By M. Clarke

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Then a search must start from P1 for a defence to the threat. Since [P,P1] did not qualify as a cycle-1 move, then if a) P is an utm node, then tally(P1) tally(P1+) < goal. b) P is a ttm node, then tally(P1) £ goal and tally(P1+) £tally(P1) for all P1 + successor nodes of P1. =>tally(P1+) £ goal. In both cases all successor moves of P1 qualify as cycle 1 moves. In order to try first the defence moves from P1 which are most likely to succeed and require the least computation, the following restriction is imposed on cycle-1 moves from P1.

This ROM also evaluates the king's centralization assuming it is an end game. The actual evaluation is a weighted sum of the king safety and king centralization selected by the stage of the game. Piece Position Every piece has a value for occupying a square. This value is maintained incrementally. Pawns are more valuable for being centralised and for being advanced. Knight mobility is statically evaluated based on the squares that they occupy. Knights and bishops are penalized for being in their original square.

Since the last time P was updated was at P- (before the descent P-> Pr) it follows that B(P-) =B(P). But then on descent from P- to P, B(P-) Sgi. This contradicts move m being the first move to falsify the lemma. The case where P is a MIN node is proved similarly. Lemma If Pr is a node that is visited on the ith and j th search, where i < j and Pr is the rth successor of P, then on descent from P to Pr: A(P)i,m£ A(P)j,m B(P)i,m£B(P)j,m where A(P)i,m is the value of A at P on the ith search on descent through move m.

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