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 (or corresponding terms) 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, direct in the sense that there is not the usual mention of ordinals. We also include an alternative road to setting up the star games, using a proof method of Buchholz, adapted by van Oostrom, resulting in a quantitative version of the star as control symbol. We conclude with a number of questions and future research directions

Proof Orders for Decreasing Diagrams

We present and compare some well-founded proof orders for decreasing diagrams. These proof orders order a conversion above another conversion if the latter is obtained by filling any peak in the former by a (locally) decreasing diagram. Therefore each such proof order entails the decreasing diagrams technique for proving confluence. The proof orders differ with respect to monotonicity and complexity. Our results are developed in the setting of involutive monoids. We extend these results to obtain a decreasing diagrams technique for confluence modulo

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 (or corresponding terms) 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, direct in the sense that there is not the usual mention of ordinals. We also include an alternative road to setting up the star games, using a proof method of Buchholz, adapted by van Oostrom, resulting in a quantitative version of the star as control symbol. We conclude with a number of questions and future research directions

Iterative Lexicographic Path Orders

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Iterative Lexicographic Path Orders

Abstract. We relate Kamin and Lévy’s original presentation of lexicographic path orders (LPO), using an inductive definition, to a presentation, which we will refer to as iterative lexicographic path orders (ILPO), based on Bergstra and Klop’s definition of recursive path orders by way of an auxiliary term rewriting sytem. Dedicated to Joseph Goguen, in celebration of his 65th birthday.