Equational presentations with ordered sorts encompass partially defined functions and subtyping information in an algebraic framework. In this work we address the problem of computing in order-sorted algebras, with very few restrictions on the allowed presentations. We adopt an algebraic framework where equational, membership and existence formulas can be expressed. A complete deduction calculus is provided to incorporate the interaction between all these formulas. The notion of decorated terms is proposed to memorize local sort information, dynamically changed by a rewriting process. A completion procedure for equational presentations with ordered sorts computes a set of rewrite rules with which not only equational theorems of the form (t = t 0 ), but also typing theorems of the for...

We prove constructively that the class of ground-confluent and decreasing conditional term rewriting systems (CTRSs) (without extra variables) coincides with the class of orthogonal and terminating, unconditional term rewriting systems (TRSs). TRSs being included in CTRSs, this result follows from a transformation from any ground-confluent and decreasing CTRS specifying a computable function f into a TRS with the mentioned properties for f . The generated TRS is ordersorted, but we outline a similar transformation yielding an unsorted TRS.

. Deterministic conditional rewrite systems are interesting because they permit extra variables on the right-hand sides of the rules. If such a system is quasi-reductive, then it is terminating and has a computable rewrite relation. It will be shown that every deterministic CTRS R can be transformed into an unconditional TRS U(R) such that termination of U(R) implies quasi-reductivity of R. The main theorem states that quasi-reductivity of R implies innermost termination of U(R). These results have interesting applications in two different areas: modularity in term rewriting and termination proofs of well-moded logic programs. 1 Introduction Conditional term rewriting systems (CTRSs) are the basis of functional logic programming; see [Han94] for an overview of this field. In CTRSs variables on the right-hand side of a rewrite rule which do not occur on the left-hand side are often forbidden. This is because it is in general not clear how to instantiate them. On the other hand, a rest...

This document explains the design and use of a termination checker tool and of a Knuth-Bendix completion tool. The termination checker tool checks whether an equational specication terminates, and the Knuth-Bendix completion tool tries to complete an equational speci- cation. These tools can be used to prove the termination or to complete order-sorted equational specications in Maude [7, 6, 4]. The tools have been written entirely in Maude and are in fact executable specications in rewriting logic [17] of the formal inference system that they implement. The fact that rewriting logic is reective [8, 3], and that Maude eciently supports reective rewriting logic computations [5, 4] is systematically exploited in the design of the tools. Contents 1

The class of Deterministic Conditional Term Rewriting Systems (DCTRSs) is of utmost importance for the tight relationships exhibited with functional programming, logic programming and inductive reasoning. However, its analysis is extremely difficult, and to date there are only very few works on the subject, each analyzing a particular aspect of DCTRSs. In this paper, we perform a thorough analysis of DCTRSs, ranging from the study of termination criteria, to new verification methods for the major properties of DCTRSs like termination and confluence, to the identification of subclasses of DTCRSs that exhibit a particularly nice behaviour. Moreover, we also address the study of modularity of DCTRSs, providing a number of new powerful results. This is particularly important, since to the best of our knowledge there is so far not a single result on the modularity of DCTRSs, and of 3-CTRSs in general. Finally, most of the analysis of the paper is performed relying on the recent tool of unra...

We consider the problem of proving termination of ordersorted rewrite systems. The dominating method for proving termination of order-sorted systems has been to simply ignore sort information, and use the techniques developed for unsorted rewriting. The problem with this approach is that many order-sorted rewrite systems terminate because of the structure of the set of sorts. In these cases the corresponding unsorted system would not terminate. In this paper we approach the problem of order-sorted termination by mapping the order-sorted rewrite system into an unsorted one suchthat termination of the latter implies termination of the former. By encoding sort information into the unsorted mapping, we are able to use general purpose termination orderings to prove termination of order-sorted rewrite systems whose termination depend on the sort hierarchy. We present a sequence of gradually stronger methods, and show that a previously published method is contained in ours as a special case. 1

In this paper we consider the lazy conditional narrowing calculus LCNC of [4]. LCNC is the extension of lnc to conditional term rewrite systems (CTRSs for short). The extension is motivated by the observation that CTRSs are much more expressive than unconditional TRSs for describing interesting problems in natural and concise way. However, the additional expressive power raises two problems: (1) confluence is insufficient to guarantee the completeness with respect to normalized solutions, and (2) the search space increases dramatically because the conditions of the applied rewrite rule are added to the current goal. In [4] several completeness results for lcnc are presented. The only result which does not assume some kind of termination assumption does not permit extra variables in the conditions and right-hand sides of the rewrite rules. In this paper we show the completeness of LCNC with leftmost selection for the class of (confluent) deterministic oriented CTRSs. Determinism was introduced by Ganzinger [1] and has proved to be very useful for the study of the (unique) termination behavior of well-moded Horn clause programs (cf. [5]).

der Technischen Fakult"at der Universit"at des Saarlandes Saarbr"ucken

. OBJ--3 [GWM + 93] is a functional programming language with first-order function types. OBJ--3 has two special features: overloading of function symbols and the possibility to order the sorts. This ordering is induced by set inclusion on the carrier sets. We call the feature to be able to order the sorts inclusion set subtyping. The algebraic semantics of OBJ--3 is based on the theory of order-sorted algebras [GM89]. Furthermore, OBJ--3 allows parameterized programming [Gog90]. However, the concepts of higher-order functions and parametric polymorphism are only emulated by parameters of OBJ--3 modules. In this paper we show how to extend OBJ--3 by parametric polymorphism in an elegant way. We call this extended language OBJ--P. In the second part of the paper we describe the operational semantics of OBJ--P. The operational semantics is a translation of OBJ--P programs into programs without overloading and subtypes. Here, we improve the approaches of Goguen, Jouannaud, and Mesegu...

. Deterministic conditional rewrite systems permit extra variables on the right-hand sides of the rules. If such a system is quasireductive or quasi-simplifying, then it is terminating and has a computable rewrite relation. This paper provides new criteria for showing quasi-reductivity and quasi-simplifyingness. In this context, another criterion from [ALS94] will be rectified and a claim in [Mar96] will be refuted. Moreover, we will investigate under which conditions the properties exhibit a modular behavior. 1 Introduction Conditional term rewriting systems (CTRSs) are the basis for the integration of the functional and logic programming paradigms; see [Han94] for an overview of this field. In these systems variables on the right-hand side of a rewrite rule which do not occur on the left-hand side are problematic because it is in general not clear how to instantiate them. On the other hand, a restricted use of these extra variables enables a more natural and efficient way of writing...