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The Nature of Computation$
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Cristopher Moore and Stephan Mertens

Print publication date: 2011

Print ISBN-13: 9780199233212

Published to Oxford Scholarship Online: December 2013

DOI: 10.1093/acprof:oso/9780199233212.001.0001

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Memory, Paths, and Games

Memory, Paths, and Games

Chapter:
(p.300) (p.301) Chapter 8 Memory, Paths, and Games
Source:
The Nature of Computation
Author(s):

Cristopher Moore

Stephan Mertens

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199233212.003.0008

Both time and memory are limited resources in the real world, but a fundamental difference between the two is that the latter can be reused. The gap between determinism and nondeterministic computation — that is, between finding a solution and checking one — is considerably smaller with memory than it is with time. This chapter explores how various amounts of memory can be used to solve complexity classes of problems either deterministically or nondeterministically. It shows how these classes highlight the difficulty of finding a path through an enormous graph, or finding a winning strategy against a clever opponent. It illustrates computational complexity in mazes, wooden puzzles, and board games thousands of years old. The chapter first discusses the computer's state space, the relationship between space-bounded computation and reachability, paths, and symmetric space. It then examines how nondeterminism works in the space-bounded case and proves that reachability is NL-complete. It also considers middle-first search and nondeterministic space, along with the word game Geography and the Asian game of Go.

Keywords:   memory, determinism, computation, computational complexity, board games, state space, reachability, nondeterminism, middle-first search, paths

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