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....only be seen once we had contracted the first two redexes. There is moreover a wish to make as many needed redexes as possible visible and even though the notion of a needed redex is undecidable, much work has been carried out in order to study some classes of needed redexes (as in [BKKS 87] Gardner 94] and [BKN 9 ] Our proposal is not only to make as many redexes as possible visible, but also to give newly visible redexes the possibility to be contracted before other ones. Firstly, this view on reduction gives an appropriate tool for the study of some programming languages. For example, in ....

Gardner, P. (1994), Discovering Needed Reductions Using Type Theory, TACS.

....only be seen once we had contracted the first two redexes. There is however a need to make as many needed redexes as possible visible and even though the notion of a needed redex is undecidable, much work has been carried out in order to study some classes of needed redexes (as in [BKKS 87] and [Gardner 94] Our proposal, is not only to make as many redexes as possible visible, but also to give newly visible redexes the possibility to be contracted before other ones. Example 1.3 In example 1.2, we may want to contract the redex based on ( z:R : Gamma)d before we have contracted any of the redexes ....

Gardner, P., Discovering Needed Reductions Using Type Theory, to appear in TACS, 1994.

....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 assignment system in the sense that using the principal type of a term one can find all the needed redexes in it (the principal type cannot be found efficiently, as one might expect) Antoy et al. AEH94] designed a needed ....

Gardner P. Discovering needed reductions using type theory. In: Proc. of the 2 d International Symposium on Theoretical Aspects of Computer Software, TACS'94, Springer LNCS, v. 789, M. Hagiya, J. C. Mitchell, eds. Sendai, 1994, p. 555-574.

....orthogonal rewrite systems is developed in Kennaway [Ken89] and Antoy Middeldorp [AnMi94] Antoy et al. AEH94] designed a needed narrowing strategy. 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 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 ....

Gardner P. Discovering needed reductions using type theory. In: Proc. of the 2 nd International Symposium on Theoretical Aspects of Computer Software, TACS'94, Sendai, M. Hagiya, J. C. Mitchell, eds., Springer LNCS, vol. 789, pp. 555-574, 1994.

....Higher Order Recursive Program Schemes one can find all needed redexes in any term, implying decidability of weak and strong normalization. Khasidashvili and Piperno [28] designed an algorithm for statically finding all inessential subterms (i.e. the garbage) in simply typeable # terms. Gardner [11] described a complete way of encoding neededness information, for the case of the # calculus, using a type assignment system in the sense that using the principal type of a term one can find all the needed redexes in it. Antoy et al. 2] designed a needed narrowing strategy. Kennaway et al. 20] ....

P. Gardner. Discovering needed reductions using type theory. In M. Hagiya and J. C. Mitchell, editors, 2 nd International Symposium on Theoretical Aspects of Computer Software, TACS'94, volume 789 of LNCS, pages 555--574. SpringerVerlag, 1994.

....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 calculus, using a type assignment system in the sense that using the principal type of a term one can find all the needed redexes in it. Antoy et al. AEH94] designed a needed narrowing strategy. In [Mar92] ....

Gardner P. Discovering needed reductions using type theory. In: Proc. of the 2 nd International Symposium on Theoretical Aspects of Computer Software, TACS'94, Springer LNCS, v. 789, M. Hagiya, J. C. Mitchell, eds. Sendai, 1994, p. 555-574.

....which are fi equivalent have the same type. Moreover, intersection types characterize normalizable terms: a term is normalizable if and only if it is typable. Intersection type systems are therefore very expressive: this is why many authors have been interested by their theory or their usage [23, 24, 21, 10, 30]. However, the price of this expressiveness is that type assignment is only semi decidable. Another drawback is the loss of the principal type property in the classical sense. As a matter of fact, in order to find all possible types of a term from a unique type, we must have more than just ....

.... [4, 8, 9, 12] and which is semantically 34 complete [1] Some papers about the principal type property complete these theoretical studies [6, 26, 35] There are also many works which use intersection types in several domains (programming languages, partial evaluation, term rewriting system) [20, 10, 23, 11, 31, 34, 14] is a non exhaustive list of these works. In [32, 33] S. van Bakel determines the essential characteristics of the intersection type systems and proposes two restrictions of system D which have the same properties as the original system. As far as we know, his work is the first real advance in ....

Philippa Gardner. Discovering needed reductions using type theory. In M. Hagiya and John C. Mitchell, editors, Proceedings of TACS'94, volume 789 of Lecture Notes in Computer Science. Springer-Verlag, 1994.

....only be seen once we had contracted the first two redexes. There is however a need to make as many needed redexes as possible visible and even though the notion of a needed redex is undecidable, much work has been carried out in order to study some classes of needed redexes (as in [BKKS 87] and [Gardner 94] Our proposal, is not only to make as many redexes as possible visible, but also to give newly visible redexes the possibility to be contracted before other ones. Example 1.3 In example 1.2, we may want to contract the redex based on ( z:R : Gamma)d before we have contracted any of the redexes ....

Gardner, P., Discovering Needed Reductions Using Type Theory, to appear in TACS, 1994.

