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Firstly, living systems have a sense of the direction of time , and therefore, their models must be irreversible. In addition to that, evolution of living systems is directed toward higher levels of complexity if complexity is associated with a number of different features. Combining Newtonian mechanics, thermodynamics and the phenomenon of instability, both of these
properties can be implemented. However, they are necessary, but not sufficient for life: there are plenty of physical processes which possess the same characteristics: chaos, turbulence, convection, etc.
The third property of living systems is the capacity to stimulate their own replication; however, even that property is not sufficient since it cannot disqualify fire or other exponentially unstable physical processes.
The fourth property can be associated with a so called "free will", or in terms of mathematical formalism, with a probabilistic evolution. Again, there are plenty of physical phenomena (chaos, Langevin models) whose evolution can be described only probabilistically.
The fifth property can be stated as the ability to perform certain transitions or motions which are not directly controlled from outside. Such an autonomy must be supported by energy flux (with low entropy input and high entropy output), and as a "side effect", it can be accompanied by information processing. Indeed, autonomous systems can converge to a limit cycle(flutter), to chaotic attractor (Lorenz model) or to a static attractor (neural nets), and that can serve as a memory.
All the five properties listed above exhibit "complexity without purpose", and that is why they are necessary, but not sufficient for life identification. We will postulate now the last property: any living system has an objective of its activity. The global objective is always to survive, but local objectives can be different as long as they contribute to the global one. From
the viewpoint of phenomenological formalism, a living system must possess its own model and interact with it to achieve the objective. For instance, it can run the model faster than real time thereby predicting its future state, compare this state with the objective and change the strategy if necessary. Actually, all the man-made controlled systems mimic this property of a
living system (however, controlled systems usually do not possess all the previous five properties mentioned above).