# Nimber-Preserving Reductions and Homomorphic Sprague-Grundy Game Encodings

arXiv: https://arxiv.org/abs/2109.05622 · HTML: https://mlsanity.com/paper/2109.05622/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 41 references that are in the map.
- First posted on arXiv 12 Sep 2021.

## Builds on (most cited first, 3 of 3)

- [Quantum Combinatorial Games: Structures and Computational Complexity](https://mlsanity.com/paper/2011.03704/) (2020)
- [The role of quantum correlations in Cop and Robber game](https://mlsanity.com/paper/1702.07932/) (2017)
- [Winning the War by (Strategically) Losing Battles: Settling the Complexity of Grundy-Values in Undirected Geography](https://mlsanity.com/paper/2106.02114/) (2021)

## Abstract (arXiv)

The concept of nimbers--a.k.a. Grundy-values or nim-values--is fundamental to combinatorial game theory. Nimbers provide a complete characterization of strategic interactions among impartial games in their disjunctive sums as well as the winnability. In this paper, we initiate a study of nimber-preserving reductions among impartial games. These reductions enhance the winnability-preserving reductions in traditional computational characterizations of combinatorial games. We prove that Generalized Geography is complete for the natural class, calIP , of polynomially-short impartial rulesets under nimber-preserving reductions, a property we refer to as Sprague-Grundy-complete. In contrast, we also show that not every PSPACE-complete ruleset in calIP is Sprague-Grundy-complete for calIP . By considering every impartial game as an encoding of its nimber, our technical result establishes the following striking cryptography-inspired homomorphic theorem: Despite the PSPACE-completeness of nimber computation for calIP , there exists a polynomial-time algorithm to construct, for any pair of games G1, G2 of calIP , a prime game (i.e. a game that cannot be written as a sum) H of calIP , satisfying: nimber(H) = nimber(G1) oplus nimber(G2).