....only be seen once we had contracted the first two redexes. There is moreover a wish to make as many needed redexes as possible visible and even though the notion of a needed redex is undecidable, much work has been carried out in order to study some classes of needed redexes (as in [BKKS 87] Gardner 94] and [BKN 9 ] Our proposal is not only to make as many redexes as possible visible, but also to give newly visible redexes the possibility to be contracted before other ones. Firstly, this view on reduction gives an appropriate tool for the study of some programming languages. For example, in ....

Gardner, P. (1994), Discovering Needed Reductions Using Type Theory, TACS.

.... specified in this vain are [17, 18, 19, 23] Our analysis belongs to the other group where subcomponents of types are annotated; further analyses in this group are [2, 8, 9, 10] Inference based methods have also been used for variations of strictness and totality analysis; examples include [5] that uses a type system with intersection types to determine neededness of redexes and [3] that studies liveness properties. Overview Section 2 presents the natural style operational semantics and the standard type inference rules for our simply typed lazy lambda calculus. Based on these ....

Philippa Gardner. Discovering needed reductions using type theory. In Proceeding of TACS'94, 1994.

....[10] is a restricted version of the first. Since it limits the possible bases that can be used in a derivation, it is not a proper extension of Curry s system: if B M :oe and the term variable x does not occur in B, then for (x:M ) only the type oe can be derived. This system is also used in [19]. Definition 1.2.3 Restricted type assignment and restricted derivations are in [10] defined by: x:oe 1 ] Delta Delta Delta [x:oe n ] M :oe ( I) a) x:M :oe 1 Delta Delta Delta oe n oe M :oe 1 : M :oe n ( I) b) M :oe 1 Delta Delta Delta oe n ( M : M :oe 1 Delta ....

....types, and to introduce a separate operation of covering, that deals with the assignment of to subterms. 6.1 Relevant intersection type assignment The next definition presents a restricted variant of the essential system, that is similar to that of Definition 1.2. 3, and is the system used in [19]. Since bases play a more significant role in this system, the presentation of the derivation rules differs from the one used above. Definition 6.1.1 i) Relevant intersection type assignment and relevant intersection derivations are defined by the following natural deduction system (where all ....

P. Gardner. Discovering Needed Reductions Using Type Theory. In M. Hagiya and J.C. Mitchell, editors, Proceedings of TACS '94. International Symposium on Theoretical Aspects of Computer Software, Sendai, Japan, volume 789 of Lecture Notes in Computer Science, pages 555--597. Springer-Verlag, 1994.

.... a set at least one member of which is contracted in every normalizing reduction. A different approach to normalization in not necessarily orthogonal rewrite systems is developed in Kennaway [Ken89] and Antoy 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. Kennaway et al. KKSV95] studied a needed strategy for infinitary OTRSs. The contribution of this paper is to develop a theory of minimal reduction in the framework of relative normalization, and to ....

Gardner P. Discovering needed reductions using type theory. In: Proc. of TACS'94, Springer LNCS, v. 789, M. Hagiya, J. C. Mitchell, eds. Sendai, 1994, p. 555-574.

....and Ramakrishnan [SeRa93] study a normalizing strategy which in each multi step contracts a necessary set of redexes. A different approach to normalization is developed in Kennaway [Ken89] and 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 ....

Gardner P. Discovering needed reductions using type theory. In: Proc. of TACS'94, Springer LNCS, v. 789, M. Hagiya, J. C. Mitchell, eds. Sendai, 1994, p. 555-574.

....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 assignment system in the sense that using the principal type of a term one can find all the needed redexes in it (the principal type cannot be found efficiently, as one might expect) Antoy et al. AEH94] designed a needed ....

Gardner P. Discovering needed reductions using type theory. In: Proc. of the 2 nd International Symposium on Theoretical Aspects of Computer Software, TACS'94, Springer LNCS, v. 789, M. Hagiya, J. C. Mitchell, eds. Sendai, 1994, p. 555-574.

....fi reduction is well defined. 4.8 Lemma The relation L on well typed preterms satisfies the Church Rosser property. 4.9 Lemma Gamma L e : T and e = L e 0 implies Gamma L e 0 : T . The proofs of the above lemmas are technical but not difficult; details can be found in the full paper [8]. We let r : e L f denote the derivation of e L f such that r is the contracted redex 3 . Also let r 1 ; r n : e 0 L e 1 L : L e n denote the finite reduction path such that r i is the redex contracted in the reduction step e i Gamma1 L e i . Let ae : e L f denote an ....

....: x Tn ; x] The details for defining when a term is contained in another term are technical, but not difficult. It is therefore possible to prove that redex r i is contained in e[f 1 ; f n ; f=x T1 ; x Tn ; x] The details of this proof can be found in the full paper [8]. 2 5.2 Corollary Let r be an allowable redex in well typed preterm e, and let ae : e a ANF (e) Then a residual of redex r is used in ae. We now show that the leftmost allowable reduction is a needed reduction, using the result of Barendregt et al. 2] that the leftmost reduction is a needed ....

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P.A. Gardner. Discovering Needed Reductions Using Type Theory, full paper, in preparation, 1993.

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Gardner, P., Discovering Needed Reductions Using Type Theory, TACS, 1994.

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Gardner, P., Discovering Needed Reductions Using Type Theory, TACS, 1994.

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