In a previous work (“Abstract Data Type Systems”, TCS 173(2), 1997), the last two authors presented a combined language made of a (strongly normalizing) algebraic rewrite system and a typed λ-calculus enriched by pattern-matching definitions following a certain format, called the “General Schema”, which generalizes the usual recursor definitions for natural numbers and similar “basic inductive types”. This combined language was shown to be strongly normalizing. The purpose of this paper is to reformulate and extend the General Schema in order to make it easily extensible, to capture a more general class of inductive types, called “strictly positive”, and to ease the strong normalization proof of the resulting system. This result provides a computation model for the combination of an algebraic specification language based on abstract data types and of a strongly typed functional language with strictly positive inductive types.

This paper extends the termination proof techniques based on reduction orderings to a higher-order setting, by adapting the recursive path ordering definition to terms of a typed lambda-calculus generated by a signature of polymorphic higher-order function symbols. The obtained ordering is well-founded, compatible with fi-reductions and with polymorphic typing, monotonic with respect to the function symbols, and stable under substitution. It can therefore be used to prove the strong normalizationproperty of higher-order calculi in which constants can be defined by higher-order rewrite rules. For example, the polymorphic version of Gödel's recursor for the natural numbers is easily oriented. And indeed, our ordering is polymorphic, in the sense that a single comparison allows to prove the termination property of all monomorphic instances of a polymorphic rewrite rule. Several other non-trivial examples are given which examplify the expressive power of the ordering.

This paper will appear in the proceedings of the 10th international conference on rewriting techniques and applications (RTA'99). c flSpringer Verlag.

Confluence of orthogonal rewriting systems can be proved using the Finite Developments Theorem. We present, in a general setting, several adaptations of this proof method for obtaining confluence of `not quite' orthogonal systems. 1. Introduction Rewriting as studied here is based on the analogy: rewriting = substitution + rules. This analogy is useful since it enables a clearcut distinction between the `designer' defined substition process, i.e. management of resources, and the `user' defined rewrite rules, of rewriting systems. For example, application of the `user' defined term rewriting rule 2 \Theta x ! x + x to the term 2 \Theta 3 gives rise to the duplication of the term 3 in the result 3 + 3. How this duplication is actually performed (for example, using sharing) depends on the `designer's' implementation of substitution. This decomposition has been shown useful in [OR94, Oos94] in the case of first-order term rewriting systems (TRSs, [DJ90, Klo92]) and higher-order term r...

Nominal rewriting is based on the observation that if we add support for alphaequivalence to first-order syntax using the nominal-set approach, then systems with binding, including higher-order reduction schemes such as lambda-calculus betareduction, can be smoothly represented. Nominal rewriting maintains a strict distinction between variables of the objectlanguage (atoms) and of the meta-language (variables or unknowns). Atoms may be bound by a special abstraction operation, but variables cannot be bound, giving the framework a pronounced first-order character, since substitution of terms for variables is not capture-avoiding. We show how good properties of first-order rewriting survive the extension, by giving an efficient rewriting algorithm, a critical pair lemma, and a confluence theorem

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Consider Adam and Eve. Count generations starting from them. Supposing that there will always be people, then it's true that for any generation X, eventually there will be people belonging to the next generation X + 1. In this paper the same result is established for the class of higher order pattern rewriting systems. 1 Introduction Consider a set of structures and a set of transformations on them specifying how a structure may be transformed into another one. Suppose the transformations are of the following form: first a structure is decomposed into substructures, next some substructure is replaced by another one, and finally the substructures are composed into a structure again. (destroy) The parts of the initial structure eliminated in the course of the transformation (i.e. the parts of the replaced substructure as well as the parts eliminated in the initial decomposition) can be thought of as being destroyed . (create) The parts of the final structure introduced in the cou...

Abstract: In the last twenty years, several approaches to higher-order rewriting have been proposed, among which Klop’s Combinatory Rewrite Systems (CRSs), Nipkow’s Higher-order Rewrite Systems (HRSs) and Jouannaud and Okada’s higher-order algebraic specification languages, of which only the last one considers typed terms. The later approach has been extended by Jouannaud, Okada and the present author into Inductive Data Type Systems (IDTSs). In this paper, we extend IDTSs with the CRS higher-order pattern-matching mechanism, resulting in simply-typed CRSs. Then, we show how the termination criterion developed for IDTSs with first-order pattern-matching, called the General Schema, can be extended so as to prove the strong normalization of IDTSs with higher-order pattern-matching. Next, we compare the unified approach with HRSs. We first prove that the extended General Schema can also be applied to HRSs. Second, we show how Nipkow’s higher-order critical pair analysis technique for proving local confluence can be applied to IDTSs. 1

We present a new criterion for confluence of (possibly) non-terminating leftlinear term rewriting systems. The criterion is based on certain strong joinability properties of parallel critical pairs . We show how this criterion relates to other well-known results, consider some special cases and discuss some possible extensions. 1 Introduction and Overview Computation formalisms which are based on rewriting systems heavily rely on the fundamental properties of termination and confluence. For terminating and confluent systems normal forms exist and are unique, irrespective of the computation (rewriting) strategy. For non-terminating but confluent systems, normal forms need not exist, however, if a normal form exists, it is still unique. More generally, any (possibly infinite) diverging computations can be joined again. In some cases, non-termination is inherently unavoidable, in other cases it may be very difficult to verify this property. Hence the problem of proving confluence (with o...

Abstract. We demonstrate how access control models and policies can be represented by using term rewriting systems, and how rewriting may be used for evaluating access requests and for proving properties of an access control policy. We focus on two kinds of access control models: discretionary models, based on access control lists (ACLs), and rolebased access control (RBAC) models. For RBAC models, we show that we can specify several variants, including models with role hierarchies, and constraints and support for security administrator review querying. 1