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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...
Relative Normalization in Deterministic Residual Structures
 In: Proc. of the 19 th International Colloquium on Trees in Algebra and Programming, CAAP'96, Springer LNCS
, 1996
"... . This paper generalizes the Huet and L'evy theory of normalization by neededness to an abstract setting. We define Stable Deterministic Residual Structures (SDRS) and Deterministic Family Structures (DFS) by axiomatizing some properties of the residual relation and the family relation on redexes in ..."
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Cited by 17 (13 self)
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. This paper generalizes the Huet and L'evy theory of normalization by neededness to an abstract setting. We define Stable Deterministic Residual Structures (SDRS) and Deterministic Family Structures (DFS) by axiomatizing some properties of the residual relation and the family relation on redexes in an Abstract Rewriting System. We present two proofs of the Relative Normalization Theorem, one for SDRSs for regular stable sets, and another for DFSs for all stable sets of desirable `normal forms'. We further prove the Relative Optimality Theorem for DFSs. We extend this result to deterministic Computation Structures which are deterministic Event Structures with an extra relation expressing selfessentiality. 1 Introduction A normalizable term, in a rewriting system, may have an infinite reduction, so it is important to have a normalizing strategy which enables one to construct reductions to normal form. It is well known that the leftmostoutermost strategy is normalizing in the calc...
Relative Normalization in Orthogonal Expression Reduction Systems
 In: Proc. of the 4 th International workshop on Conditional (and Typed) Term Rewriting Systems, CTRS'94, Springer LNCS
, 1994
"... . We study reductions in orthogonal (leftlinear and nonambiguous) Expression Reduction Systems, a formalism for Term Rewriting Systems with bound variables and substitutions. To generalise the normalization theory of Huet and L'evy, we introduce the notion of neededness with respect to a set of r ..."
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Cited by 11 (10 self)
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. We study reductions in orthogonal (leftlinear and nonambiguous) Expression Reduction Systems, a formalism for Term Rewriting Systems with bound variables and substitutions. To generalise the normalization theory of Huet and L'evy, we introduce the notion of neededness with respect to a set of reductions \Pi or a set of terms S so that each existing notion of neededness can be given by specifying \Pi or S. We imposed natural conditions on S, called stability, that are sufficient and necessary for each term not in Snormal form (i.e., not in S) to have at least one Sneeded redex, and repeated contraction of Sneeded redexes in a term t to lead to an Snormal form of t whenever there is one. Our relative neededness notion is based on tracing (open) components, which are occurrences of contexts not containing any bound variable, rather than tracing redexes or subterms. 1 Introduction Since a normalizable term, in a rewriting system, may have an infinite reduction, it is important to...
Discrete Normalization and Standardization in Deterministic Residual Structures
 In ALP '96 [ALP96
, 1996
"... . We prove a version of the Standardization Theorem and the Discrete Normalization (DN) Theorem in stable Deterministic Residual Structures, which are Abstract Reduction Systems with axiomatized residual relation, and model orthogonal rewrite systems. The latter theorem gives a strategy for construc ..."
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Cited by 10 (3 self)
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. We prove a version of the Standardization Theorem and the Discrete Normalization (DN) Theorem in stable Deterministic Residual Structures, which are Abstract Reduction Systems with axiomatized residual relation, and model orthogonal rewrite systems. The latter theorem gives a strategy for construction of reductions L'evyequivalent (or permutationequivalent) to a given, finite or infinite, regular (or semilinear) reduction, based on the neededness concept of Huet and L'evy. This and other results of this paper add to the understanding of L'evyequivalence of reductions, and consequently, L'evy's reduction space. We demonstrate how elements of this space can be used to give denotational semantics to known functional languages in an abstract manner. 1 Introduction Long ago, Curry and Feys [CuFe58] proved that repeated contraction of leftmostoutermost redexes in any normalizable term eventually yields its normal form, even if the term possesses infinite reductions as well. The reaso...
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 ...
Relative Normalization in Stable Deterministic Residual Structures
 Z. Khasidashvili and J. Glauert
, 1996
"... This paper generalizes the Huet and L'evy theory of normalization by neededness to an abstract setting. We define Stable Deterministic Residual Structures (SDRS) and Deterministic Family Structures (DFS) by axiomatizing some properties of the residual relation and the family relation on redexes in a ..."
