J. Glauert and Z. Khasidashvili. Relative normalization in orthogonal expression reduction systems. In N. Dershowitz and N. Lindenstrauss, editors, Workshop on Conditional (and Typed) Term Rewriting Systems, CTRS'94, volume 968 of Lecture Notes in Computer Science, pages 144-- 165. Springer Verlag, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....origin tracking, which was introduced in Klop [Klo90] Several variants of this notion have been studied, sometimes with applications that are similar to the ones described in this paper. We mention the work of Boudol [Bou85] Khasidashvili [Kha90, Kha93] Maranget [Mar92] Glauert Khasidashvili [GK94] and van Oostrom [Oos97a] 2 A distinctive feature is that our presentation makes extensive use of L evy labels (see Section 6) The method of origin tracking gives rise to perspicuous proofs of some well known classical theorems. In Section 7 we prove in some detail the Genericity Lemma in ....

J. Glauert and Z. Khasidashvili. Relative normalization in orthogonal expression reduction systems. In N. Dershowitz and N. Lindenstrauss, editors, Workshop on Conditional (and Typed) Term Rewriting Systems, CTRS'94, volume 968 of Lecture Notes in Computer Science, pages 144-- 165. Springer Verlag, 1994.

.... difficult task, and much work has been done to improve and strengthen the theory of finding needed redexes initiated in [10] An alternative concept of neededness, that of essentiality, was independently developed in [11] for the l calculus and later extended to orthogonal ERSs (OERSs) see e.g. [9, 12]) Essential are subterms that have descendants under any reduction of the given term, and in particular all essential redexes are needed. The main difference is that essentiality (but not neededness) makes sense for all subterms, e.g. for bound occurrences of variables, and we will take ....

....take advantage of this fact in our algorithm for finding (all, not just one) essential subterms in hyperbalanced terms. Definition 4. 1 A subterm N M is called essential if it has a descendant under any reduction starting from M, and is called inessential otherwise [11] The following lemma from [9] shows that inessential subterms do not contribute to the normal form of a term in an OERS, and they can be replaced with arbitrary terms without effecting the normal form. Inessential subterms of a term form garbage , and the lemma states that the garbage can safely be collected, i.e. replaced ....

[Article contains additional citation context not shown here]

GLAUERT, J.R.W., KHASIDASHVILI, Z., 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, 968:144-165, 1994.

....showed that in Persistent OERSs, where redex creation is limited, one can find all needed redexes in any term. Gardner [Gar94] described a complete way of encoding neededness information using a type assignment system. Kennaway et al. KKSV95] studied needed strategies for infinitary OTRSs. In [GlKh94], the present authors found natural conditions on a set S of terms, called stability, that are necessary and sufficient for the following Relative Normalization (RN) theorem to hold: each S normalizable term not in S (not in S normal form) has at least one S needed redex, and repeated contraction ....

....G s of s (which consists of terms to which s is reducible) is stable and regular, then the task can be reduced to construction of a minimal , or rather a least (w.r.t. L evy s ordering Theta on reductions) G s normalizing reduction [GlKh94a] since it must end at s. However, as we already know [GlKh94], G s need not be stable, mainly because of syntactic accidents [L ev80] For example, the graph fI(x) xg of s = I(x) in the TRS with the rule I(x) x, is not stable, since t = I(I(x) can be reduced to s by contracting either redex. Neither of the two redexes in t is G s needed, and neither ....

[Article contains additional citation context not shown here]

Glauert J.R.W., Khasidashvili Z. Relative Normalization in Orthogonal Expression Reduction Systems. In: Proc. of the 4 th International workshop on Conditional (and Typed) Term Rewriting Systems, CTRS'94, Jerusalem, N. Dershowitz, N. Lindenstrauss, eds., Springer LNCS, vol. 968, pp. 144-165, 1994.

....for infinitary OTRSs. A different approach to normalization is developed in Kennaway [Ken89] and Antoy and Middeldorp [AnMi94] Antoy et al. AEH94] design a needed narrowing strategy. Gardner [Gar94] describes a complete way of encoding neededness information using a type assignment system. In [GlKh94], the present authors address the question of normalization relative to a desired set of final terms, considering the properties that a set S of terms must possess in order for the neededness theory of Huet and L evy still to make sense. This work is done in the context of orthogonal Expression ....

