A step-by-step ordered and finite set of prescribed operations along an optimal path, whose use permits the solution of a specific class of problems.
Any algorithm implies determinism.
J.Z. YOUNG gives a definition related to computation: "A program by which a complicated calculation is reduced to a long series of simple ones" (adding: "that a digital computer can perform": see hereafter the comment on artificial – or natural – neural networks).(1978, p.289)
G. KLIR states: "The intuitive notion of an algorithm was formalized in several ways, including formalizations based on the concepts of TURING machines, MARKOV algorithms, and recursive functions, which were all proved to be equivalent" (1991, p.127)
An algorithm may be quite complex and contain a number of subordinated routines and instructions for their eventual use. However, the TURING machine is, at least in principle, a universal representation of all possible algorithms. The potential and global use of an algorithm is strictly limited to its content, which necessarily reflects an implicit and specific representation of its field of application.
GÖDEL's incompleteness concept is thus valid for any algorithm.
As to algorithms relations to natural or artificial intelligence, it can be argued that they are merely acquired properties of brains or computers. In the latter's case, they must be introduced in the machine as a product of natural intelligence. It is now however conceivable that artificial neural networks could become able to construct complex algorithms by learning.
According to R.W. FULLER and P. PUTNAM, this is precisely what the nervous system does: "… (the) basic operation of the nervous system: establishing a relative dominance among the members of a class of competitive motor acts" (1967, p.103)
Knowledge itself seems to be stabilized through algorithms. R. FISCHER, quoting H.M. COLLINS, writes: "… the discovery is generally re-constructed as if it followed an algorithm, a narrative receipt, with the experience of contingency, trial and error written off as irrelevant "side effects" (1992, p.229).
According to the Russian psychologist L. VYGOTSKY, children use private speech (i.e. speak to themselves) as a self-narrative way to acquire new skills, until the moment they dominate it. (L. BERK – 1994, p.60)
This possibly explains why knowledge acquisition through teaching is so different from discovery and self-training, and sometimes quite stultifying.