Further progress in the application of computers to many practical fields seems to depend heavily on the success in implementing learning and inductive processes within machines. For example, to develop a consultation system for medical or plant disease diagnosis, prognosis and decision making in general, it is very desirable, perhaps even necessary, to be able to'teach' the system through examples of correct and/or incorrect decisions, rather than by precisely describing the decision process in its full generality and then transforming this description into a computer program. A similar situation exists in computer chess. The development of computer programs playing at the master level (especially the end games) seems to be a formidable task if the programs are not eventually able to learn and improve on their decision making rules through the specific examples of games, rather than by being explicitly told all the rules. Due to easy access to human knowledge about chess and the relative simplicity of testing the results, chess is one of the most attractive testing domains for inductive inference programs.

Machine intelligence problems are sometimes defined as those problems which (i) computers can't yet do, and (ii) humans can. We shall further consider how much "knowledge" about a finite mathematical function can, on certain assumptions, be credited to a computer program. Although our approach is quite general, we are really only interested in programs which evaluate "semihard" functions, believing that the evaluation of such functions constitutes the defining aspiration of machine intelligence work. If a function is less hard than "semihard," then we can evaluate it by pure algorithm (trading space for time) or by pure lookup (making the opposite trade), with no need to talk of knowledge, advice, machine intelligence, or any of those things. We call such problems "standard." If however the function is "semihard," then we will be driven to construct some form of artful compromise between the two representations: without such a compromise the function will not be evaluable within practical resource limits. If the function is harder than "semihard," i.e. is actually "hard," then no amount of compromise can ever make feasible its evaluation by any terrestrial device.

The alpha-beta technique for searching game trees is analyzed, in an attempt to provide some insight into its behavior. The first portion of this paper is an expository presentation of the method together with a proof of its correctness and a historical discussion. The alpha-beta procedure is shown to be optimal in a certain sense, and bounds are obtained for its running time with various kinds of random data.

Research in computer chess has been active for over three decades. Over that period, computer chess has fallen from the position of being a prominent research application in artificial intelligence to a peripheral area. In this paper, we take a retrospective look at what has been accomplished, in order to understand where the field is today and where it is headed tomorrow. Whereas the past has often been clouded by engineering passing as science, misspent effort for short-term gains, and research results with little applicability to other domains, there is evidence that computer chess is emerging from the shadow of its past and may now be recapturing some of its lost stature in the research world.

Since 1967 there has again been great interest in chess programming. This paper demonstrates that the structure of today's most successful programs cannot be extended to play Master level chess. Certain basic requirements of a Master player's performance are shown to be outside the performance limits to which a program of this type could be extended. The paper also examines a basic weakness in the tree-searching model approach when applied to situations that cannot be searched to completion. This is the Horizon Effect, which causes unpredictable evaluation errors due to an interaction between the static evaluation function and the rules for search termination. The outline of a model of chess playing that avoids the Horizon Effect and appears extendable to play Master level chess is presented, together with some results already achieved In IJCAI-73: THIRD INTERNATIONAL JOINT CONFERENCE ON ARTIFICIAL INTELLIGENCE, 20-23 August 1973, Stanford University Stanford, California, pp. 77-85

Dept. of Computer Science, Carnegie-Mellon University. "Many game-playing programs must search very large game trees. Use of the alpha-beta pruning algorithm instead of the simple minimax search reduces by a large factor the number of bottom positions which must be examined in the search. An analytical expression for the expected number of bottom positions examined in a game tree using alpha-beta pruning is derived, subject to the assumptions that the branching factor N and the depth D of the tree are arbitrary but fixed, and the bottom positions are a random permutation of ND unique values. A simple approximation to the growth rate of the expected number of bottom positions examined is suggested, based on a Monte Carlo simulation for large values of N and D. The behavior of the model is compared with the behavior of the alpha-beta algorithm in a chess playing program and the effects of correlation and non-unique bottom position values in real game trees are examined."

Russian Mathematical Surveys, 25, 221-262

The game of Go invites analysis. The rules seem few and simple, suggesting that the game may have helpful theorems. Tens of millions of people play and skill has developed over centuries to extraordinary levels. Thus, computer analysis can be tested against analysis by highly skilled human players. We study M × N boards, rather than the usual 19 × 19.

Russian Mathematical Surveys, 25, 221-262

BOXES is the name of a computer program. This is what the chess player does when he lumps together large numbers of positions as being'similar' to each other, by neglecting the strategically irrelevant features in which they differ. The resultant small game can be said to be a'model' of the large game. To give a brutally extreme example, consider a specification of chess positions so incomplete as to map from the viewpoint of White the approximately 1050 positions of the large game on to the seven shown in Figure 1. Even this simple classification may have a role in the learning of chess.