those levels that are unnecessary because the quantities can be found by other listed levels. There must be only the minimum required information.

For example, we do not need to include the level for the volume of a cube if that can be found by it's length, width and, height which are already listed. It also does not matter which level, out of a choice of several, is included if it can describe the domain without the others. As an example of this, a circle can be described by any one of several criteria, radius, diameter, curvature or, area. Only one would be necessary and to include more than one in a measurement of complexity would bring redundancy.

SYSTEMS AND REPETITION

A so-called "system" is a domain consisting of at least two sub-domains. Such a system would be more complex than would the complexity of the separate sub-domains added together. This is simply because in the system, not only do the sub-domains have to be described but it also has to be described by additional levels how the sub-domains relate to each other by location or function in the larger domain.

Thus, we could say that the combining of sub-domains into a larger domain increases complexity by the number of levels it would require to describe how the sub-domains relate to each other in the larger domain.

On the other hand, any type of repetition in a large domain reduces complexity. In any arrangement of identical units, the complexity of the unit counts only once. An arrangement of twenty identical cars is only a little bit more complex than one of the cars would be. In addition, even if the cars were not identical, the arrangement could be described by first describing one car and then describing how the other cars differ from the first, if that would require less information, and thus fewer levels, than describing each car individually.

So, we could say that in any system or domain consisting of sub-domains, any repetition or similarity between sub-domains decreases complexity.

Remember that complexity consists of the minimum amount of information that can describe a domain, any redundancy must be eliminated for a true measure of complexity. The domain of which we are measuring complexity is defined by the relevance for which we are taking the measurement.

MAXIMUM AND MINIMUM COMPLEXITY

In any system or domain, there is a minimum and a maximum possible complexity. If you arrange pennies on a table top, you lower the complexity by simply arranging them in a circle. You maximize the complexity by spelling out words with the pennies in single file. We go back to minimum again by scattering the pennies in a meaningless, random pattern.

Basically the number of units, such as pennies, multiplied by the possible arrangement of each, gives us the potential complexity of a system. To reach the maximum possible complexity, the arrangement must have no repetition or randomness. Motion, or potential motion, in a system will obviously drastically increase complexity. Theoretically, there can be neither zero nor infinite complexity in the universe.

INTRICACY

Complexity can also be expressed by the ratio of actual complexity to the maximum or minimum possible complexity. At this point, we should also compare and contrast intricacy and complexity. Intricacy, as opposed to complexity, is just such a ratio. A system that is intricate will be closer to it's maximum potantial complexity than a system that is not intricate, regardless of how complex it is.

A lawnmower engine may be more complex than a watch mechanism but the watch is more intricate because, given the amount of metal in the watch, because it is much closer to it's maximum possible complexity. A comparison of intricacy requires a common denominator of the given, while complexity does not.

RANDOMNESS

The patterns of complexity have no effect on it's quantification, only the numbers of levels. Neither does the amount or quantity of each level have any effect on the complexity of a system, only the number of levels. A ton of sugar is no more complex than a pound of sugar as long as it has no special arrangement.

In taking a measurement of complexity, it is necessary to define or understand what will be considered as randomness. Such as the meteorologist being unconcerned with documenting the patterns of the crystals in each snowflake. He would leave that undocumented and it would be considered as randomness. Thus, randomness is a form of irrelevance.

This is the reason that we have never gotten around to quantifying complexity, that not all levels are relevant to whatever reason we would be taking the measurement of complexity.

Anything that is done to an object, such as cutting it into pieces, that requires more levels to be manifested, will increase complexity. This is only if it is deemed necessary to describe the dimensions and gaps between the pieces, that they are not to be considered as random. Likewise, anything that reduces the number of levels reduces complexity.

Randomness is not complexity. Random arrangements are not to be classified as complexity. Randomness is a state in which any arrangement has about as much meaning as any other arrangement, given the purpose for which that the complexity is being measured. Thus, any random permutations can be considered as of equal complexity. In pure randomness, positions can be expressed as a concentration, the actual positions or arrangements of sub-domains are not expressed and are not considered as increasing complexity.

A fundamental rule of the quantification of complexity is that, if the number of levels necessary to describe a system can be determined either by a direct description of that system or a description of the forces that created that system, the complexity of that system is the lesser number of levels. Thus, we could say that a system can be effectively described by a description of the forces that brought it about, but only if it can be done with less information than a direct description of the system. The rule is that the complexity of a system is the least information necessary to describe it.

