Adaptive system

An adaptive system is a set of interacting or interdependent entities, real or abstract, forming an integrated whole that together are able to respond to environmental changes or changes in the interacting parts, in a way analogous to either continuous physiological homeostasis or evolutionary adaptation in biology. Feedback loops represent a key feature of adaptive systems, such as ecosystems and individual organisms; or in the human world, communities, organizations, and families. Adaptive systems can be organized into a hierarchy.

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

[math]\displaystyle{ P(S \rightarrow S'|E)\gt P(S \rightarrow S') }[/math]

Let [math]\displaystyle{ S }[/math] be an arbitrary system subject to changes in time [math]\displaystyle{ t }[/math] and let [math]\displaystyle{ E }[/math] be an arbitrary event that is a stimulus for the system [math]\displaystyle{ S }[/math]: we say that [math]\displaystyle{ S }[/math] is an adaptive system if and only if when t tends to infinity [math]\displaystyle{ (t\rightarrow \infty) }[/math] the probability that the system [math]\displaystyle{ S }[/math] change its behavior [math]\displaystyle{ (S\rightarrow S') }[/math] in a time step [math]\displaystyle{ t_0 }[/math] given the event [math]\displaystyle{ E }[/math] is equal to the probability that the system change its behavior independently of the occurrence of the event [math]\displaystyle{ E }[/math]. In mathematical terms:

1. - [math]\displaystyle{ P_{t_0}(S\rightarrow S'|E) \gt P_{t_0}(S\rightarrow S') \gt 0 }[/math]
2. - [math]\displaystyle{ \lim_{t\rightarrow \infty} P_t(S\rightarrow S' | E) = P_t(S\rightarrow S') }[/math]

Thus, for each instant [math]\displaystyle{ t }[/math] will exist a temporal interval [math]\displaystyle{ h }[/math] such that:

[math]\displaystyle{ P_{t+h}(S\rightarrow S' | E) - P_{t+h}(S\rightarrow S') \lt P_t(S\rightarrow S' | E) - P_t(S\rightarrow S') }[/math]

Benefit of self-adjusting systems

In an adaptive system, a parameter changes slowly and has no preferred value. In a self-adjusting system though, the parameter value “depends on the history of the system dynamics”. One of the most important qualities of self-adjusting systems is its “adaptation to the edge of chaos” or ability to avoid chaos. Practically speaking, by heading to the edge of chaos without going further, a leader may act spontaneously yet without disaster. A March/April 2009 Complexity article further explains the self-adjusting systems used and the realistic implications. Physicists have shown that adaptation to the edge of chaos occurs in almost all systems with feedback.

See also

• Autopoiesis
• Adaptive immune system
• Artificial neural network
• Complex adaptive system
• Diffusion of innovations
• Ecosystems
• Gaia hypothesis
• Gene expression programming
• Genetic algorithms
• Learning
• Neural adaptation

References

• "Adaptation, Anticipation and Rationality in Natural and Artificial Systems: Computational Paradigms Mimicking Nature". Natural Computing 8 (4): 757–775. 2009. doi:10.1007/s11047-008-9096-6.