Solving Go under Positional Superko: Sound Proof Transfer and New Board Values

Sami Kharma, Florian Schintke

Positional superko (PSK) forbids recreating any earlier whole-board position. It is part of the Chinese rules as written and of the rules used in computer Go. Yet almost all solved Go boards were computed under a simpler basic-ko rule set. PSK makes solving hard: legality depends on the whole game history, and this breaks the three tools that make solving tractable, namely transposition tables (the graph-history interaction problem), static life-and-death recognition, and safe move pruning. We present Sanko, a parallel depth-first proof-number solver that restores all three soundly. Its core is a cheap proof-transfer theorem: a stored proof stays valid on a new path unless a board that the proof's winner must create has already occurred. Sanko proves 5x5 Go in 16 minutes with six threads, where the previous state of the art needed 5.25 hours with six processes, and in about a minute and a half on one 192-thread machine. It also establishes the first PSK values of twenty-three further boards, eleven of which were unsolved under any rules. Superko changes game-theoretic values: 4x7, a full-board win under basic ko, is worth only 12 points under PSK, and we pinpoint the move at…