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Perpetuality and Uniform Normalization in Orthogonal Rewrite Systems
 INFORMATION AND COMPUTATION
"... We present two characterizations of perpetual redexes, which are redexes whose contractions retain the possibility of infinite reductions. These characterizations generalize and strengthen existing criteria for the perpetuality of redexes in orthogonal Term Rewriting Systems and the calculus due ..."
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Cited by 7 (2 self)
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We present two characterizations of perpetual redexes, which are redexes whose contractions retain the possibility of infinite reductions. These characterizations generalize and strengthen existing criteria for the perpetuality of redexes in orthogonal Term Rewriting Systems and the calculus due to Bergstra and Klop, and others. To unify our results with those in the literature, we introduce Contextsensitive Conditional Expression Reduction Systems (CCERSs) and prove confluence for orthogonal CCERSs. We then define a perpetual onestep reduction strategy which enables one to construct minimal (w.r.t. Levy's permutation ordering on reductions) infinite reductions in orthogonal CCERSs. We then prove (1) perpetuality (in a specific context) of a redex whose contraction does not erase potentially infinite arguments, which are possibly finite (i.e., strongly normalizable) arguments that may become infinite after a number of outside steps, and (2) perpetuality (in every con...
Minimal Relative Normalization in Orthogonal Expression Reduction Systems
 In Proc. of the 16 th International Conference on Foundations of Software Technology and Theoretical Computer Science, FST&TCS'96, Springer LNCS
, 1996
"... . In previous papers, the authors studied normalization relative to desirable sets S of `partial results', where it is shown that such sets must be stable. For example, the sets of normal forms, headnormalforms, and weak headnormalforms in the calculus, are all stable. They showed that, for an ..."
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Cited by 5 (2 self)
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. In previous papers, the authors studied normalization relative to desirable sets S of `partial results', where it is shown that such sets must be stable. For example, the sets of normal forms, headnormalforms, and weak headnormalforms in the calculus, are all stable. They showed that, for any stable S, Sneeded reductions are Snormalizing. This paper continues the investigation into the theory of relative normalization. In particular, we prove existence of minimal normalizing reductions for regular stable sets of results. All the above mentioned sets are regular. We give a sufficient and necessary criterion for a normalizing reduction (w.r.t. a regular stable S) to be minimal. Finally, we establish a relationship between relative minimal and optimal reductions, revealing a conflict between minimality and optimality: for regular stable sets of results, a term need not possess a reduction that is minimal and optimal at the same time. 1 Introduction The Normalization Theorem in ...
Standardization and Evaluation in Combinatory Reduction Systems
, 2000
"... A rewrite system has standardization i for any rewrite sequence there is an equivalent one which contracts the redexes in a standard order. Standardization is extremely useful for finding normalizing strategies and proving that a rewrite system for a programming language is sound with respect to the ..."
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Cited by 4 (1 self)
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A rewrite system has standardization i for any rewrite sequence there is an equivalent one which contracts the redexes in a standard order. Standardization is extremely useful for finding normalizing strategies and proving that a rewrite system for a programming language is sound with respect to the language's operational semantics. Although for some rewrite systems the standardorder can be simple, e.g., lefttoright or outermostfirst, many systems need a more delicate order. There are abstract notions of standard order which always apply, but proofs (often quite dicult) are required that the rewrite system satis es a number of axioms and not much guidance is provided for finding a concrete order that satisfies the abstract definition. This paper gives a framework based on combinatory reduction systems (CRS's) which is general enough to handle many programming languages. If the CRS is orthogonal and fully extended and a good redex ordering can be found, then a standard order is obtain...
Perpetuality and Uniform Normalization
 In Proc. of the 6 th International Conference on Algebraic and Logic Programming, ALP'97
, 1997
"... . We define a perpetual onestep reduction strategy which enables one to construct minimal (w.r.t. L'evy's ordering \Theta on reductions) infinite reductions in Conditional Orthogonal Expression Reduction Systems. We use this strategy to derive two characterizations of perpetual redexes, i.e., redex ..."