....Family Structure (DFS) as a DRS with a very liberal notion of family relation [L ev78, L ev80] and a contribution relation on families, expressing the notion of (at least one member of) a family to be needed to create another family. For DFSs, the proof of the RN theorem for all stable S from [GlKh94] works perfectly. All existing definitions of family relation in the literature [L ev78, Klo80, KeSl89, Mar92, AsLa93, Oos96] satisfy the above axioms. For DFSs we show that a strategy that contracts, in an arbitrary order, only redexes that belong to S needed families, but which need not be ....

[Article contains additional citation context not shown here]

Glauert J.R.W., Khasidashvili Z. Relative Normalization in Orthogonal Expression Reduction Systems. In: Proc. of the 4 th International workshop on Conditional (and Typed) Term Rewriting Systems, CTRS'94, Jerusalem, N. Dershowitz, N. Lindenstrauss, eds., Springer LNCS, vol. 968, pp. 144-165, 1994.

....S needed strategy is normalizing. Although we claim that our definition of stability is the most natural that has been proposed, it is possible to weaken the conditions for stability and still retain the concept of relative neededness. For example, an alternative definition is explored in [13, 12] where a stable set S is closed under unneeded expansion and closed under parallel moves : for any t ## S, any P : t ## o # S, and any Q : t ## e not containing terms in S, the final term of P Q, the residual of P under Q, is in S. Minimal and Optimal Relative Normalization We ....

....stable sets S. In Section 8 we establish a Relative Standardization Theorem. In Section 9, we prove the Relative Optimality Theorem. Finally, in Section 10, we relate relative optimal and minimal reductions. The conclusions appear in Section 11. The concepts introduced in Section 3 build on [13], but here the definitions use only the residual concept while the earlier definitions were based on a concept of 4 descendants for subterms and components. The stability concept is simpler, and hence slightly less general. The labelling system in Section 4 extends unpublished work by Kennaway ....

[Article contains additional citation context not shown here]

J. R. W. Glauert and Z. Khasidashvili. On relative normalization in orthogonal expression reduction systems. In N. Dershowitz and N. Lindenstrauss, editors, 4 th International Workshop on Conditional (and Typed) Term Rewriting Systems, CTRS'94, volume 968 of LNCS, pages 144--165. Springer-Verlag, 1994.

....essentiality, and in section 9, reductions standard w.r.t. regular stable sets. In section 10, we prove the Relative Optimality theorem. Finally, in section 11, we relate relative optimal and minimal reductions. The conclusions appear in section 12. The results in Sections 3 6 are published as [GlKh94], there we in addition study relative neededness and essentiality properties of components of terms in OERSs, which are occurrences of contexts. This requires introduction of notions of descendants and residuals for components. Technical Report SYS C94 06 UEA Norwich, UK Minimal and Optimal ....

....Proposition 7.2. 8 Relative Essentiality In this section, we introduce the relative notion of essentiality, relate it to relative neededness, and give a characterization of R erased redexes in terms of relative neededness and essentiality. First we generalize some properties of essential subterms [Kha93, GlKh94] to the relative case. Definition 8.1 We call s t S essential, written ES S (s; t) if it has a P descendant for at least one S normalizing P starting from t, and call it S inessential, written IE S (s; t) otherwise. Lemma 8.1 (1) Let e s t and ES S (e; t) Then ES S (s; t) 2) Let s be ....

Glauert J.R.W., Khasidashvili Z. On 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, vol. 968, N. Dershowitz, N. Lindenstrauss, eds. Jerusalem, 1994, p. 144-165.

....general [KG97c, KG98] In order to make the introduced concepts computationally more meaningful, we relativize them w.r.t. the semantics one may be interested in. For example, in the calculus, one might be interested in computing normal forms, head normal forms, weak head normal forms, etc. In [GK94], we have characterized all reasonable sets of finite (partial) results as stable sets S of terms, and have shown that w.r.t. stable sets S, S needed reductions are S normalizing (i.e. end at a term in S if the initial term is reducible to a term in S) This allows us to ignore S unneeded ....

....results concerning relative normalization in DFSs from [GK96, KG98] and prove the Optimal Decomposition Theorem: an optimal computation of a term t in a DFS, relative to a stable set of results S, can be decomposed into optimal computations of S independent redex sets of t. Definition 5. 1 ([GK94]) Let S be a set of terms in an SDRS R. 1) We call a redex u t S needed if at least one residual of it is contracted in any reduction from t to a term in S, and call it S unneeded otherwise. 2) We call a set S of terms stable if: a) S is closed under parallel moves: for any t 62 S, any P : ....

Glauert J.R.W., Khasidashvili Z. 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, vol. 968, N. Dershowitz, N. Lindenstrauss eds. Jerusalem, 1994, pp. 144-165.

