Published online by Cambridge University Press: 28 February 2011
The reduced canonical form of a game G, denoted by Ḡ, is the simplest game infinitesimally close to Ḡ. Reduced canonical forms were introduced by Calistrate [2], who gave a simple construction for computing Ḡ. We provide a new correctness proof of Calistrate’s algorithm, and show that his techniques generalize to produce a family of reduction operators. In addition, we introduce a completely new construction of Ḡ, motivated by Conway’s original canonical-form construction.
Although canonical forms sometimes reveal substantial information about the structure of combinatorial games, they are often too complicated to be of any great use. Many of the most interesting games—including Clobber, Amazons, and Hare and Hounds—give rise to some massively complex canonical forms even on relatively small boards. In such cases, a method for extracting more specific information is highly desirable. The familiar temperature theory, and the theory of atomic weights for all-small games, can be viewed as efforts to address this problem.
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