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Cited by 4 (2 self)
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. We define a perpetual onestep reduction strategy which enables one to construct minimal (w.r.t. L'evy's ordering \Theta on reductions) infinite reductions in Conditional Orthogonal Expression Reduction Systems. We use this strategy to derive two characterizations of perpetual redexes, i.e., redexes whose contractions retain the existence of infinite reductions. These characterizations generalize existing related criteria for perpetuality of redexes. We give a number of applications of our results, demonstrating their usefulness. In particular, we prove equivalence of weak and strong normalization (the uniform normalization property) for various restricted calculi, which cannot be derived from previously known perpetuality criteria. 1 Introduction The objective of this paper is to study sufficient conditions for uniform normalization, UN, of a term in an orthogonal (first or higherorder) rewrite system, and for the UN property of the rewrite system itself. Here a term is UN if ei...
Unique Normal Form Property of HigherOrder Rewriting Systems
, 1996
"... . Within the framework of HigherOrder Rewriting Systems proposed by van Oostrom, a sufficient condition for the unique normal form property is presented. This requires neither leftlinearity nor termination of the system. 1 Introduction Several frameworks of rewriting systems for higherorder expr ..."
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Cited by 3 (1 self)
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. Within the framework of HigherOrder Rewriting Systems proposed by van Oostrom, a sufficient condition for the unique normal form property is presented. This requires neither leftlinearity nor termination of the system. 1 Introduction Several frameworks of rewriting systems for higherorder expressions have been proposed [Klo80, Nip91, MN94, LS93, KvO95]. Van Oostrom and van Raamdonk proposed a framework of HigherOrder Rewriting Systems (HORSs) [vO94, vOvR94, vR96], capable of unifying the existing theory of rewriting, e.g., Combinatory Reduction Systems (CRSs) [Klo80], (another variation of) Higherorder Rewriting Systems (HRSs) by Nipkow [Nip91], and Term Rewriting Systems (TRSs). They also presented a sufficient condition for the ChurchRosser property of HORSs by introducing a notion corresponding to orthogonality (i.e., nonoverlap and leftlinearity) of TRSs. The framework of HORSs is characterised by the clear separation of replacement with rewrite rules and matching/subst...
Two Applications of Standardization and Evaluation in Combinatory Reduction Systems
, 2000
"... We present two worked applications of a general framework that can be used to support reasoning about the operational equality relation defined by a programming language semantics. The framework, based on Combinatory Reduction Systems, facilitates the proof of standardization theorems for programmin ..."
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Cited by 1 (1 self)
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We present two worked applications of a general framework that can be used to support reasoning about the operational equality relation defined by a programming language semantics. The framework, based on Combinatory Reduction Systems, facilitates the proof of standardization theorems for programming calculi. The importance of standardization theorems to programming language semantics was shown by Plotkin [Plo75]: standardization together with confluence guarantee that two terms equated in the calculus are semantically equal. We apply the framework to the λ_νcalculus and to an untyped version of the λ^CILcalculus. The latter is a basis for an intermediate language being used in a compiler.
A de Bruijn notation for higherorder rewriting (Extended Abstract)
 In Proceedings of the 11th International Conference on Rewriting Techniques and Applications (RTA'00
, 2000
"... ) Eduardo Bonelli 1;2 , Delia Kesner 2 , Alejandro R'ios 1 1 Departamento de Computaci'on  Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Pabell'on I, Ciudad Universitaria (1428), Buenos Aires, Argentina. febonelli,riosg@dc.uba.ar 2 LRI (UMR 8623)  Bat 490, Unive ..."
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) Eduardo Bonelli 1;2 , Delia Kesner 2 , Alejandro R'ios 1 1 Departamento de Computaci'on  Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Pabell'on I, Ciudad Universitaria (1428), Buenos Aires, Argentina. febonelli,riosg@dc.uba.ar 2 LRI (UMR 8623)  Bat 490, Universit'e de ParisSud, 91405 Orsay Cedex, France. kesner@lri.fr Abstract. We propose a formalism for higherorder rewriting in de Bruijn notation. This notation not only is used for terms (as usually done in the literature) but also for metaterms, which are the syntactical objects used to express general higherorder rewrite systems. We give formal translations from higherorder rewriting with names to higherorder rewriting with de Bruijn indices, and viceversa. These translations can be viewed as an interface in programming languages based on higherorder rewrite systems, and they are also used to show some properties, namely, that both formalisms are operationally equivalent, and th...