Adaptive system

Artificial adaptive systems include robots with control systems that utilize negative feedback to maintain desired states.

The law of adaptation

The law of adaptation may be stated informally as:

Every adaptive system converges to a state in which all kind of stimulation ceases.

Formally, the law can be defined as follows:

Given a system $S$ , we say that a physical event $E$ is a stimulus for the system $S$ if and only if the probability $P (S \to S^{'} | E)$ that the system suffers a change or be perturbed (in its elements or in its processes) when the event $E$ occurs is strictly greater than the prior probability that $S$ suffers a change independently of $E$ :

P (S \to S^{'} | E) > P (S \to S^{'})

Let $S$ be an arbitrary system subject to changes in time $t$ and let $E$ be an arbitrary event that is a stimulus for the system $S$ : we say that $S$ is an adaptive system if and only if when t tends to infinity $(t \to \infty)$ the probability that the system $S$ change its behavior $(S \to S^{'})$ in a time step $t_{0}$ given the event $E$ is equal to the probability that the system change its behavior independently of the occurrence of the event $E$ . In mathematical terms:

- $P_{t_{0}} (S \to S^{'} | E) > P_{t_{0}} (S \to S^{'}) > 0$
- $lim_{t \to \infty} P_{t} (S \to S^{'} | E) = P_{t} (S \to S^{'})$

Thus, for each instant $t$ will exist a temporal interval $h$ such that:

P_{t + h} (S \to S^{'} | E) - P_{t + h} (S \to S^{'}) < P_{t} (S \to S^{'} | E) - P_{t} (S \to S^{'})

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The law of adaptation

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