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On Higher Order Recursive Program Schemes
 In: Proc. of the 19 th International Colloquium on Trees in Algebra and Programming, CAAP'94
"... . We define Higher Order Recursive Program Schemes (HRPSs) by allowing metasubstitutions (as in the calculus) in righthand sides of function and quantifier definitions. A study of several kinds of similarity of redexes makes it possible to lift properties of (first order) Recursive Program Schemes ..."
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Cited by 20 (16 self)
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. We define Higher Order Recursive Program Schemes (HRPSs) by allowing metasubstitutions (as in the calculus) in righthand sides of function and quantifier definitions. A study of several kinds of similarity of redexes makes it possible to lift properties of (first order) Recursive Program Schemes to the higher order case. The main result is the decidability of weak normalization in HRPSs, which immediately implies that HRPSs do not have full computational power. We analyze the structural properties of HRPSs and introduce several kinds of persistent expression reduction systems (PERSs) that enjoy similar properties. Finally, we design an optimal evaluation procedure for PERSs. 1 Introduction Higher Order Recursive Program Schemes (HRPSs) are recursive definitions of functions, predicates, and quantifiers, considered as rewriting systems. Similar definitions are used when one extends a theory by introducing new symbols [16]. 9aA , (øa(A)=a)A and 9!aA , 9aA 8a8b(A (b=a)A ) a = b) a...
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 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...
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 ...
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...
Relating Conflictfree Stable Transition and Event Models (Extended Abstract)
, 1997
"... We describe an eventstyle (or poset) semantics for conflictfree rewrite systems, such as the calculus, and other stable transition systems with a residual relation. Our interpretation is based on considering redexfamilies as events. It treats permutationequivalent reductions as representing th ..."
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Cited by 4 (1 self)
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We describe an eventstyle (or poset) semantics for conflictfree rewrite systems, such as the calculus, and other stable transition systems with a residual relation. Our interpretation is based on considering redexfamilies as events. It treats permutationequivalent reductions as representing the same concurrent computation. Due to erasure of redexes, Event Structures are inadequate for such an interpretation. We therefore extend the Prime Event Structure model by axiomatizing permutationequivalence on finite configurations in two different ways, for the conflictfree case, and show that these extended models are equivalent to known stable transition models with axiomatized residual and family relations.
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...