On the General Dead-Ending Universe of Partizan Games

Abstract

The universe $\mathcal{E}$ of dead-ending partizan games has emerged as an important structure in the study of mis\`ere play. Here we attempt a systematic investigation of the structure of $\mathcal{E}$ and its subuniverses. We begin by showing that the dead-ends exhibit a rich "absolute" structure, in the sense that they behave identically in any universe in which they appear. We will use this result to construct an uncountable family of dead-ending universes and show that they collectively admit an uncountable family of distinct comparison relations. We will then show that whenever the ends of a universe $\mathcal{U} \subset \mathcal{E}$ are computable, then there is a constructive test for comparison modulo $\mathcal{U}$. Finally, we propose a new type of generalized simplest form that works for arbitrary universes (including universes that are not dead-ending), and that is computable whenever comparison modulo $\mathcal{U}$ is computable. In particular, this gives a complete constructive theory for subuniverses of $\mathcal{E}$ with computable ends. This theory has been implemented in cgsuite as a proof of concept. As an application of these results, we will characterize the universe generated by mis\`ere Domineering, and we will compute the mis\`ere simplest forms of $2 \times n$ Domineering rectangles for small values of $n$.