Z. Khasidashvili and J.R.W. Glauert. Discrete normalization and standardization in deterministic residual structures. In M. Hanus and M. Rodrguez-Artalejo, editors, Proceedings of the Fifth International Conference on Algebraic and Logic Programming, (ALP '96), volume 1139 of Lecture Notes in Computer Science, pages 135--149. Springer, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....left normal orthogonal systems. The methods presented above are also applicable to ordinary standardisation, as shown in [47] The introduction of parallel standardisation is of more recent origin ( 16] and its further development was triggered by the axiomatic approaches to standardisation of [11,19,34]. iii) Termination of parallel standardisation by inversion is essentially more di#cult than termination of parallel standardisation by selection, corresponding to selection sort (see e.g. 48] Section 3.1 for selection sort, and [47] for selection (parallel) standardisation ) This is ....

Z. Khasidashvili and J.R.W. Glauert. Discrete normalization and standardization in deterministic residual structures. In M. Hanus and M. Rodrguez-Artalejo, editors, Proceedings of the Fifth International Conference on Algebraic and Logic Programming, (ALP '96), volume 1139 of Lecture Notes in Computer Science, pages 135--149. Springer, 1996.

....for various reasons: a) Some methods are too weak, e.g. the method of Klop only handles left normal rule sets [Klo80] Any method which only supports the traditional left to right order will usually be too weak. b) Other methods are too abstract. There are very general syntax free frameworks [GLM92, KG96], but they do not reduce the burden signi cantly in comparison with the hand crafted approach because the programming language theorist still has to prove their system satis es the various axioms, which is very tedious to do correctly. What seems to be needed is a framework that is abstract enough ....

.... that result to prove the context lemma [JM96] Suzuki used Klop s second method to prove standardization for conditional term rewriting systems [Suz96] There is some more discussion of Klop s second method in [vO96] Khasidashvili and Glauert proved something they call abstract standardization [KG96] and what they call relative standardization [GK] Standardization has been used extensively for validating the consistency of an operational semantics with a calculus by Plotkin, Felleisen, Ariola, Friedman, Hieb, Muller, and others not listed [Plo75, FF89, FH92, Mul92, AF97] The method of ....

Zurab Khasidashvili and John Glauert. Discrete normalization and standardization in deterministic residual structures. In ALP '96 [ALP96], pages 135-149.

....weak, e.g. the method of Klop [Klo80] only supports rule sets where evaluation can proceed from left to right. Some more recent methods are suciently general but are at such an abstract level that they provide little help with the proof burden. These methods include various syntax free frameworks [GLM92, KG96]. They do not signi cantly reduce the burden in comparison with the hand crafted approach because the programming language theorist still has to prove that their system satis es the various axioms. In practice, this is very dicult problem in its own right. 1.1 Contributions of this Paper In this ....

.... that result to prove the context lemma [JM96] Suzuki used Klop s second method to prove standardization for conditional term rewriting systems [Suz96] There is some more discussion of Klop s second method in [vO96] Khasidashvili and Glauert proved something they call abstract standardization [KG96] and what they call relative standardization [GK] Standardization has been used extensively for validating the consistency of an operational semantics with a calculus by Plotkin, Felleisen, Ariola, Friedman, Hieb, Muller, and others not listed [Plo75, FF89, FH92, Mul92, AF97] The method of ....

Zurab Khasidashvili and John Glauert. Discrete normalization and standardization in deterministic residual structures. In ALP '96 [ALP96], pages 135-149.

....this, but its results are largely negative: the axioms given there are insu#cient to establish transfinite analogues of many basic theorems concerning finitary abstract reduction systems, such as Newman s Lemma. We are currently working on a more expressive set of axioms, based on those given in [12] for finitary systems. 4.1 Axioms for abstract systems A transfinite abstract reduction system consists of a complete ultrametric space T , a set Red of reductions, and functions source : Red # T , target : Red # T , # : Red # N , # : Red # N , and a residual function to be described ....

Z. Khasidashvili and J.R.W. Glauert. Discrete normalization and standardization in deterministic residual structures. In Proc. 5th International Conference on Algebraic and Logic Programming, Aachen, 1996.

....the left most redex is always needed, and this yields another proof of the normalization theorem. Similar results were shown by Huet and L evy [26] in their early study of neededness in the context of orthogonal term rewriting systems, and much has been done since in various contexts see [36] for references to some papers. Similar results were discovered independently by Khasidashvili [31] see also [33, 35] in particular, the proof of Theorem 7.41 can be viewed as a special case of a proof due to Khasidashvili [31] For more on normalization, see [38, 58] 7.7. Conservation from ....

