Roel Bloo, Fairouz Kamareddine, and Rob Nederpelt. The Barendregt cube with definitions and generalised reduction. Information and Computation, 126(2):123--143, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....as we have said, ## uses a de Bruijn nameless notation of variables. 5 A di#erent solution proposed by Bloo in [2] is to introduce substitutions in contexts and to deal with these extended contexts via additional typing rules. This approach is similar to type systems with definitions [41, 3], where closures are typeable, but substitutions are not considered as typeable objects. We discuss this approach in the last section. When we consider annotated substitutions, the system may lose the subject reduction property due to the non left linear rule (SCons) 1[S] A (# S. For ....

....and reduction rules. In this way, the term (m (l x) x : 0] is ill typed, just as the term (#x:nat. m (l x) 0) is. The let in structure has a more complex behavior. It provides a mechanism for definitions in the language. Formal presentations of type systems with definitions are given in [41, 3]. Some type theories extended with explicit substitutions have been proposed: The Simple Type Theory [1, 27, 8, 21, 6] the Second Order Type Theory [1] the Martin Lof Type Theory [43] the Calculus of Constructions [39] and Pure Type Systems [2] Except for the simply typed version of ## in ....

R. Bloo, F. Kamareddine, and R. Nederpelt, The Barendregt cube with definitions and generalised reduction, Information and Computation, 126 (1996), pp. 123--143.

....is formulated for pure type systems, and applies therefore for arbitrary type systems as those of the cube. A higher order unification procedure for systems in the cube is a particular payoff. It is well known that definitions, i.e. let in expressions, are problematic in dependent type systems [24, 4]. Two approaches have been used to extend the calculus with definitions. Severi and Poll [24] consider definitions as terms and extend the reduction relation to unfold definitions during the typing process. Bloo et al. [4] do not extend the syntax of terms (although they use a different notation ....

....i.e. let in expressions, are problematic in dependent type systems [24, 4] Two approaches have been used to extend the calculus with definitions. Severi and Poll [24] consider definitions as terms and extend the reduction relation to unfold definitions during the typing process. Bloo et al. [4] do not extend the syntax of terms (although they use a different notation for terms called item notation) but they consider definitions as part This research is supported by NASA under award No NAG 2 1227, and DARPA under Contract No. F30602 96 C 0282. This research was supported by the ....

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R. Bloo, F. Kamareddine, and R. Nederpelt. The Barendregt cube with definitions and generalised reduction. Inf. and Computation, 126(2):123--143, 1 May 1996.

....As a consequence of the subject matter, admissibility proofs also give weak inner most normalisation (WIN) as a corollary. 5 The significance of this should not be underestimated as SR for some explicit substitution calculi has proven itself to be an unexpectedly hard result to come by, cf. [6, 7, 32]. We will not go into further detail but only mention that the culprit is dependent types and that they very much are part of our future work. 1.5 Related work Several papers have dealt with issues similar to what we are concerned with. They have done so in roughly speaking two different ways: ....

....by T we obtain: e k ( e 2 ) n e 1 ) 0 e) k 1 e 2 ) n e 1 ) This reduction is remarkable in that the abstractor has its body enlarged with e while making sure not to capture any free variables of e. Such a reduction bears some resemblance to so called generalised reduction [7]. A further study of this issue is left for future work. 5 (Derel) and Garbage Collection We will now take a closer look at the interaction between Cuts and (Derel) We will especially focus on occurrences of the (Derel) rule above which specific levels of the context contain place holders. More ....

Roel Bloo, Fairouz Kamareddine, and Rob Nederpelt. The Barendregt cube with definitions and generalised reduction. Information and Computation, 126(2):123--143, 1 May 1996.

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Bloo, R., Kamareddine, F., Nederpelt, R., The Barendregt Cube with definitions and generalised reduction, Computing Science Note 94/34, Eindhoven University of Technology, Department of Mathematics and Computing Science, 1994.

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R. Bloo, F. Kamareddine, and R. Nederpelt. The Barendregt Cube with Definitions and Generalised Reduction. To appear in Information and Computation, 1996.

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R. Bloo, F. Kamareddine, and R. Nederpelt. The Barendregt Cube with Definitions and Generalised Reduction. Information and Computation, 126(2):123--143, 1996.

....as in Figure 2. cffi) P x ) bffi) Q y ) dffi) R z ) affi) z Figure 2: Term reshuffling in item notation [BKN 9x] shows that ; fi (the reflexive transitive closure of ; fi ) is a generalisation of fi . We then show that with ; fi satisfies all the desirable typing properties. BKN 9y] extends the Barendregt cube with this generalised reduction and shows that all the above properties hold for this extension. Moreover, BKN 9x] shows that term reshuffling is correct. In particular, we show that accommodated with term reshuffling TS, satisfies the following: 1. Reshuffling a ....

....the expression. One of the advantages of the definition let x : A be a in b over ( x:A :b)a is that it is convenient to have the freedom of substituting only some of the occurrences of an expression in a given formula. Another advantage is that defining x to be a in b can be used to type b. BKN 9y] introduces definitions to Barendregt s cube and shows that Church Rosser, Subject Reduction, Unicity of Typing and Strong Normalisation all hold for this extension. ....

