L'evy J.-J. An algebraic interpretation of the fiK-calculus; and an application of a labelled -calculus. TCS, vol. 2, no. 1, pp. 97-114, 1976.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....from the alphabet and a special constant to denote undefined (in some sense) terms. And the reductions P F t are those that compute these trees. Examples of semantic trees are Nivat s trees for Recursive Program Schemes [Niv75] Bohm trees [Bar84] and L evy Longo trees (called also lazy trees [L ev76, Lon83, AbOn93]) in the calculus. Thus, our DN theorem characterizes computations of a term which converge to the value of the term, where value can be defined in various ways. We will describe an abstract approach to semantics of orthogonal systems, and show how the usual semantics of orthogonal systems can ....

....the rewrite alphabet and the distinguished constant symbol which represents undefined or meaningless terms; and is the ordering induced by s for any s 2 T 1 . The tree cpo was first defined for Recursive Program Schemes by Nivat [Niv75] and was later extended to the calculus by L evy [L ev76] (the so called L evy Longo trees or Lazy trees [Lon83, AbOn93] and Barendregt [Bar84] the so called Bohm trees) and to TRSs by Boudol [Bou85] In Nivat s and Boudol s semantics, undefined are terms which cannot be reduced to a root stable form a term that cannot be reduced to a redex. In ....

L'evy J.-J. An algebraic interpretation of the fiK-calculus; and an application of a labelled -calculus. TCS, vol. 2, no. 1, pp. 97-114, 1976.

....usually compute minimal reductions, and it is natural to ask whether optimality can be achieved while retaining minimality. The prime example is the leftmost outermost strategy computing the so called principal hnf and whnf of a term, and used in constructions of Bohm [Bar84] and L evy Longo [L ev76, Lon83] (also called lazy [AbOn93] trees, respectively. These trees represent the values of the term according to different semantics Bohm semantics and L evy or lazy semantics, respectively. Clearly this property of minimality is not useful for full normal forms, but full normal forms are rarely ....

L'evy J.-J. An algebraic interpretation of the fiK-calculus; and an application of a labelled -calculus. TCS 2(1):97-114, 1976.

....fully lazy. But this contradicts a result of Ong (see [2] that is fully lazy. 4 Lazy approximants and L evy Longo trees In this section we start relating the observational preorder m with an intensional representation ( 1 ) of terms due to L evy, and studied by Longo [10] and Ong [13] In [8], L evy introduced a refinement of Wadsworth s notion of syntactic approximant [15] suited for the lazy calculus where x Omega must be distinguished from Omega Gamma Then L evy defined an interpretation of the calculus, based on this notion of lazy approximant , and he showed that the ....

....L evy s preorder on approximants: A vL B if and only if A is a prefix of B, where the prefix ordering is the precongruence on approximants generated by Omega A. Using the Proposition 3. 5, it is easy to see that A B ) A m B The Church Rosser property has the following consequence (see [8]) Lemma 4.2 For any M 2 , the set A(M) is directed with respect to the prefix preorder, namely 8A 0 ; A 00 2 A(M) 9A 2 A(M) A 0 A A 00 A 1 by intensional we mean relying on the syntactic shape of the terms, up to fi conversion . INRIA The discriminating power of ....

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J.J. L'evy. An algebraic interpretation of the fiK-calculus; and an application of a labelled -calculus. Theoretical Computer Science, 2(1):97 -- 114, 1976.

....studied a similar question, where the may testing v is replaced by weak bisimulation, denoted here . He showed see [23, 24] and especially [25] from which we shall borrow some results that this semantics M N on terms coincides with equality in a semantics given by L evy in [13] for the calculus. In [15] Longo gave a suggestive presentation of L evy s interpretation, by means of what are now called L evy Longo trees, a refinement of the well known Bohm trees (see [4] suited for the weak calculus where any divergent term as Omega = x xx) x xx) is different from x ....

....M v N , for all contexts C; C[M ] C[N ] This is the semantics introduced by Abramsky, who called it applicative bisimulation , see [2, 3] One should note that, although is not taken as a computation rule, it is semantically valid. More generally, one has: M = fi N ) M N In [13], L evy defined an interpretation of the weak calculus, based on a refinement of Wadsworth s notion of syntactic approximant [26] as follows: Definition 3.1 (L evy s Interpretation) The set L of (lazy) approximants, ranged over by A; B; Delta Delta Delta, is the least subset of containing x ....

