Quantifying Complexity

Complexity matters. This will hopefully become evident through the course of our discussion, but for now let's accept the principle that, in a great many situations, the extent to which something is complicated can be hugely important.

For a mathematician or scientist, a natural step after identifying something important is to attempt to quantify it, in the hopes of determining some of its properties. Us humans actually have a decent intuitive sense of different quantities of complexity. For example, we could all agree that Mozart's 40th is more complex than Twinkle Twinkle Little Star, or that sovling a crossword puzzle is more complex than tying a shoe. Other comparisons are less clear: Is a horse a more complex animal than a lion? Is China's economy more complex than India's?

Complexity researchers have identified several different ways that complexity can be quantified. These measures roughly fall into three categories:

• Variety - the complexity of an object can be quantified in terms of the number of actions it can take or the number of states in which it can exist.
  
• Descriptive complexity - the complexity of an object can be quantified in terms of the length of the shortest complete description of that object
  
• Algorithmic complexity - the complexity of a process can be quantified in terms of the number of steps or amount of time required to complete that process.
  

These three categories can be linked mathematically, which supports the idea that they are three expressions of the same concept rather than three different concepts. However, none of these can be unambiguously applied to the real world. For example, the number of actions or states of an object can be difficult to quantify. How many different actions can a human take? Descriptive complexity notions are dependent on the language used to describe something, and on what counts as a "complete" description. Similarly, algorithmic complexity notions depend on how a process is broken into tasks. This is not to say that the above quantification schemes are useless; just that care should be used in applying them and the values they give should be seen as approximate.

This approximateness is a problem for many scientists, who are used to dealing with the exact. How can we apply our analytical tools to a quantity which can never be precisely measured?

The way forward, in my opinion, is as follows. We (complex systems researchers) will investigate abstract models in which complexity can be mathematically quantified. The goal of investigating such models will be to discover laws of complexity which may carry over to the real world. At the same time, these laws must be checked against real-world experiment and observation. Because of the semi-fuzzy nature of complexity, the laws we discover will likely not be quantitative (e.g. F=ma or e=mc^2), but qualitative (e.g. "energy is conserved.")

In the near future, we will investigate two examples of such laws: Occam's Razor (the simplest explanation is the likeliest) and Ashby's Law (an organism must be as complex as its environment.) In the meantime, can anyone think of other qualitative laws of complexity/complex systems?