The Reduced Canonical Form of a Game
Dan Calistrate · 1997
Cooling by *, followed by the elimination of the stars, is used to define an operator G → Ḡ on short games, having the following properties: G is the simplest game infinitesimally close to G; the operator is a homomorphism; it can be used for recursive calculations, provided that the games involved are not in a “strictly cold” form. 1. Introduction We will use the classical definitions and facts about two-person, perfect information combinatorial games with the normal winning convention, as developed in Winning Ways [Berlekamp et al. 1982] and On Numbers and Games [Conway 1976]. We recapitulate them briefly. Formally, games are constructed recursively as ordered pairs ﹛ Γ L \ Γ R ﹜, where Γ L and Γ R are sets of games, called, respectively, the set of Left options and the set of Right options from G. We will restrict ourselves to short games, that is, games where the sets of options Γ L and Γ R are required to be finite in this recursive definition. The basis for this recursion is the game ﹛∅ |∅ ﹜, which is called 0. We will often let G L and G R represent typical Left and Right options of a game G, and write G = ﹛G L | G R ﹜. Two games G and H are identical, or have the same form, if they have identical sets of left options and identical sets of right options. In this case we write G = H. Whenever the distinction between the value and the form of a game is essential, we will specify it; otherwise, by G we will mean the form of G.