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47
Inductive Data Type Systems
 THEORETICAL COMPUTER SCIENCE
, 1997
"... 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 λcalculus enriched by patternmatching definitions following a certain format, called the “General Schema”, w ..."
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Cited by 44 (10 self)
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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 λcalculus enriched by patternmatching 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.
The HigherOrder Recursive Path Ordering
 FOURTEENTH ANNUAL IEEE SYMPOSIUM ON LOGIC IN COMPUTER SCIENCE
, 1999
"... This paper extends the termination proof techniques based on reduction orderings to a higherorder setting, by adapting the recursive path ordering definition to terms of a typed lambdacalculus generated by a signature of polymorphic higherorder function symbols. The obtained ordering is wellfoun ..."
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Cited by 44 (10 self)
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This paper extends the termination proof techniques based on reduction orderings to a higherorder setting, by adapting the recursive path ordering definition to terms of a typed lambdacalculus generated by a signature of polymorphic higherorder function symbols. The obtained ordering is wellfounded, compatible with fireductions 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 higherorder calculi in which constants can be defined by higherorder 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 nontrivial examples are given which examplify the expressive power of the ordering.
HigherOrder Rewriting
 12th Int. Conf. on Rewriting Techniques and Applications, LNCS 2051
, 1999
"... This paper will appear in the proceedings of the 10th international conference on rewriting techniques and applications (RTA'99). c flSpringer Verlag. ..."
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Cited by 20 (1 self)
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This paper will appear in the proceedings of the 10th international conference on rewriting techniques and applications (RTA'99). c flSpringer Verlag.
Nominal rewriting
 Information and Computation
"... Nominal rewriting is based on the observation that if we add support for alphaequivalence to firstorder syntax using the nominalset approach, then systems with binding, including higherorder reduction schemes such as lambdacalculus betareduction, can be smoothly represented. Nominal rewriting ma ..."
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Cited by 19 (7 self)
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Nominal rewriting is based on the observation that if we add support for alphaequivalence to firstorder syntax using the nominalset approach, then systems with binding, including higherorder reduction schemes such as lambdacalculus betareduction, can be smoothly represented. Nominal rewriting maintains a strict distinction between variables of the objectlanguage (atoms) and of the metalanguage (variables or unknowns). Atoms may be bound by a special abstraction operation, but variables cannot be bound, giving the framework a pronounced firstorder character, since substitution of terms for variables is not captureavoiding. We show how good properties of firstorder rewriting survive the extension, by giving an efficient rewriting algorithm, a critical pair lemma, and a confluence theorem
Developing Developments
, 1994
"... 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 ..."
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Cited by 18 (2 self)
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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 firstorder term rewriting systems (TRSs, [DJ90, Klo92]) and higherorder term r...
Termination and confluence of higherorder rewrite systems
 In Proc. RTA ’00, volume 1833 of LNCS
, 2000
"... Abstract: In the last twenty years, several approaches to higherorder rewriting have been proposed, among which Klop’s Combinatory Rewrite Systems (CRSs), Nipkow’s Higherorder Rewrite Systems (HRSs) and Jouannaud and Okada’s higherorder algebraic specification languages, of which only the last on ..."
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Cited by 14 (8 self)
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Abstract: In the last twenty years, several approaches to higherorder rewriting have been proposed, among which Klop’s Combinatory Rewrite Systems (CRSs), Nipkow’s Higherorder Rewrite Systems (HRSs) and Jouannaud and Okada’s higherorder 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 higherorder patternmatching mechanism, resulting in simplytyped CRSs. Then, we show how the termination criterion developed for IDTSs with firstorder patternmatching, called the General Schema, can be extended so as to prove the strong normalization of IDTSs with higherorder patternmatching. 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 higherorder critical pair analysis technique for proving local confluence can be applied to IDTSs. 1
Finite Family Developments
"... Associate to a rewrite system R having rules l → r, its labelled version R ω having rules l ◦ m+1 → r • , for any natural number m m ∈ ω. These rules roughly express that a lefthand side l carrying labels all larger than m can be replaced by its righthand side r carrying labels all smaller than o ..."
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Cited by 13 (6 self)
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Associate to a rewrite system R having rules l → r, its labelled version R ω having rules l ◦ m+1 → r • , for any natural number m m ∈ ω. These rules roughly express that a lefthand side l carrying labels all larger than m can be replaced by its righthand side r carrying labels all smaller than or equal to m. A rewrite system R enjoys finite family developments (FFD) if R ω is terminating. We show that the class of higher order pattern rewrite systems enjoys FFD, extending earlier results for the lambda calculus and first order term rewrite systems.
Term rewriting for access control
 In Proc. DBSec’2006, volume 4127 of LNCS
, 2006
"... 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 model ..."
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Cited by 12 (4 self)
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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
Confluence without Termination via Parallel Critical Pairs
 In Proceedings of the 21st International Colloquium on Trees in Algebra and Programming (CAAP'96
, 1996
"... We present a new criterion for confluence of (possibly) nonterminating 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 wellknown results, consider some special cases and disc ..."
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Cited by 10 (3 self)
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We present a new criterion for confluence of (possibly) nonterminating 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 wellknown 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 nonterminating 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, nontermination is inherently unavoidable, in other cases it may be very difficult to verify this property. Hence the problem of proving confluence (with o...