A Study of Solving Life-and-Death Problems in Go Using Relevance-Zone Based Solvers

Abstract

This paper analyzes the behavior of solving Life-and-Death (L&D) problems in the game of Go using current state-of-the-art computer Go solvers with two techniques: the Relevance-Zone Based Search (RZS) and the relevance-zone pattern table. We examined the solutions derived by relevance-zone based solvers on seven L&D problems from the renowned book Life and Death Dictionary written by Cho Chikun, a Go grandmaster, and found several interesting results. First, for each problem, the solvers identify a relevance-zone that highlights the critical areas for solving. Second, the solvers discover a series of patterns, including some that are rare. Finally, the solvers even find different answers compared to the given solutions for two problems. We also identified two issues with the solver: (a) it misjudges values of rare patterns, and (b) it tends to prioritize living directly rather than maximizing territory, which differs from the behavior of human Go players. We suggest possible approaches to address these issues in future work.

Solving Games with Relevance-Zone Based Solvers

Relevance-Zone Based Search (RZS) is a goal-oriented solving framework designed for exact analysis of life-and-death problems in Go. Instead of searching the entire board, the solver dynamically identifies a relevance zone (RZ), which contains all board intersections that are essential for proving the final outcome.

p_a

p_b′

p_c′

p_d′

p_*

p_b

p_c

p_d

An example illustrating relevance zones in Go. The shaded regions indicate automatically identified relevance zones.

For the example above, if Black plays at 1 as in \(p_{b'}\), \(p_{c'}\), and \(p_{d'}\), White can reply at 2 to achieve unconditional life (UCA), resulting in \(p_b\), \(p_c\), and \(p_d\), respectively. Since these positions are UCA for White stones, the corresponding relevance zones are denoted as \(z_b\), \(z_c\), and \(z_d\), shown as shaded intersections. The same relevance zones apply to the preceding positions \(p_{b'}\), \(p_{c'}\), and \(p_{d'}\). Stone configurations within these zones form relevance-zone (RZ) patterns. Because Black’s move at 1 lies outside \(z_b\), it is irrelevant to the outcome and can be treated as a null move. Consequently, White can always win by replying at E2 against any Black move outside \(z_b\). The relevance zone \(z_a\) is the union of \(z_b\), \(z_c\), and \(z_d\). During search, moves played outside the relevance zone are guaranteed not to affect the winning strategy and can therefore be safely ignored. This allows the solver to dramatically reduce the search space while preserving correctness. Unlike traditional approaches, relevance zones are discovered automatically during search, without any human-designed restriction of the board area.

Problem Analysis

We analyze the solving behavior of relevance-zone based Go solvers on seven classic 19×19 life-and-death problems. Instead of reporting benchmark performance on multiple games, this website highlights three representative problems selected from the seven analyzed in the paper, illustrating how the solver identifies critical regions (relevance zones), diverges from book solutions, and exhibits neural network misjudgment.

We highlight three representative problems from the solved set, each illustrating a distinct phenomenon. (Refer to the paper for the remaining problems.)

• Automatically expanded relevance zones that go beyond the area suggested by human intuition.
• Alternative correct answers that differ from the book’s solution.
• Neural misjudgment on rare patterns that must be corrected by search.

Automatically Expanded Relevance Zone

In this problem, the solver generates an unexpectedly large relevance zone compared to human intuition. Although the final life is local, intermediate tactical exchanges force the solver to include additional board areas to maintain correctness. This illustrates why automatic relevance-zone discovery can replace manual restriction of the search region.

Initial position

Black appears to need only a small area to form two eyes

However, the solution requires much more space to ensure life

Relevance Zone (RZ) of the problem

Alternative Correct Solution Different from the Book Answer

In this problem, the solver discovers an alternative correct solution B that differs from the book’s answer A. Although the book presents a unique solution, the solver identifies another valid line that also guarantees unconditional life, revealing additional tactical possibilities beyond those documented in the literature.

Initial position

The solution following A

The solution following B

The RZ of the problem

Neural Misjudgment of Rare Patterns

The neural value estimation can be highly inaccurate on rare patterns. Despite severe early misjudgment, the solver still proves the correct outcome through search, and pattern reuse accelerates the correction of Q-values across related positions.

Initial position

A move sequence involving a rare pattern called ishinoshita

The network outputs a value of 0.99 from White’s perspective, whereas the position is actually Black's winning move

The RZ of the problem

Downloads

All solution trees for the seven problems can be downloaded here.
They can be viewed with our specialized SGF viewer (Windows only), which supports board coloring for enhanced visualization: green indicates the Relevance-Zone, and red indicates the winning move for the player attempting to live.