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The Longest Perpetual Reductions in Orthogonal Expression Reduction Systems
 In: Proc. of the 3 rd International Conference on Logical Foundations of Computer Science, LFCS'94, A. Nerode and Yu.V. Matiyasevich, eds., Springer LNCS
, 1994
"... We consider reductions in Orthogonal Expression Reduction Systems (OERS), that is, Orthogonal Term Rewriting Systems with bound variables and substitutions, as in the calculus. We design a strategy that for any given term t constructs a longest reduction starting from t if t is strongly normaliza ..."
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Cited by 18 (8 self)
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We consider reductions in Orthogonal Expression Reduction Systems (OERS), that is, Orthogonal Term Rewriting Systems with bound variables and substitutions, as in the calculus. We design a strategy that for any given term t constructs a longest reduction starting from t if t is strongly normalizable, and constructs an infinite reduction otherwise. The Conservation Theorem for OERSs follows easily from the properties of the strategy. We develop a method for computing the length of a longest reduction starting from a strongly normalizable term. We study properties of pure substitutions and several kinds of similarity of redexes. We apply these results to construct an algorithm for computing lengths of longest reductions in strongly persistent OERSs that does not require actual transformation of the input term. As a corollary, we have an algorithm for computing lengths of longest developments in OERSs. 1 Introduction A strategy is perpetual if, given a term t, it constructs an infinit...
Perpetuality and Strong Normalization in Orthogonal Term Rewriting Systems
 In: Proc. of 11 th Symposium on Theoretical Aspects of Computer Science, STACS'94
, 1994
"... . We design a strategy that for any given term t in an Orthogonal Term Rewriting System (OTRS) constructs a longest reduction starting from t if t is strongly normalizable, and constructs an infinite reduction otherwise. For some classes of OTRSs the strategy is easily computable. We develop a metho ..."
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Cited by 10 (5 self)
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. We design a strategy that for any given term t in an Orthogonal Term Rewriting System (OTRS) constructs a longest reduction starting from t if t is strongly normalizable, and constructs an infinite reduction otherwise. For some classes of OTRSs the strategy is easily computable. We develop a method for finding the least upper bound of lengths of reductions starting from a strongly normalizable term. We give also some applications of our results. 1 Introduction It is shown in O'Donnell [12] that the innermost strategy is perpetual for orthogonal term rewriting systems (OTRSs). That is, contraction of innermost redexes gives an infinite reduction of a given term whenever such a reduction exists. In fact, a strategy that only contracts redexes that do not erase any other redex is perpetual. Moreover, one can even reduce redexes whose erased arguments are strongly normalizable (Klop [10]). For the lambdacalculus, a more subtle perpetual strategy was invented in Barendregt et al. [1]. H...
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...
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...
Perpetual Reductions and Strong Normalization in Orthogonal Term Rewriting Systems
, 1993
"... We design a strategy that for any given term t in an Orthogonal Term Rewriting System (OTRS) constructs a longest reduction starting from t if t is strongly normalizable, and constructs an infinite reduction otherwise. We define some classes of OTRSs for which the strategy is easily computable. We ..."
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Cited by 3 (2 self)
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We design a strategy that for any given term t in an Orthogonal Term Rewriting System (OTRS) constructs a longest reduction starting from t if t is strongly normalizable, and constructs an infinite reduction otherwise. We define some classes of OTRSs for which the strategy is easily computable. We develop a method for finding the least upper bound of lengths of reductions starting from a strongly normalizable term. We give also some applications of our results. AMS Subject Classification (1991): 68Q42. CR Subject Classification (1991): F4.2, F4.1. Keywords & Phrases: orthogonal term rewriting systems, perpetual reductions, strong normalization, reduction strategies. Note: Part of this work was completed during an enjoyable visit of the author at CWI in the summer of 1993. 1. Introduction It is shown in O'Donnell [9] that the innermost strategy is perpetual for orthogonal term rewriting systems (OTRSs). That is, contraction of innermost redexes gives an infinite reduction of a...