H. P. Barendregt, J. R. Kennaway, J. W. Klop, and M. R. Sleep. Needed reduction and spine strategies for the lambda calculus. Inform. and Comput. , 75(3):191--231, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....a spine redex (the bottommost one) When no spine redices exist, the expression is said to be in head normal form. Notice that it is always safe to reduce a spine redex of e, in the sense that if e reduces to a head normal form then this reduction involves the reduction of all spine redices of e [BKKS87, theorem 4.9] A reduction which reduces the topmost spine redex is termed a normal order reduction. Lazy evaluation One can work with DAG s instead of trees, i.e. allow a subexpression to be shared among several expressions. When the normal order strategy is applied, this amounts to using ....

....graph reductions and whose Result 3 states that sound contractions (reducing sound nodes ) are optimal contractions , a sound node loosely speaking being one the ancestors of which are never reduced before the node itself is the paper then gives criteria for being sound. Also, BKKS87] investigates the issue of detecting redices in a # expression the reduction of which is needed in order to arrive at (head) normal form. 127 3. lookup rules, which model the retrieval of a previously stored result; 4. folding rules, which increase sharing by identifying nodes with identical ....

H.P. Barendregt, J.R. Kennaway, J.W. Klop, and M.R. Sleep. Needed reduction and spine strategies for the lambda calculus. Information and Computation, 75:191--231, 1987. 247

....edge a b where (c; r) c 1 ; r 1 c 2 ) r 2 ) if r 1 c 2 (c; r) c 1 (c 2 r 1 ) r 2 ) otherwise Transitive closure is performed using this de nition of composition. The de nition of CR edges and the operation of composition bear a remarkable resemblance to the KSL algorithm of [BKKS87]; the cost and resource correspond to the K and L components respectively and the composition operation is a simpli ed version of the operation on KSL triples. Now if there is an edge a b, then we can conclude that there is a computation path from to (b) b . In ....

Henk Barendregt, J R Kennaway, J W Klop, and M R Sleep. Needed reduction and spine strategies for the lambda calculus. Information and Computation, 74:191-231, 1987.

....a needed redex, and that repeated contraction of needed redexes in t leads to its normal form whenever there is one; we refer to it as the Normalization Theorem. They also defined the class of strongly sequential OTRSs where a needed redex can efficiently be found in any term. Barendregt et al. [BKKS87] generalized the concept of neededness to the A calculus. They studied neededness not only w.r.t. normal forms, but also w.r.t. head normal forms a redex is head needed if its residuals are contracted in each reduction to a head normal form. The authors proved correctness of the two This work ....

.... simpler proofs of correctness of the essential strategy in OTRSs and the A calculus, which generalize straightforwardly to all Orthogonal Expression Reduction Systems (OERSs) Kennaway and Sleep [KeS189] used a generalization of Lvy s labelling fox the A calculus [Lv78] to adapt the proof from [BKKS87] to the case of Klop s OCRSs [Klo80] which can also be applied to OERSs. Khasidashvili [Kha94] showed that in Persistent OERSs, where redex creation is limited, one can find all needed redexes in any term. Gard ner [Gar94] described a complete way of encoding neededhess information using a type ....

[Article contains additional citation context not shown here]

Barendregt H. P., Kennaway J. R., Klop J. W., Sleep M. R. Needed Reduc- tion and spine strategies for the lambda calculus. Information and Computation, v. 75, no. 3, 1987, p. 191-231.

....a (c;r) b where (c; r) c 1 ; r 1 c 2 ) r 2 ) if r 1 c 2 (c; r) c 1 (c 2 r 1 ) r 2 ) otherwise Transitive closure is performed using this definition of composition. The definition of CR edges and the operation of composition bear a remarkable resemblance to the KSL algorithm of [6]; the cost and resource correspond to the K and L components respectively and the composition operation is a simplified version of the operation on KSL triples. Now if there is an edge a (0;r) b, then we can conclude that there is a computation path from 1 (a) a 0 to 1 (b) ....

H. Barendregt, J. R. Kennaway, J. W. Klop, and M. R. Sleep. Needed reduction and spine strategies for the lambda calculus. Information and Computation, 74:191--231, 1987.

....2:sieve(filter(2,from(3) which is rootstable as its root symbol : is a constructor. It is well known that in the calculus the leftmostoutermost redex in a term that is not yet root stable is root needed, but it is undecidable whether an arbitrary redex is root needed (cf. Barendregt et al. [3] where these statements are proved for the related notion of neededness with respect to head normal forms) For orthogonal term rewriting things are more complicated because the leftmost outermost redex is in general not root needed. Nevertheless, we will prove that every non root stable term has ....

