Food for thought!

[Michael Sherwin]

"The algorithm that we are to work with rest on two principles. The first is that both sides in the end have to strive for material gain (this is not in the rules of the game, but is true in practice). It follows that everything begins with the attack, and the attack defines the trajectory (the path) of the attacking piece. A chess game may be thought of as a debate between the two sides; the attack is an assertion, the defence a negation. The assertion can bring forth a negation, which defines the path of defence (the negation)."

"Can there be a defence (a negation) without an attack (an assertion)? Probably not. True, it seems to be generally felt that a piece can "protect" another even in the absence of an attack. For example, in Fig. 1, the king at b2 "protects" the Queen at a2. In this position, according to the rules of the game, Black may not attack the White Queen, so this is all merely a combination of words. The King at b2 is in fact not defending the Queen at a2, since there is no attack on the Queen and there cannot be one."

"We may negate not only an assertion but also a negation; that is, we may not only defend (if there is an attack) but we may also hinder a defence.

"The second principle amounts to saying that we must find an approximate solution by limiting the problem. If we are to consider all possible attacks, the problem becomes impossibly difficult to solve. We have to limit the problem and look at only a few attacks and the corresponding negations. In other words, we must establish a horizon and deal with only those attacks that fall within it. Within the limits of the horizon the problem will be (or should be) solved exactly."

"We said earlier that we would consider some attacks only, not all, and the negations connected with them."

"Suppose we think of a simile--I hope that chess players will not be insulted. How can a dog count her litter? She has to know how to count: if "counting" were not included in her "algorithm"she would never notice a dissappearance of her offspring. But even if the method were at hand, without some limitation of the task she could never keep up with the count: her canine capabilities would not be equal to an infinite litter-count (say, a dozen puppies). She counts at a very low level: one, two, three, and "many." The disappearance of one puppy out of five would not be noticed! But for the preservation of the canine race the ability she has is sufficient."

"We must define the notion of "many" for chess. Then we shall have to calculate only a limited number of functions and the task becomes real."

"In any given position, the more the number of half-moves that we have to make to get an attacking piece to the square where the enemy waits, the more complex the problem, and visa versa. The longer the path, i.e., the greater the number of squares to be stopped on, the more complex the battle becomes. Thus the conclusion comes to us of its own accord--to limit the problem by inspecting only those pieces that can reach an enemy-held square in no more than a specified number of half moves ..."

"An attack falling within the horizon is included in the mathematical calculations--otherwise, it is not."

"We could prove that all the devices used in chess war--attack (from both sides, blockade (by either side), and (mutual) retreat--can all be accounted for in an initial mathematical map constructed on an attack contained within four half moves." -- Botvinnik!