Massively Parallel Proof-Number Search for Impartial Games and Beyond

TL;DR

This paper tackles the challenge of scaling Proof-Number Search to massively parallel distributed systems for impartial games, using Sprouts as a benchmark. It introduces PNS-PDFPN, a two-level parallel framework that shares intermediate results, and extends PNS/DFPN with Grundy numbers to exploit impartial-game structure. The work demonstrates substantial speedups on up to 1024 cores (208.6× baseline, up to 332.97× with Grundy-number synchronization) and achieves state-of-the-art results in Sprouts, solving 42 new positions and producing proofs far beyond prior capability. The findings indicate a domain-agnostic approach to large-scale game solving with potential extensions to other combinatorial domains and RL-informed search strategies, while keeping Sprouts conjecture open.

Abstract

Proof-Number Search is a best-first search algorithm with many successful applications, especially in game solving. As large-scale computing clusters become increasingly accessible, parallelization is a natural way to accelerate computation. However, existing parallel versions of Proof-Number Search are known to scale poorly on many CPU cores. Using two parallelized levels and shared information among workers, we present the first massively parallel version of Proof-Number Search that scales efficiently even on a large number of CPUs. We apply our solver, enhanced with Grundy numbers for reducing game trees, to the Sprouts game, a case study motivated by the long-standing Sprouts Conjecture. Our solver achieves a significantly improved 332.9$\times$ speedup when run on 1024 cores, enabling it to outperform the state-of-the-art Sprouts solver GLOP by four orders of magnitude in runtime and to generate proofs 1,000$\times$ more complex. Despite exponential growth in game tree size, our solver verified the Sprouts Conjecture for 42 new positions, nearly doubling the number of known outcomes.

Figures (11)

• Figure 1: Estimated game tree complexity of Sprouts (with and without Grundy numbers) compared to other games.
• Figure 2: A scheme of a NAND tree with Grundy numbers. We use $C_1, \dots, C_m$ for the children of $P_k$ and , , for decomposable, atomic, and Grundy nodes, respectively, and for nodes that can be atomic or decomposable.
• Figure 3: Performance of PNS implementations in GLOP and SPOTS and of DFPN($C$) in SPOTS on the 29-spot position, where $C$ is the capacity of the transposition table.
• Figure 4: Comparison of PNS-PNS and PNS-DFPN, including the impact of retention of Grundy numbers (GN) in workers and their synchronization (GN Sync). All variants are run on the 29-spot position with 100 iterations, 100 updates, no grouping, and no second-level parallelization.
• Figure 5: Impact of GN synchronization (GN Sync), grouping (Gr), second-level parallelization (Th), and the child-ordering heuristic (Hr) on the final speedup of PNS-PDFPN relative to DFPN($5 \cdot 10^7$), which computed for $293\pm41.7$ minutes. Measured on the 47-spot position.

Theorems & Definitions (3)

• Theorem 1: The Sprague--Grundy Theorem
• Theorem 2
• proof : Proof of Theorem \ref{['thm-dfpn']}