H.P. Barendregt, J.R. Kennaway, J.W. Klop, and M.R. Sleep, Needed Reduction and Spine Strategies for the Lambda Calculus, Information and Computation 75(3), pp. 191--231, 1987.

....case every created redex has a strictly smaller degree w.r.t. the lexicographic ordering on pairs (l; h) Further, it is easy to see that the degrees of residuals are not greater than the degrees of their ancestor redexes, and the theorem follows from Lemma 3.6. 4 Static garbage collection In [6], a theory of needed reduction is developed for the l calculus, extending a similar theory of normalization by neededness for orthogonal Term Rewriting Systems (OTRSs) in [10] The main result of the theory is that every term not in normal form has a needed redex one whose residual is ....

BARENDREGT H.P., KENNAWAY J.R., KLOP J.W., SLEEP M.R., Needed Reduction and spine strategies for the lambda calculus. Information and Computation, 75(3):191-231, 1987.

....[HL91a, HL91b] To aid in nding needed redexes, they devised the notions of sequentiality and strong sequentiality. Klop and Middeldorp provide a quite readable discussion of strong sequentiality [KM91] Barendregt, Kennaway, Klop, and Sleep raised the idea of needed redexes to the calculus [BKKS87]. Glauert, Khasidashvili, N ocker, and Middeldorp have all written about normalization to sets of terms that are not exactly normal forms [GK, N oc94, Mid97] Van Raamsdonk showed that the outermost fair (multistep) strategy is normalizing for some HORS s [vR96] Kennaway, Antoy, and Middeldorp ....

H. P. Barendregt, J. R. Kennaway, Jan Willem Klop, and M. R. Sleep. Needed reduction and spine strategies for the lambda calculus. Inf. & Comput., 75(3):191-231, 1987.

....[HL91a, HL91b] To aid in nding needed redexes, they devised the notions of sequentiality and strong sequentiality. Klop and Middeldorp provide a quite readable discussion of strong sequentiality [KM91] Barendregt, Kennaway, Klop, and Sleep raised the idea of needed redexes to the calculus [BKKS87]. Glauert, Khasidashvili, N ocker, and Middeldorp have all written about normalization to sets of terms that are not exactly normal forms [GK, N oc94, Mid97] Van Raamsdonk showed that the outermost fair (multistep) strategy is normalizing for some HORS s [vR96] Kennaway, Antoy, and Middeldorp ....

H. P. Barendregt, J. R. Kennaway, Jan Willem Klop, and M. R. Sleep. Needed reduction and spine strategies for the lambda calculus. Inf. & Comput., 75(3):191-231, 1987.

....a general normalizing strategy, called the needed strategy, for Orthogonal Term Rewriting Systems (OTRSs) They showed that any term t not in normal form has a needed redex, and that repeated contraction of needed redexes in t leads to its normal form whenever there is one. Barendregt et al. [BKKS87] studied neededness w.r.t. normal forms and also head normal forms in the calculus, proving correctness of the two needed strategies for computing normal forms and head normal forms, respectively. Maranget [Mar92] studied a strategy that computes a weak head normal form of a term in an OTRS. ....

Barendregt H. P., Kennaway J. R., Klop J. W., Sleep M. R. Needed Reduction and spine strategies for the lambda calculus. Information & Computation, vol. 75, no. 3, pp. 191-231, 1987.

....be contracted in any reduction to normal form. Huet and L evy showed that any term not in normal form has a needed redex, and that repeated contraction of needed redexes leads to its normal form whenever there is one. This fundamental work has been extended in several directions. Barendregt et al. [BKKS87], Maranget [Mar92] and Nocker [Nok94] study neededness w.r.t. head normal forms, weak head normal forms, and constructor head normal forms, respectively. Sekar and Ramakrishnan [SeRa93] study normalization via necessary set of redexes. Kennaway et al. KKSV95] study a needed strategy for ....

....systems, defined via labelled orthogonal TRSs. All these are stable, and can be shown to be so just using an appropriate notion of descendant which assigns the contractum to the contracted redex, as defined in [Kha92] for ERSs labelling is not necessary. The descendant notion defined e.g. in [BKKS87] is essentially the residual notion, and is not useful for this purpose. 3 Relative Normalization for regular stable sets In this section, we prove that, for any regular stable set of terms S in a stable DRS R, an S normal form of an S normalizable term can be found by contracting S needed ....