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Cited by 3 (3 self)
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This paper generalizes the Huet and L'evy theory of normalization by neededness to an abstract setting. We define Stable Deterministic Residual Structures (SDRS) and Deterministic Family Structures (DFS) by axiomatizing some properties of the residual relation and the family relation on redexes in an Abstract Reduction System. We present two proofs of the Relative Normalization Theorem, one for SDRSs for regular stable sets, and another for DFSs for all stable sets of desirable `normal forms'. We further prove the Relative Optimality Theorem for DFSs. We extend this result to deterministic Computation Structures which are deterministic Prime Event Structures with an extra relation expressing (in)essentiality of events. A version of this paper appears in Proc. of CAAP'96 [GlKh96]. c fl J. Glauert & Z. Khasidashvili, UEA Norwich, 1996 1 Supported by the Engineering and Physical Sciences Research Council of Great Britain under grant GR/H 41300 1 Introduction A normalizable term, i...
Minimal and Optimal Relative Normalization in Orthogonal Expression Reduction Systems
 J. Logic & Comput
, 1996
"... We study normalization relative to desirable sets S of `normal forms' by generalizing Huet&L'evy theory of `normalization by neededness'. We impose natural conditions on S, called stability, that are sufficient and necessary for each term not in Snormal form (i.e., not in S) to have at least one S ..."
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Cited by 3 (0 self)
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We study normalization relative to desirable sets S of `normal forms' by generalizing Huet&L'evy theory of `normalization by neededness'. We impose natural conditions on S, called stability, that are sufficient and necessary for each term not in Snormal form (i.e., not in S) to have at least one Sneeded redex, and repeated contraction of Sneeded redexes in a term t to lead to an Snormal form of t whenever there is one. Further, we prove existence of minimal normalizing reductions for regular stable sets of normal forms. For example, the sets of normal forms, headnormalforms, and weak headnormalforms, in the calculus, are all stable and regular. Finally, we generalize L'evy's Optimality theorem to the case of all stable sets of normal forms, and establish a relationship between relative minimal and optimal reductions, revealing a conflict between minimality and optimality of a reduction  for regular stable sets of normal forms, a term need not posses a reduction that is minim...
1 Contextsensitive Conditional Reduction Systems
"... We introduce Contextsensitive Conditional Expression Reduction Systems (CERS) by extending and generalizing the notion of conditional TRS to the higher order case. We justify our framework in two ways. First, we define orthogonality for CERSs and show that the usual results for orthogonal systems ( ..."
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Cited by 1 (0 self)
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We introduce Contextsensitive Conditional Expression Reduction Systems (CERS) by extending and generalizing the notion of conditional TRS to the higher order case. We justify our framework in two ways. First, we define orthogonality for CERSs and show that the usual results for orthogonal systems (finiteness of developments, confluence, permutation equivalence) carry over immediately. This can be used e.g. to infer confluence from the subject reduction property in several typed λcalculi possibly enriched with patternmatching definitions. Second, we express several proof and transition systems as CERSs. In particular, we give encodings of Hilbertstyle proof systems, Gentzenstyle sequentcalculi, rewrite systems with rule priorities, and the πcalculus into CERSs. This last encoding is an important example of real contextsensitive rewriting. ○c
The Geometry of Conflictfree Reduction Spaces
, 1998
"... We investigate mutual dependencies of subexpressions of a computable expression, in orthogonal rewrite systems, and identify conditions for their independent computation, concurrently. To this end, we introduce concepts familiar from ordinary Euclidean Geometry (such as basis, projection, distance, ..."
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We investigate mutual dependencies of subexpressions of a computable expression, in orthogonal rewrite systems, and identify conditions for their independent computation, concurrently. To this end, we introduce concepts familiar from ordinary Euclidean Geometry (such as basis, projection, distance, etc.) for reduction spaces. We show how a basis at an expression can be constructed so that any reduction starting from that expression can be decomposed as the sum of its projections on the axes of the basis. To make the concepts computationally more relevant, we relativize them w.r.t. stable sets of results (such as the set of normal forms, headnormal forms, and weakheadnormal forms, in the calculus), and show that an optimal concurrent computation of an expression w.r.t. S consists of optimal computations of its Sindependent subexpressions. All these results are obtained in the framework of Deterministic Family Structures, which are Abstract Reduction Systems with an axiomatized resi...