Star Games and Hydras

The recursive path ordering is an established and crucial tool in term rewriting to prove termination. We revisit its presentation by means of some simple rules on trees equipped with a 'star' as control symbol, signifying a command to make that tree (or term) smaller in the order being defined. This leads to star games that are very convenient for proving termination of many rewriting tasks. For instance, using already the simplest star game on finite unlabeled trees, we obtain a very direct proof of termination of the famous Hydra battle. We also include an alternative road to setting up the star games, using a proof method of Buchholz, and a quantitative version of the star as control symbol. We conclude with a number of questions and future research directions.