Path ordering (term rewriting)
Path ordering is a family of term ordering techniques used in term rewriting systems to prove termination. These orderings are based on precedence, a partial or total order defined on the function symbols appearing in the terms being rewritten. The core idea is to define an ordering on terms such that if a rewrite rule l → r is oriented so that l > r according to the path ordering, then repeated application of the rewrite rule will necessarily terminate.
Path orderings work by recursively comparing terms based on the structure of the terms and the precedence of the function symbols at the root. Several variations of path orderings exist, each with its own specific comparison rules:
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Recursive Path Ordering (RPO): RPO compares two terms f(s1,...,sn) and g(t1,...,tm) based on the precedence of f and g. It considers cases where f > g, f = g, or f < g according to the precedence relation, and recursively compares subterms. Different variants of RPO exist, such as RPO with status (RPOS), which takes into account the status of function symbols (e.g., left-to-right or multiset), affecting how subterms are compared.
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Lexicographic Path Ordering (LPO): LPO is a specific variant of RPO. When the top function symbols of the terms being compared are equal, i.e., f = g, LPO compares the lists of subterms lexicographically based on a specified order.
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Semantic Path Ordering (SPO): Semantic Path Ordering uses a semantic interpretation of the terms to compare them. This allows for more flexible comparisons than purely syntactic approaches like RPO and LPO. SPO assigns a value from a well-founded domain to each term based on the semantic interpretation, and compares terms based on these values.
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Knuth-Bendix Ordering (KBO): While technically not a path ordering, the Knuth-Bendix Ordering is another important technique for proving termination of term rewriting systems. It combines symbol weights with lexicographic comparison.
The choice of path ordering and precedence relation depends on the specific term rewriting system being analyzed. Finding a suitable precedence that orients all rewrite rules is often a crucial step in proving termination. Path orderings are a powerful tool for automated termination analysis and are widely used in automated theorem proving and related areas. The strength of these orderings lies in their ability to effectively handle complex term structures and their relatively efficient computability.