[2016-07-30]
Conway games [2016-07-30]
Conway gamesL and R are left and right options of G.
G = {L1, L2, β¦ Ln | R1, R2 β¦ Rm }
0 = ({} , {}) = {|}
1 = ({0}, {}} = {0|}
-1 = ({} , {0}) = {|0}
n = {n - 1|}
1/2 = {0|1}
Conway induction: ICGN. 0 satisfies automatically.
Proof: infinite sequence of games not satisfying the property, contradiciton.
Conway induction implies DGC
Finitely many positions: short
Same moves: impartial game
Abelian group:
The set of all Conway games forms a partial order with respect to the comparison operations:
A basic theorem shows that all games may be put in a canonical form, which allows an easy test for equality. The canon
ical form depends on two types of simplification:
2,β¦}} and R1>=R2, then G={GL|{R2,β¦}}.
(R1L)},G(L2),β¦}|GR}.
G is said to be in canonical form if it has no dominated options or reversible moves. If G and H are both in canonical
form, they both have the same sets of left and right options and so are equal.
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