Theorem (Levin): Let n be the number of plies in a tree, and let b be the number of branches a every branch point. Then the number of terminal points on the tree is

However, if the best possible advantage is take of the alpha-beta heuristic then the number of terminal points that need to be examined is for odd n,

and for even n,

which can be reformulated for both cases using ceil and floor functions:

Chess programs catch some of the human chess playing abilities but rely on the limited effective branching of the chess move tree. The ideas that work for chess are inadequate for go. Alpha-beta pruning characterizes human play, but it wasn't noticed by early chess programmers - Turing, Shannon, Pasta and Ulam, and Bernstein. We humans are not very good at identifying the heuristics we ourselves use. Approximations to alpha-beta used by Samuel, Newell and Simon, McCarthy. Proved equivalent to minimax by Hart and Levin, independently by Brudno. Knuth gives details.

## Table of Contents

Home * People * Michael LevinMichael Levin,an American computer scientist, in the 60s affiliated with Massachusetts Institute of Technology, and involved in the initial development of Lisp within the group of John McCarthy. The 1961 memo on Alpha-Beta by Daniel Edwards and Timothy Hart

^{[1]}, contains a Theorem by Michael Levin, the well known formula of the number of leaf nodes that need to be examined in Alpha-Beta.## Theorem

Theorem(Levin): Letbe the number of plies in a tree, and letnbe the number of branches a every branch point. Then the number of terminal points on the tree isbHowever, if the best possible advantage is take of the alpha-beta heuristic then the number of terminal points that need to be examined is for odd

n,and for even

n,which can be reformulated for both cases using ceil and floor functions:

## Quotes

## Alpha-Beta

Quote by John McCarthy fromHuman-Level AI is harder than it seemed in 1955^{[2]}:Chess programs catch some of the human chess playing abilities but rely on the limited effective branching of the chess move tree. The ideas that work for chess are inadequate for go. Alpha-beta pruning characterizes human play, but it wasn't noticed by early chess programmers - Turing, Shannon, Pasta and Ulam, and Bernstein. We humans are not very good at identifying the heuristics we ourselves use. Approximations to alpha-beta used by Samuel, Newell and Simon, McCarthy. Proved equivalent to minimax by Hart and Levin, independently by Brudno. Knuth gives details.## LISP

Quote by John McCarthy inFrom LISP 1 to LISP 1.5^{[3]}:Many people participated in the initial development of LISP, and I haven't been able to remember all their contributions and must settle, at this writing, for a list of names. I can remember Paul W. Abrahams, Robert Brayton, Daniel Edwards, Patrick Fischer, Phyllis Fox, Saul Goldberg, Timothy Hart, Louis Hodes, Michael Levin, David Luckham, Klim Maling, Marvin Minsky, David Park, Nathaniel Rochester of IBM, and Steve Russell.## See also

## Selected Publications

1962) LISP 1.5 Programmer's Manual. The M.I.T. Press, second edition (1985) available as pdf reprint^{[4]}1963).Primitive Recursion, AIM-055, reprint available from DSpace at MIT1964).LISP Exercises, AIM-064, reprint available from DSpace at MIT1964).Syntax of the New Language, AIM-068, reprint available from DSpace at MIT1964).New Language Storage Conventions, AIM-069, reprint available from DSpace at MIT## References

1961).The Alpha-Beta Heuristic, AIM-030, reprint available from DSpace at MIT1979). History of Lisp, From LISP 1 to LISP 1.5## What links here?

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