....corresponding normal forms. In [Mar92] Maranget al..so studied a strategy that computes a weak head normal form of a term in an OTRS. Normalization w.r.t. another interesting set of normal forms , that of constructor head normal forms in constructor OTRSs, is studied by Nocker [Nok94] In [GlKh94], the present authors studied normalization with respect to any desired set of final terms, and found the sufficient and necessary properties, called This work was supported by the Engineering and Physical Sciences Research Council of Great Britain under grant GR H41300 stability, that a set ....

....Several interesting formalisms have been introduced later [Kha92, Nip93, OR94] We refer to van Raamsdonk [Raa96] for a survey. Here we use a system of higher order rewriting, Expression Reduction Systems (ERSs) defined in [Kha92] under the name of CRSs) the present formulation follows [GlKh94] and is simpler. Definition 2.1 Let Sigma be an alphabet, comprising variables, denoted by x; y; z; function symbols, also called simple operators; and operator signs or quantifier signs. Each function symbol has an arity k 2 N , and each operator sign oe has an arity (m; n) with m;n ....

[Article contains additional citation context not shown here]

Glauert J.R.W., Khasidashvili Z. Relative Normalization in Orthogonal Expression Reduction Systems. In: Proc. of CTRS'94, Springer LNCS, vol. 968, N. Dershowitz, N. Lindenstrauss, eds. Jerusalem, 1994, p. 144-165.

....in Antoy and Middeldorp [AnMi94] Antoy et al. AEH94] designed a needed narrowing strategy. Gardner [Gar94] described a complete way of encoding neededness information using a type assignment system, in the calculus. Kennaway et al. KKSV95] studied needed strategies for infinitary OTRSs. In [GlKh94], the present authors found natural conditions on a set S of terms, in an OERS, called stability, that are necessary and sufficient for the following Relative Normalization (RN for short) theorem to hold: each S normalizable term not in S (not in S normal form) has at least one S needed redex, ....

....G s of s (which consists of terms to which s is reducible) is stable and regular, then the task can be reduced to construction of a minimal , or rather a least (w.r.t. L evy s ordering Theta on reductions) G s normalizing reduction [GlKh94a] since it must end at s. However, as we already know [GlKh94], G s need not be stable, mainly because of syntactic accidents [L ev80] For example, the graph fI(x) xg of s = I(x) in the TRS with the rule I(x) x, is not stable, since t = I(I(x) can be reduced to s by contracting either redex. Neither of the two redexes in t is G s needed, and neither ....

[Article contains additional citation context not shown here]

Glauert J.R.W., Khasidashvili Z. Relative Normalization in Orthogonal Expression Reduction Systems. In: Proc. of CTRS'94, Springer LNCS, vol. 968, N. Dershowitz, N. Lindenstrauss, eds. Jerusalem, 1994, p. 144-165.

....[SeRa90] study normalization via necessary set of redexes. Kennaway et al. KKSV96] study a needed strategy for infinitary OTRSs. A different approach to normalization is developed in Kennaway [Ken89] and Antoy and Middeldorp [AnMi94] Antoy et al. AEH94] design a needed narrowing strategy. In [GlKh94], the present authors address the question of normalization relative to a desired set of final terms, considering the properties that a set of terms must possess in order for the neededness theory of Huet and L evy still to make sense. This work is This work was supported by the Engineering ....

....Family Structure (DFS) as a DRS with a very liberal notion of family relation [L ev78, L ev80] and a contribution relation on families, expressing the notion of (at least one member of) a family to be needed to create another family. For DFSs, the proof of the RN theorem for all stable S from [GlKh94] works perfectly. An advantage of the first RN theorem is that checking for Berry s stability is much simpler than constructing a sound family relation. For example, the calculus [L ev78, L ev80] orthogonal TRSs [Mar92] Interaction Systems [AsLa93] and orthogonal CRSs (and ERSs) Klo80, ....

[Article contains additional citation context not shown here]

Glauert J.R.W., Khasidashvili Z. 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, vol. 968, N. Dershowitz, ed. Jerusalem, 1994, p. 144-165.

No context found.

J. Glauert and Z. Khasidashvili. Relative normalization in orthogonal expression reduction systems. In N. Dershowitz and N. Lindenstrauss, editors, Workshop on Conditional (and Typed) Term Rewriting Systems, CTRS'94, volume 968 of Lecture Notes in Computer Science, pages 144-- 165. Springer Verlag, 1994.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC