Normal programs P produce the same output y each time they receive a particular input x. Learning programs are capable of improving their performance so that they may produce different (better) results on second or later times that they receive the same input x.
They achieve this by being able to alter their internal state, q. In effect, they are computing a function of two arguments, P(x | q) = y. When the program is in learning mode, the program computes a new state q' as well as the output y, as it executes.
In the case of supervised learning, in order to construct q', one needs a set of inputs xi and corresponding target outputs zi (i.e. you want P(xi | q) = zi when learning is complete). The new state function L is computed as:
L(P, q, ((x1,z1), ..., (xn, zn))) = q'
See also unsupervised learning, observation language, and hypothesis language.
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