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. Key words: Recursive path ordering, first-order theory, ground reducibility

In principle termination analysis is easy: find a well-founded partial order and prove that calls decrease with respect to this order. In practice this often requires an oracle (or a theorem prover) for determining the well-founded order and this oracle may not be easily implementable. Our approach circumvents some of these problems by exploiting the inductive definition of algebraic data types and using pattern matching as in functional languages. We develop a termination analysis for a higher-order functional language; the analysis incorporates and extends polymorphic type inference and axiomatizes a class of well-founded partial orders for multipleargument functions (as in Standard ML and Miranda). Semantics is given by means of operational (natural-style) semantics and soundness is proved; this involves making extensions to the semantic universe and we relate this to the techniques of denotational semantics. For dealing with the partiality aspects of the soundness proof it suffice...

We investigate, for the case of unary function symbols, polynomial orderings on term algebras, that is reduction orderings determined by polynomial interpretations of the function symbols. Any total reduction ordering over unary function symbols can be characterised in terms of numerical invariants determined by the ordering alone: we show that for polynomial orderings these invariants, and in some cases the ordering itself, are essentially determined by the degrees and leading coefficients of the polynomial interpretations. Hence any polynomial ordering has a much simpler description, and thus the apparent complexity and variety of these orderings is less than it might seem at first sight.

. For a string rewriting system, it is known that termination by a simplification ordering implies multiple recursive complexity. This theoretical upper bound is, however, far from having been reached by known examples of rewrite systems. All known methods used to establish termination by simplification yield a primitive recursive bound. Furthermore, the study of the order types of simplification orderings suggests that the recursive path ordering is, in a broad sense, a maximal simplification ordering. This would imply that simplifying string rewrite systems cannot go beyond primitive recursion. Contradicting this intuition, we construct here a simplifying string rewriting system whose complexity is not primitive recursive. This leads to a new lower bound for the complexity of simplifying string rewriting systems. Introduction Rewriting systems serve as a model of computation in many fields of theoretical computer science, for instance automated theorem proving, algebraic specificat...

This paper is concerned with developing similar results for terms. The first main section considers how we may assign numerical invariants to orders on terms, and hence establish Pn as a classifying space for term orders over n non-constant function symbols. The second concerns a general framework for ordering terms by counting embedded patterns: we construct a large class of new term orders and show how our method subsumes earlier constructions. A final section looks at the recursive path order in the light of our results. We now explain our results in more detail

This paper is concerned with developing similar results for terms. The first main section considers how we may assign numerical invariants to orders on terms, and hence establish 72n as a classifying space for term orders over n non-constant function symbols. The second concerns a general framework for ordering terms by counting embedded patterns: we construct a large class of new term orders and show how our method subsumes earlier constructions. A final section looks at the recursire path order in the light of our results. We now explain our results in more detail