[Article contains additional citation context not shown here]

Barendregt H. P., Kennaway J. R., Klop J. W., Sleep M. R. Needed Reduction and spine strategies for the lambda calculus. Information & Computation, vol. 75, no. 3, pp. 191-231, 1987.

....a needed redex, and that repeated contraction of needed redexes in t leads to its normal form whenever there is one; we refer to it as the Normalization Theorem. They also defined the class of strongly sequential OTRSs where a needed redex can efficiently be found in any term. Barendregt et al. [BKKS87] generalized the concept of neededness to the calculus. They studied neededness not only w.r.t. normal forms, but also w.r.t. head normal forms a redex is head needed if its residuals are contracted in each reduction to a head normal form. The authors proved correctness of the two needed ....

.... which generalize straightforwardly to all Orthogonal Expression Reduction Systems (OERSs) Kha90] and Klop s Orthogonal Combinatory Reduction Systems (OCRSs) Klo80] Kennaway and Sleep [KeSl89] used a generalization of L evy s labelling for the calculus [L ev78] to adapt the proof from [BKKS87] to the case of Klop s OCRSs, which can also be applied to OERSs. Khasidashvili [Kha94] 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, for the case of the ....

[Article contains additional citation context not shown here]

Barendregt H. P., Kennaway J. R., Klop J. W., Sleep M. R. Needed Reduction and spine strategies for the lambda calculus. Information and Computation, v. 75, no. 3, 1987, p. 191-231.

....approach used to avoid recomputation is to consider alternative evaluation strategies. If the expression to reduce is (lx.E)F we know that the whnf of the body E will be needed and so it is safe to reduce E prior to the b reduction. This computation rule belongs to the so called spine strategies [3]. It never takes more reductions than normal order and may save duplication of work. A revealing example, taken from [7] is the reduction of A n I where the family of l expressions A i is defined by A 0 = lx.x I and A n = lh. lw.w h (w w) A n 1 . A n I is reduced using the call by name weak ....

H.P. Barendregt, J.R. Kennaway, J.W. Klop and M.R. Sleep. Needed reduction and spine strategies for the lambda calculus. Information and Computation, Vol. 75, pp. 191-231, 1987.

....approach used to avoid recomputation is to consider alternative evaluation strategies. If the expression to reduce is (lx.E)F we know that the whnf of the body E will be needed and so it is safe to reduce E prior to the b reduction. This computation rule belongs to the so called spine strategies [4]. It never takes more reductions than normal order and may prevent duplication of work. A revealing example, taken from [8] is the reduction of A n I where the family of l expressions A i is defined by A 0 = lx.I and A n = lh. lw.w h (w w) A n 1 . The expression A n I is reduced using the ....

H.P. Barendregt, J.R. Kennaway, J.W. Klop and M.R. Sleep. Needed reduction and spine strategies for the lambda calculus. Information and Computation, Vol. 75, pp. 191-231, 1987.

....then so does N . Then proceed as in Corollary 7.42. ut The normalization theorem is due to Curry and Feys [15] Barendregt [2] infers the normalization theorem from the standardization theorem, and uses both of these theorems to prove normalization of quasi leftmost reductions. Barendregt et al. [5] define a fi redex Delta to be needed in a term M , if Delta (or a residual of Delta) is contracted in every reduction of M to normal form. They then show that every term not in normal form has at least one needed redex, and that a reduction strategy that contracts only needed redexes is ....

....use F1 , can be obtained by proving directly that F l is perpetual in I using the fundamental lemma of perpetuality, rather than inferring this from F l = F1 and perpetuality of F1 . Slight variations of this technique are due to Curry and Feys [15] and to van Raamsdonk [58] Barendregt et al. [5] show that leftmost reduction paths have maximal length among all reduction paths in which only needed redexes are contracted, and that in I all redexes are needed. This gives another proof that in I , F l is maximal and thereby perpetual. Acknowledgments. Thanks to Zurab Khasidashvili, Vincent ....

H.P Barendregt, J.R. Kennaway, J.W. Klop, and M.R. Sleep. Needed reduction and spine strategies for the lambda calculus. Information and Computation, 75(3):191--231, 1987.

....of e requires reduction of e 0 . For such an operational definition, strictness and neededness only differ subtly they agree in all circumstances except that an expression which never terminates is always strict in any subexpression but may not need a given subexpression. Barendregt et al. [1] discuss neededness in the calculus in some detail. When defining neededness formally the problem becomes a little more subtle in that evaluation order which is invisible in the standard semantics may become visible in a neededness semantics. 2 Typically, however, we wish to abstract from such ....