Random scattering could be described by describing the forces that brought about the scattering. An obvious way to see if complexity can be reduced is to look for a formula that can describe a system. A formula can thus be described as a tool to reduce complexity.

NUMERICAL AND VERBAL EXPRESSION

In the quantification and expression of complexity, it is only numerical levels that count, not verbal descriptions. Words are a human creation, the real universe operates by numbers, which form the levels by which we are going to measure complexity. Descriptions are for our use only.

Our verbal definitions of systems are fuzzy and imprecise and are not the same as absolute numerical definitions. To measure complexity, the levels are represented by numbers or statements equivalent to numbers, no matter how a description is stated.

LEVELS OF OBSERVATION

The reason for relevance and irrelavance in measurements of complexity, including randomness is that we can observe our sorroundings from many different levels. Even the nature of our observation of complexity is thus expressed in levels. The lowest level from which we can make observations is that of sub-atomic particles. The highest level is to observe the universe as a whole all at once.

You may notice that the higher we get in our observation level, the more is considered as random. At the lowest level of observation, nothing is random, the position of every electron in every atom counts. At the highest level, the level of the entire universe, almost everything is random such as the orbits of planets around stars.

Thus at the highest level of observation, the entire universe is no more complex than a single atom at the lowest observation level but there is far more randomness. We can see that the amount of randomness, the number of levels that do not need to be included in the measurement, in a measurement of complexity will be proportional to the height of our observation level.

Of course, we see this way because we are human beings. The way God sees the universe, there is the absence of any irrelevance or randomness in the view at the lowest observational level when he views the universe at the highest observational level, that of the entire universe. A formula can be described as the reduction in complexity at a lower level of observation by viewing the system from a higher level.

COMPARISON OF COMPLEXITY

This is why our present expressions of complexity are so subjective. So many present levels are irrelevant to our measurements. However, in considering whether we should begin quantifying complexity, we must also consider that the main purpose for which human beings would do so would not be so much for direct measurement but for comparison purposes.

Measurement of comparitive complexity is much simpler than measurement of absolute complexity. We do not need to know or measure everything about the systems whose complexity we are measuring and comparing as long as the levels that we do not measure are the same for both and the unmeasured levels will cancel each other out.

We can never be sure of the real complexity of any system or domain because we do not know all there is to know. But it is logical to believe that any undiscovered properties in measurement of complexity for the purpose of comparison will cancel out. Any inefficiencies in measurement and expression will also cancel out since the main purpose of measurement of complexity will be the comparison of the complexity of two or more systems.

THE LIMITS OF COMMUNICATION

I now think that I understand how we have managed to miss the tremendous benefit that could come with being able to put a number on complexity, which is what "quantification" means. The reason is not complex at all, in fact it is very simple.

Suppose that you had an object of some kind and wanted someone far away to understand the object. You would most likely take a photo or make a drawing of the object and send it to them. This is where the problem lies, for it is the use of visual representation which conceals complexity from us.

Suppose now that you only had words with which to describe the object to someone who was completely unfamiliar with it. The length of the text required to thoroughly describe the object, without comparing it with any other objects, would be an effective measure of the complexity of the object.

Take an object and thoroughly describe it, or the aspects of it that are relevant, by sentences without illustration or comparison and contrast to other objects. The number of sentences, or more precisely the number of descriptions in the sentences, since some sentences are compound, is actually equal to the complexity of the object.

The use of numbers in description makes it even easier to quantify complexity. Each number used to describe the object represents one unit of it's complexity. This is easier to deal with since it requires more care to parse verbal descriptions.

Similarity between objects is found when, in an effort to describe an object or event or situation, it becomes more efficient to describe it by starting with the description of a similar object (or event or situation) and then describing only what is different, rather than starting a description from the beginning. The degree of similarity can easily be expressed as the proportion of the description saved by starting with the description of the similar object (or event or situation) for which a description already exists. This means that the complexity of an object, such as a vase or rocket, can be described in terms of variation from standard geometric shapes such as circle and rectangle.

My conclusion is that if humans had never developed drawing, painting, or other such artwork, we would have long since been expressing complexity in numbers, instead of vague and general descriptions. The lack of illustration would have been a great disadvantage, but the quantification of complexity would have been of immeasurable benefit. Computer coding makes it somewhat easier to quantify complexity since everything, including illustrations, are written as lines of code.