[Article contains additional citation context not shown here]

J.J. L'evy. An algebraic interpretation of the fiK-calculus; and an application of a labelled -calculus. Theoretical Computer Science, 2(1):97 -- 114, 1976.

....usually compute minimal reductions, and it is natural to ask whether optimality can be achieved while retaining minimality. The prime example is the leftmost outermost strategy computing the so called principal hnf and whnf of a term, and used in constructions of Bohm [Bar84] and L evy Longo [L ev76, Lon83] (also called lazy [AbOn93] trees, respectively. These trees represent the values of the term according to different semantics Bohm semantics and L evy or lazy semantics, respectively. Clearly this property of minimality is not useful for full normal forms, but full normal forms are rarely ....

L'evy J.-J. An algebraic interpretation of the fiK-calculus; and an application of a labelled -calculus. TCS 2(1):97-114, 1976.

....the rewrite alphabet and the distinguished constant symbol which represents undefined or meaningless terms; and is the ordering induced by s for any s 2 T 1 . The tree cpo was first defined for Recursive Program Schemes by Nivat [Niv75] and was later extended to the calculus by L evy [L ev76] (the so called L evy Longo trees or Lazy trees [Lon83, AbOn93] and Barendregt [Bar84] the Bohm trees) and to TRSs by Boudol [Bou85] In Nivat s and Boudol s semantics, undefined are terms which cannot be reduced to a root stable form a term that cannot be reduced to a redex. In Bohm ....

L'evy J.-J. An algebraic interpretation of the fiK-calculus; and an application of a labelled -calculus. TCS 2(1):97-114, 1976.

....studied a similar question, where the may testing v is replaced by weak bisimulation, denoted here . He showed see [23, 24] and especially [25] from which we shall borrow some results that this semantics M N on terms coincides with equality in a semantics given by L evy in [13] for the calculus. In [15] Longo gave a suggestive presentation of L evy s interpretation, by means of what are now called L evy Longo trees, a refinement of the wellknown Bohm trees (see [4] suited for the weak calculus where any divergent term such as Omega = x xx) x xx) is different from ....

....C; C[M ] C[N ] This was shown in [3] to be equivalent to the coinductively defined applicative bisimulation introduced by Abramsky in [2] One should note that, although is not taken as a computation rule, it is semantically valid. More generally, one has: M = fi N ) M N In [13], L evy defined an interpretation of the weak calculus, based on a refinement of Wadsworth s notion of syntactic approximant [26] as follows: Definition 3.1 (L evy s Interpretation) The set L of (lazy) approximants, ranged over by A; B; Delta Delta Delta, is the least subset of containing ....

[Article contains additional citation context not shown here]

J.J. L'evy. An algebraic interpretation of the fiK-calculus; and an application of a labelled -calculus. Theoretical Computer Science, 2(1):97 -- 114, 1976.

....fully lazy. But this contradicts a result of Ong (see [2] that is fully lazy. 4 Lazy approximants and L evy Longo trees In this section we start relating the observational preorder m with an intensional representation ( 1 ) of terms due to L evy, and studied by Longo [11] and Ong [14] In [9], L evy introduced a refinement of Wadsworth s notion of syntactic approximant [17] suited for the lazy calculus where x Omega must be distinguished from Omega Gamma Then L evy defined an interpretation of the calculus, based on this notion of lazy approximant , and he showed that the ....

....the prefix ordering is the precongruence on approximants generated by Omega A. Using the Proposition 3.5, it is easy to see that A B ) A m B It should be clear that fi reduction increases the direct approximation. Then the Church Rosser property has the following easy consequence (see [9]) Lemma 4.2 For any M 2 , the set A(M ) is directed with respect to the prefix preorder, namely 8A 0 ; A 00 2 A(M ) 9A 2 A(M ) A 0 A A 00 A Moreover, it is easy to see that A(M ) is in fact an ideal, that is it is downward closed with respect to the prefix ordering: A 2 A(M ) ....

[Article contains additional citation context not shown here]

J.J. L'evy. An algebraic interpretation of the fiK-calculus; and an application of a labelled -calculus. Theoretical Computer Science, 2(1):97 -- 114, 1976.

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