H. P Barendregt, J. R Kennaway, J. W Klop and M. R Sleep. Needed Reduction and Spine Strategies for the Lambda Calculus. In Information and Control, vol. 73, 1987.

....of a principally typed normalisable term. Possible applications of these results include strictness and sharing analysis for functional programming languages, and a reduction strategy for well typed terms which satisfies L evy s notion of optimal reduction. 1 Introduction Barendregt et al. [2] show that the identification of the needed redexes in a term is an undecidable problem. A redex r in term e is needed if a residual of r is contracted in every reduction of e to normal form. For example, in the term (x: y:z)x) Ie) the redexes (y:z)x and (x: y:z)x) Ie) are needed, whereas the ....

....we are able to distinguish the allowable redexes. We prove that allowable redexes are needed redexes, and, for normalisable terms which have been principally typed, prove that all needed redexes are allowable. The problem of identifying needed redexes has also been addressed by Barendregt et al. [2]. They distinguish the so called (generalised) spine redexes, and show that these form a proper subset of the set of needed redexes. Since this approach does not identify all needed redexes, it is unlikely that these results can be used for the applications we have in mind. Needed reductions are ....

[Article contains additional citation context not shown here]

H.P. Barendregt, J.R. Kennaway, J.W. Klop and M.R. Sleep. Needed Reduction and Spine Strategies for the Lambda Calculus, Information and Computation, Vol. 75, pp 191--231, 1987.

....but more efficient, implementation of call by name. Given a particular operational semantics, such as normal order evaluation specified above, we say that a program function f needs its i th argument if evaluating f(e 1 ; e k ) always causes the evaluation of e i . Barendregt et al. [2] discuss neededness for the calculus. Singh [13] introduces a denotational analogue of neededness by likening strictness to differentiation. Obviously a function which needs its i th argument can safely have that argument evaluated before the call thereby transforming call by need to ....

Barendregt, H.P., Kennaway, J.R., Klop, J.W., and Sleep, M.R. Needed Reduction and Spine Strategies for the Lambda Calculus, Information and Control, vol. 73, 1987.

....proved that any term t not in normal form, in an OTRS, has a needed redex, and that contraction of needed redexes in a normalizable term results in a normal form. Here a redex u in t is needed if some residual of it is contracted in every normalizing reduction starting from t. Barendregt et al. [BKKS87] applied the neededness notion to the calculus, and studied neededness not only w.r.t. normal forms, but also w.r.t. head normal forms. The authors proved correctness of the two needed strategies for computing corresponding normal forms. In [Mar92] Maranget al..so studied a strategy that computes ....

Barendregt H. P., Kennaway J. R., Klop J. W., Sleep M. R. Needed Reduction and spine strategies for the lambda calculus. Inf.& Comp., 75(3):191-231, 1987.

....strategy, for Orthogonal Term Rewriting Systems (OTRSs) They showed that any term t not in normal form has a needed redex, and that repeated contraction of needed redexes in t leads to its normal form whenever there is one. This seminal work has been extended in many directions. Barendregt et al. [BKKS87] studied neededness w.r.t. normal forms as well as w.r.t. head normal forms, in the calculus, proving correctness of the two needed strategies for computing normal forms and head normal forms, respectively. Kennaway and Sleep [KeSl89] generalized the needed strategy to orthogonal Combinatory ....

Barendregt H. P., Kennaway J. R., Klop J. W., Sleep M. R. Needed Reduction and spine strategies for the lambda calculus. Inf.& Comp., 75(3):191-231, 1987.

....has to be contracted in any reduction to normal form. Huet and L evy showed that any term not in normal form has a needed redex, and that repeated contraction of needed redexes leads to its normal form whenever there is one. This work has been extended in several directions. Barendregt et al. [BKKS87], Maranget [Mar92] and Nocker [Nok94] study neededness w.r.t. head normal forms, weak head normal forms, and constructor head normal forms, respectively. Sekar and Ramakrishnan [SeRa90] study normalization via necessary set of redexes. Kennaway et al. KKSV96] study a needed strategy for ....

....of terms in some DRS. R will denote a regular stable set. For simplicity, we only consider stable sets that are closed under reduction; obviously, closure under reduction implies closure under parallel moves. The most appealing examples of stable sets are normal forms [HuL e91] headnormal forms [BKKS87], weak head normal forms in an OTRS (a partial result is in [Mar92] and constructor head normal forms for constructor TRSs [Nok94] All the above sets are closed under reduction, and are regular. Other examples include weak head normal forms (up to garbage collection, modulo a congruence) in ....

