Iterative path orders

In Dershowitz’s original presentation recursive path orders (RPO) are defined by means of recursion. In Bergstra and Klop’s presentation RPOs are operationally defined by means of a set of term rewrite rules. To distinguish the two presentations, we will refer to the latter as iterative path orders (IPO). We show that IPOs as originally defined by Bergstra and Klop are strictly weaker than RPOs, but that a simple modification of the former yields equivalent orders. Different versions of IPO are obtained by varying the set of defining rewrite rules. We focus on the lexicographic variant (ILPO), which is shown to be equivalent to the lexicographic path orders as found in the literature. A proof-theoretic analysis of ILPO is given, yielding for instance decidability. It is then shown that the method can be extended to full RPO, allowing some function symbols to have lexicographic and other ones multiset status. Other variants, such as embedding and homeomorphic embedding can be obtained by suitably restricting the set of rules of IPO.