Semi-Strongly solved: a New Definition Leading Computer to Perfect Gameplay

Abstract

Strong solving of perfect-information games certifies optimal play from every reachable position, but the required state-space coverage is often prohibitive. Weak solving is far cheaper, yet it certifies correctness only at the initial position and provides no formal guarantee for optimal responses after arbitrary deviations. We define semi-strong solving, an intermediate notion that certifies correctness on a certified region R: positions reachable from the initial position under the explicit assumption that at least one player follows an optimal policy while the opponent may play arbitrarily. A fixed tie-breaking rule among optimal moves makes the target deterministic. We propose reopening alpha-beta, a node-kind-aware Principal Variation Search/Negascout scheme that enforces full-window search only where semi-strong certification requires exact values and a canonical optimal action, while using null-window refutations and standard cut/all reasoning elsewhere. The framework exports a deployable solution artifact and, when desired, a proof certificate for third-party verification. Under standard idealizations, we bound node expansions by O(d b^(d/2)). On 6x6 Othello (score-valued utility), we compute a semi-strong solution artifact supporting exact value queries on R and canonical move selection. An attempted strong enumeration exhausts storage after exceeding 4x10^12 distinct rule-reachable positions. On 7x6 Connect Four (win/draw/loss utility), an oracle-value experiment shows that semi-strong certification is 9,074x smaller than a published strong baseline under matched counting conventions. Semi-strong solving provides an assumption-scoped, verifiable optimality guarantee that bridges weak and strong solving and enables explicit resource-guarantee trade-offs.