Barendregt H. P., Kennaway J. R., Klop J. W., Sleep M. R. Needed Reduction and spine strategies for the lambda calculus. Information and Computation, v. 75, no. 3, 1987, p. 191-231.

....a needed redex, and that repeated contraction of needed redexes in t leads to its normal form whenever there is one; we refer to it as the Normalization Theorem. They also defined the class of strongly sequential OTRSs where a needed redex can efficiently be found in any term. Barendregt et al. [BKKS87] generalized the concept of neededness to the calculus. They studied neededness not only w.r.t. normal forms, but also w.r.t. head normal forms a redex is head needed if its residuals are contracted in each reduction to a head normal form. The authors proved correctness of the two This ....

.... simpler proofs of correctness of the essential strategy in OTRSs and the calculus, which generalize straightforwardly to all Orthogonal Expression Reduction Systems (OERSs) Kennaway and Sleep [KeSl89] used a generalization of L evy s labelling for the calculus [L ev78] to adapt the proof from [BKKS87] to the case of Klop s OCRSs [Klo80] which can also be applied to OERSs. Khasidashvili [Kha94] 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 ....

[Article contains additional citation context not shown here]

Barendregt H. P., Kennaway J. R., Klop J. W., Sleep M. R. Needed Reduction and spine strategies for the lambda calculus. Information and Computation, v. 75, no. 3, 1987, p. 191-231.

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H. P. Barendregt, J. R. Kennaway, J. W. Klop, and M. R. Sleep. Needed reduction and spine strategies for the lambda calculus. Inform. and Comput. , 75(3):191--231, 1987.

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H.P. Barendregt, J.R. Kennaway, J.W. Klop, and M.R. Sleep. Needed reduction and spine strategies for the lambda calculus. Technical Report CS-R8621, CWI, Postbus 4079, 1009 AB Amsterdam, The Netherlands, May 1986.

....of strongly sequential OTRSs where a needed redex can be found e#ciently in any term. # Work undertaken at UEA with the partial support of the Engineering and Physical Sciences Research Council of Great Britain under grant GR H 41300. 1 Extending the concept of Neededness Barendregt et al. [7] generalize the concept of neededness to the # calculus. They study neededness not only w.r.t. normal forms, but also w.r.t. head normal forms a redex is head needed if its residuals are contracted in each reduction to a headnormal form. They prove correctness of the two needed strategies for ....

....sometimes treated as synonymous in the literature. 2 A multistep contracts a set of redexes simultaneously. 2 in S. For example, Huet and Levy neededness [18] is neededness w.r.t. the set NF of normal forms, Maranget neededness [35] is neededness w.r.t. all fair reductions, headneededness [7] is neededness w.r.t. the set of head normal forms, root neededness [40] is neededness w.r.t. the set of root stable forms, etc. We impose a natural condition on S, stability, which guarantees that each term not in S normal form (i.e. not in S) has at least one S needed redex, such that ....

[Article contains additional citation context not shown here]

H. P. Barendregt, J. R. Kennaway, J. W. Klop, and M. R. Sleep. Needed reduction and spine strategies for the lambda calculus. Information and Computation, 75(3):191--231, 1987.

No context found.

H.P. Barendregt, J.R. Kennaway, J.W. Klop, and M.R. Sleep. Needed reduction and spine strategies for the lambda calculus. Technical Report CS-R8621, CWI, Postbus 4079, 1009 AB Amsterdam, The Netherlands, May 1986.

....; 101 ; 111 ; but I = 2 Ter( 001 ) Ter( 101 ) And the term consisting of an infinite string of abstractions x 0 : x 1 : x 2 : 13 is absent in 001 , but is present in 101 and 111 . Also any term which would have an infinite dl branch (a spine in the sense of Barendregt et al. [BKKS87]) is absent in 001 . See the Remark 10.2 below) lx 2 x 1 l x 0 l . lx 2 x 1 l x 0 l . b) a) Figure 52: Example of an infinite d branch and an infinite dl branch It turns out that the three infinitary calculi 001 ; 101 ; 111 are the natural home resorts ....

H. P. Barendregt, J. R. Kennaway, J. W. Klop, and M. R. Sleep. Needed reduction and spine strategies for the lambda calculus. Inform. and Comput. , 75(3):191--231, 1987.

No context found.

Barendregt, H.P. et al.: Needed Reduction and Spine Strategies for the Lambda Calculus. Information and Computation 75 (1987) 191--231.

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H.P. Barendregt, J.R. Kennaway, J.W. Klop, and M.R. Sleep. Needed reduction and spine strategies for the lambda calculus. Information and Computation, 75:191--231, 1987.

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