**6. Conclusion**

This work examines the systems described by networks that are: autonomous, complex, finite, discreet, directed, functioning, deterministic, and designed as the Kauffman network (Boolean), but with the admission of more than two (*s* ≥ 2) equally probable signal variants. The number *K* of the node inputs in a given network is fixed. Parameters *s* and *K* have values (*s* = 2, *K* = 4 or *s* = 4, *K* = 3), which in the case of fully random networks give unambiguous chaos.

Half-chaos is the state of such a system in which small disturbances cause both small and large 'damage' (changes in the system's functioning) occurring statistically similarly often. As a reminder, in the chaotic system there are only large damages and in the ordered - only very small. The studied half-chaotic systems are not fully random, but they have typical characteristics indicating a full randomness.

The evidence presented in the work indicates that half-chaos is an experimental fact. Its basic mechanism is based on a short attractor, but it is too weak a condition for modeling adaptive evolution. Much more adequate (to describe the purposeful systems) the half-chaos variant depends on in-ice-modularity. The simplest way to obtain such a state is starting from an easily attainable system with a point attractor, but it has been shown that it is possible to build such a state based on its description recognized in the evolution started from the point attractor. These are 'small lakes of (nodes) activity' in 'ice' (the area of the network where nodes do not change their states) - a picture similar to the one described by Kauffman (in the network parameters space) of the 'liquid' area at the boundary of the ordered ('frozen / solid') area and chaotic ('gas'). The Kauffman model is the basis of the famous hypothesis "life on the edge of chaos", however, this model strongly limits the parameters allowed for modeling life to *K* = 2 and *s* = 2 and their immediate vicinity called phase transition. Half-chaos allows a much larger range of these parameters, many estimates indicate such a need.

The experiments used a constant random structure (mainly scale-free and Erdős-Rényi random networks, but also others), random initial states of nodes and random functions, but despite the maintenance of characteristics indicating the randomness of the function, they were non-randomly correlated with states. Evolutionary

variability concerned node functions. However, half-chaos was also observed in experiments, where the network grew by random addition and removing of nodes. This testifies to the more general nature of the discovered phenomenon of half-chaos.

Acceptance (as an evolutionary change) of a disturbance that gives great damage leads to ordinary chaos, which practically does not return to half-chaos. This is the elimination model - death. On the other hand, acceptance of a disturbance that gives small damage is enough to remain in half-chaos. This feature is: 'evolutionary stability of half-chaos', it is one of the most important, added to the definition of half-chaos. It creates a natural criterion of identity of the evolving object. The distinction between small and large damage is natural because they create separate peaks in the distribution of damage size separated by a large gap, in which there are practically no counts.

The discovery of half-chaos radically changes the vision of the dynamics of the studied systems. The famous Kauffman's hypothesis 'life on the edge of chaos' is strongly reinterpreted to 'life evolve in half-chaos of not fully random systems' and the analogy to phase transition is substituted by the comparison of half-chaos to 'superheated liquid'. Strong limitations contrary to the observation, on the parameters of modeling of purposeful systems are removed.
