# The Theory of Total Unary RPO Is Decidable

## Abstract

The Recursive Path Ordering (*rpo*) is a syntactic ordering on terms that has been widely used for proving termination of term-rewriting systems [7, 20]. How to combine term-rewriting with ordered resolution and paramodulation is now well-understood and it has been successfully applied in many theorem-proving systems [11, 16, 21]. In this setting an ordering such as rpo is used both to orient rewrite rules and to select maximal literals to perform inferences on. In order to further prune the search space the ordering requirements on conditional inferences are better handled when they are treated as constraints [12, 18]. Typically a non-orientable equation *s* = *t* will be split as two constrained rewrite rules: *s* → *t* | *s* > *t* and *t* → *s* | *t* > *s*. Such constrained rules are useless when the constraint is unsatisfiable. Therefore it is important for the efficiency of automated reasoning systems to investigate decision procedures for the theory of terms with ordering predicates.

Other types of constraints can be introduced too such as disunification constraints [1]. It is often the case that they can be expressed with ordering constraints (although this might be inefficient).

We prove that the first-order theory of the recursive path ordering is decidable in the case of unary signatures with total precedence. This solves a problem that was mentioned as open in [6]. The result has to be contrasted with the undecidability results of the lexicographic path ordering [6] for the case of symbols with arity ≥ 2 and total precedence and for the case of unary signatures with partial precedence. We recall that lexicographic path ordering (lpo) and the recursive path ordering and many other orderings such as [13, 10] coincide in the unary case.

Among the positive results it is known that the existential theory of total lpo is decidable [3, 17]. The same result holds for the case of total rpo [8, 15].

The proof technique we use for our decidability result might be interesting by itself. It relies on encoding of words as trees and then on building a tree automaton to recognize the rpo relation.

## Keywords

Recursive path ordering first-order theory ground reducibility tree automata ordered rewriting## Preview

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