Trans-algorithmic nature of learning in biological systems

Abstract

Learning ability is a vitally important, distinctive property of biological systems, which provides dynamic stability in non-stationary environments. Although several different types of learning have been successfully modeled using a universal computer, in general, learning cannot be described by an algorithm. In other words, algorithmic approach to describing the functioning of biological systems is not sufficient for adequate grasping of what is life. Since biosystems are parts of the physical world, one might hope that adding some physical mechanisms and principles to the concept of algorithm could provide extra possibilities for describing learning in its full generality. However, a straightforward approach to that through the so-called physical hypercomputation so far has not been successful. Here an alternative approach is proposed. Biosystems are described as achieving enumeration of possible physical compositions though random incremental modifications inflicted on them by active operating resources (AORs) in the environment. Biosystems learn through algorithmic regulation of the intensity of the above modifications according to a specific optimality criterion. From the perspective of external observers, biosystems move in the space of different algorithms driven by random modifications imposed by the environmental AORs. A particular algorithm is only a snapshot of that motion, while the motion itself is essentially trans-algorithmic. In this conceptual framework, death of unfit members of a population, for example, is viewed as a trans-algorithmic modification made in the population as a biosystem by environmental AORs. Numerous examples of AOR utilization in biosystems of different complexity, from viruses to multicellular organisms, are provided.

Appendix: Impossibility of programmatic increase in complexity

Below is a proof that an algorithm cannot increase its complexity. It is based on the notion of Kolmogorov complexity of a given object’s binary code, which can be defined as the length of the shortest program that produces the code (Li and Vitanyi 2008).

Assume opposite, namely, that there exists a program $\hbox {P}_{1}$ of complexity ${L}_{1}$ such that it somehow can produce a binary sequence encoding a program $\hbox {P}_{2}$ of complexity ${L}_{2}>{L}_{1}$. By definition, the complexity of a given program is equal to the minimal length of a (perhaps different) program that computes the same function. Therefore, there is a program P of length ${L}_{1}$ that computes the same function as $\hbox {P}_{1}$, meaning that it can produce $\hbox {P}_{2}$. This, however, directly contradicts the above assumption that the complexity of $\hbox {P}_{2}$ is greater than ${L}_{1}$. End of proof.

The above proof might seem to contradict the fact that a relatively simple program can enumerate all programs as finite binary sequences, including ones that more complex than the program itself. However, production of a specific binary sequence means that the program stops after having produced it, for which adding a suitable stopping criterion to the above program is required. That criterion can be viewed as a constructive descriptor of the desired binary sequence, the addition of which adds complexity to the enumerating program.

A conclusion that programmatic increase in complexity is impossible has been also obtained by Hernández-Orozco et al. (2016) in their modeling of open-ended evolution and adaptability in terms of computable dynamic systems.