# Difficulty and measures of difficulty

The concept of complexity has been intensively developed not only within the framework of **systems research** (i.e. research on the development of a systemic approach), but also in many other scientific and even engineering disciplines/theories. There is no final agreement on what complexity is, and it is not foreseen: different projects and different roles in projects are interested in different types of complexity. Seth Lloyd has compiled some examples of definitions for complexity measures^{[1]}. He attributed all these definitions to attempts to answer three questions:

**1.** **How difficult is it to describe the system?** This is usually measured in bits spent on representing the description. Complexity measures here would be information, entropy, algorithmic complexity or algorithmic content of information, maximum length of description, Fisher information, Rényi entropy, code length (prefix-free, Huffman, Shannon-Fano, error-correcting, Hamming), Chernoff information, dimension, fractal dimension, Lempel-Ziv complexity.

**2.** **How difficult is it to create a system?** Complexity as the difficulty of creation is measured in terms of time, energy, dollars, etc. Complexity measures here are computational complexity, time complexity, space complexity, information-based complexity, logical depth, thermodynamic depth, cost, crypticity.

**3. What is the degree of organization?** There can be two variants:

**a)** **"effective complexity"**, the difficulty of describing the organizational structure, whether corporate, chemical, cellular. Here are some in English: Metric Entropy; Fractal Dimension; Excess Entropy; Stochastic Complexity; Sophistication; Effective Measure Complexity; True Measure Complexity; Topological epsilon-machine size; Conditional Information; Conditional Algorithmic Information Content; Schema length; Ideal Complexity; Hierarchical Complexity; Tree subgraph diversity; Homogeneous Complexity; Grammatical Complexity.

**b)** **the amount of information that needs to be exchanged between parts of the system** due to such organizational structure: Algorithmic Mutual Information; Channel Capacity; Correlation; Stored Information; Organization.

There are concepts that are not complexity concepts, but closely related: Long-Range Order; Self-Organization; Complex Adaptive Systems; Edge of Chaos. There are also completely alternative measures of complexity (for example, based on the speed of description of objects being evaluated, rather than the volume of this description^{[2]}).

The situation with the concept of "complexity" is very characteristic of the systemic approach: the words used in it seem quite "everyday" and have clear and intuitively understandable meanings from childhood. But no, these words suddenly turn out to be terms behind which hide very different concepts, with these concepts working the most diverse logical or quantitative models of the most diverse schools of thought, quantitative measurements are carried out of their very different characteristics for very different purposes.

**In the context of our course**, **we will consider a system as complex if it consists of a sufficiently large number of elements, so large that** **one individual agent cannot** **assess all connections and interactions** **with the precision required for** **rational** **project activities.**

For example, when it comes to human agents, a project involving the creation and development of an organizational system may easily have five hundred people. This is more than enough for one mind to assess all the connections and interactions of these people, even if one mind keeps records in their computerized exocortex. Such an organizational system is inherently complex.

It is equally difficult for one agent (even a collective one) to grasp all the connections and interactions in a smartphone system, in a nuclear power plant system, and many other engineering systems—extended enterprises in such projects unite the work of various autonomous individual enterprises. Someone designs the processor chip for the smartphone, someone manufactures the chip, someone designs the smartphone, someone designs marketing activities—various enterprises are involved, employing many smart individuals, each pursuing their own interests, some of which are negotiated. It is all complex, extremely complex.

For our educational purposes in systemic thinking, this informal understanding of complexity is quite sufficient, but you should remember that there are formal understandings of complexity, there are numerous complexity theories from which these formal understandings have emerged.

In physical theories of evolutionary complexity growth in biological systems, the main definition goes in a different direction: the number of operations needed to create a specific biological object^{[3]}.

Systems thinking does not provide any "objective answers" to questions about systems. These answers always depend on which role from what interest is asking and which role from what interest is answering, which conflicts of system levels we are trying to overcome, what ethics we adhere to.

**There is no step-by-step algorithm in systems thinking that leads to the right answer** **(and this is not surprising: remember about the unstructured nature: a huge number of different answers provide a huge number of approximately equally suboptimal quality** **solutions), there is no sequence of steps guaranteeing some acceptable result of this thinking, there are no** **"typical cases".** **There are always nuances that make even the most** **"typical"** **completely atypical.**

Systems thinking is complex, it takes a long time to learn—it resembles some "unobjective higher mathematics," more or less typical reasoning about a set of objects with types prescribed by the systemic approach. Systems thinking doesn't even bother to give precise definitions to its concepts: different agents try to adapt these concepts to their very different current interests and roles.

Despite all this uncertainty, **the concepts of systems thinking allow compact and simple description of the complex world in order to manage collective attention in** **complex** **project situations!** Alternative approaches (for example, reductionism or holism—when you do not explicitly use systemic levels to organize people's attention in the project) turn out to be much, much worse.

Complexity is notoriously difficult to define precisely, and no single, universal definition has been agreed upon (24--26), although it seems to be commonly held that "when we see it, we know it." The definitions of complexity that appear to be meaningful in biology involve, depending on the level of analysis, the number of evolutionarily conserved nucleotide sites in genomes and the number of genes or functional components in organisms or suborganismal functional systems, as well as the hierarchical organization of biological entities, be it functional networks and pathways, cells, organisms, or communities (20--23, 27). Perhaps the most general definition, pathway complexity, deriving from the concept of algorithmic complexity in mathematics, includes the number of steps required to create a given object. It has been proposed that entities with a pathway complexity above a certain threshold can only originate from biological processes (23). And further references in the work "Physical foundations of biological complexity," where this quote is taken from, https://www.pnas.org/doi/full/10.1073/pnas.1807890115 ↩︎