# Taking-and-merging games as rewrite games

## TL;DR

This paper studies a class of rewrite games called taking-and-merging games, analyzing their structure, language properties of losing positions, and undecidability results, while connecting to open problems in combinatorial game theory.

## Contribution

It introduces taking-and-merging games, provides conditions for their losing positions to form regular or context-free languages, and proves undecidability results for general rewrite games.

## Key findings

- Losing positions can form regular or context-free languages under certain conditions.
- Decidability of winning positions is lost in more general rewrite games.
- Open questions relate to the periodicity of Grundy functions in octal games.

## Abstract

This work is a contribution to the study of rewrite games. Positions are finite words, and the possible moves are defined by a finite number of local rewriting rules. We introduce and investigate taking-and-merging games, that is, where each rule is of the form a^k->epsilon. We give sufficient conditions for a game to be such that the losing positions (resp. the positions with a given Grundy value) form a regular language or a context-free language. We formulate several related open questions in parallel with the famous conjecture of Guy about the periodicity of the Grundy function of octal games. Finally we show that more general rewrite games quickly lead to undecidable problems. Namely, it is undecidable whether there exists a winning position in a given regular language, even if we restrict to games where each move strictly reduces the length of the current position. We formulate several related open questions in parallel with the famous conjecture of Guy about the periodicity of the Grundy function of octal games.