Z. Khasidashvili and J. Glauert. Discrete normalization and standardization in deterministic residual structures. In S. Tison, editor, Algebraic and Logic Programming, volume 1139 of Lecture Notes in Computer Science, pages 135--149. Springer-Verlag, 1996.

....of orthogonal systems precisely correspond to family reductions in the original system. Part of this work was supported by the Engineering and Physical Sciences Research Council of Great Britain under grant GR H Recent advances in the abstract study of orthogonal rewrite systems [Sta89, GLM92, GK96, KG96, KG97] allow us to address the problem in an entirely abstract setting. Stable Deterministic Residual Structures (SDRSs) and Deterministic Family Structures (DFSs) model computation in orthogonal rewrite systems. DRSs are Abstract Rewrite Systems with an axiomatized residual relation; DFSs are DRSs ....

....Residual Structures (SDRSs) and Deterministic Family Structures (DFSs) model computation in orthogonal rewrite systems. DRSs are Abstract Rewrite Systems with an axiomatized residual relation; DFSs are DRSs where in addition the concept of redex family is axiomatized. Stable DRSs allow [GK96, KG96] proofs of analogs of the Normalization and Standardization Theorems [Bar84] In DFSs one can further prove L evy s Optimality Theorem [L ev80] and the Unique Families Lemma [GK96] The latter states that any family can be contracted at most once in a family reduction, and corresponds to the fact ....

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Khasidashvili Z., Glauert J. R. W. Discrete normalization and Standardization in Deterministic Residual Structures. ALP'96, Springer LNCS, vol. 1139, M. Hanus, M. Rodr'iguez-Artalejo, eds. 1996, p.135-149.

.... of redex family in the literature [L ev78, L ev80, KS89, Mar92, AL94, Oos96] Important standard results such as the Standardization and Normalization Theorems of Curry and Feys [Bar84] or L evy s Optimality Theorem [L ev78, L ev80] can be proven using only the abstract framework of DFSs [GK96, KG96]. Furthermore, by using the DFS axioms alone, one can interpret DFSs into non duplicating, also called affine, DFSs with zig zag as the family relation, by interpreting family reduction multi steps in the former as reduction steps in the latter [KG97c, KG98] This allows us to reduce studying the ....

.... under L evy equivalence, we work with standard reductions, which in DFSs are reductions in which later steps do not erase the preceding ones (we ignore outside in and left to right order of contraction of redexes, as there are no such concepts in DFSs, and these are actually inappropriate) [KG96]. Finally, we remark that DFSs are an important transition model in their own right. In addition to proving the syntactic results mentioned above, they have been successfully used in defining a Prime Event Structure [Win89] style semantics [KG97b, KG98a] and a computational semantics [Kha98] for ....

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Khasidashvili Z., Glauert J. R. W. Discrete normalization and Standardization in Deterministic Residual Structures. In proc. of the 5 th International Conference on Algebraic and Logic Programming, ALP'96, Springer LNCS, vol. 1139, M. Hanus, M. Rodr'iguezArtalejo, eds. 1996, pp.135-149.

....Physical Sciences Research Council of Great Britain under grant GR H SDRSs are Abstract Rewrite Systems with an axiomatized residual relation, which model all orthogonal rewrite systems. Standard important results like the Standardization and Normalization theorems can already be proven in SDRSs [GK96, KG96]. Furthermore, via Deterministic Family Structures, DFSs [GK96] which are SDRSs with an axiomatized family relation on redexes, one can prove optimality results of L evy [L ev80] and achieve Prime Event Structure [Win89] style semantics for orthogonal rewrite systems in a uniform way [KG97a] ....

....of the technical difficulties come from the erasure of redexes in SDRSs. To cope with the erasure problems, and to have (most of the) concepts invariant under L evy equivalence, we work with standard reductions, which in SDRSs are reductions in which later steps do not erase the preceding ones [KG96]. If the SDRS is duplicating, concepts like restriction of P to a redex set U cannot be defined correctly for arbitrary P we need P to be a family reduction, that is, a multi step reduction contacting all members of a (zig zag) family in parallel, in every multi step. However, as we have ....

[Article contains additional citation context not shown here]

Khasidashvili Z., Glauert J. R. W. Discrete normalization and Standardization in Deterministic Residual Structures. In proc. of ALP'96, Springer LNCS, vol. 1139, M. Hanus, M. Rodr'iguez-Artalejo, eds. 1996, p.135-149.

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