Bloo, R., Kamareddine, F. and Nederpelt, R., The Barendregt Cube with definitions and generalised reduction, submitted for publication.

....only separated by the redex ( x :N)P but by many redexes (ordinary and generalised) There are other reasons for using generalised reduction than those mentioned above. KN 95] showed that generalised reduction makes more redexes visible and hence allows for more flexibility in reducing a term. BKN 9y] showed that with generalised reduction one may indeed avoid size explosion without the cost of a longer reduction path and that calculus can be elegantly extended with definitions which result in shorter type derivation. All the research mentioned above is a living proof for the importance and ....

....is a living proof for the importance and usefulness of generalised reduction (from now on, fi e ) For this reason, properties of this reduction must be studied. Confluence of fi e is a direct consequence of the fact that M = fi N , M = fi e N . Subject reduction for fi e has been established in [BKN 9y] with the condition that explicit definitions must be added for some systems of the cube) Strong Normalisation of fi e has been established for the whole Cube (with or without definitions) in [BKN 9y] One important property however, the Preservation of Strong Normalisation (PSN) of fi e has ....

[Article contains additional citation context not shown here]

Bloo, R., Kamareddine, F., Nederpelt, R., The Barendregt Cube with Definitions and Generalised Reduction, Computing Science Note, University of Glasgow, Computing Science department, 1994. To appear in Information and Computation.

....Subject Reduction in and is proved in the following: Lemma 4.8 (Shuffle Lemma for and ) Gamma L s 1 (Affi)s 2 B : C ( Gamma L s 1 s 2 (Affi)B : C where s 2 is well balanced and the binding variables in s 2 are not free in A. Proof: For a detailed proof, the reader is referred to [BKN 94y] Informally, the reason for this lemma to be true for and is that in these systems, for any legal term of the form (P Pi x )Q, x = 2 FV (Q) this is not true for the other systems of the cube because of the mixing of levels that comes with the rules ( 2) and (2; Therefore none of the ....

....92] to show Strong Normalisation for with extended reduction. Here we shall extend the flexible proof of [Geuvers 94] We do not give the full details, but we only give a rough outline of the adaptations that had to be made to the proof in [Geuvers 94] For details, the reader is referred to [BKN 94y] The proof holds for any relation of Section 3, for and any reduction relation which is CR, contains fi and is such that the least equivalence relation closed under is the same as = fi . Lemma 6.3 (Soundness of ) If A; B 2 T are legal terms such that A = fi B then there is a path ....

Bloo, R., Kamareddine, F., Nederpelt, R. (1994), The Barendregt Cube with Definitions and Generalised Reduction, Computing Science Note 94/34, Eindhoven University of Technology, Department of Mathematics and Computing Science.

....2. By reshuffling terms so that matching ffi s and s occur adjacently. Hence Figure 1 will be redrawn as in Figure 2. 26] shows that ; fi (the reflexive transitive closure of ; fi ) is a generalisation of fi . We then show that with ; fi satisfies all the desirable typing properties. [27] extends the Barendregt cube with this generalised reduction and shows that all the above properties hold for this extension. Moreover, 26] shows that term reshuffling is correct. In particular, we show that accommodated with term reshuffling TS, satisfies the following: cffi) P x ) bffi) ....

....about the expression. One of the advantages of the definition let x : A be a in b over ( x:A :b)a is that it is convenient to have the freedom of substituting only some of the occurrences of an expression in a given formula. Another advantage is that defining x to be a in b can be used to type b. [27] introduces definitions to Barendregt s cube and shows that Church Rosser, Subject Reduction, Unicity of Typing and Strong Normalisation all hold for this extension. Acknowledgements I am grateful for the useful comments of Richard Oehrle and Aarne Ranta. ....

Bloo, R., Kamareddine, F. and Nederpelt, R., The Barendregt Cube with definitions and generalised reduction, submitted for publication.

....calculus rather than to Category Theory. In fact, we believe that as and ffi are operators of the calculus whose behaviour is well understood, oe, and should also be treated similarly. This approach of treating the calculus via items has proven advantageous in our various extensions as in [BKN 95] KN 95] and [KN 96b] KN 93] provides an account of explicit substitution which is used to discuss local and global substitution and reduction. No semantics is provided for that account and the precision of this paper is not assumed there. The reduction rules however of the present paper are ....

Bloo, R., Kamareddine, F., Nederpelt, R., The Barendregt Cube with Definitions and Generalised Reduction, to appear in Information and Computation.

....free calculus in this section and said that our resulting reduction is Church Rosser (CR) One might ask what will happen if we use this extended reduction in type systems. In other words, if we extend the cube of Section 1 with this reduction, do we get al..l the original properties of the cube In [3], we studied the cube with this general reduction and we obtained that all the properties of the cube including Strong Normalisation SN, except Subject Reduction SR, still hold with this general reduction. We did find however that if definitions are also added to the cube, then SR holds. The ....

R. Bloo, F. Kamareddine and R. Nederpelt, The Barendregt Cube with Definitions and Generalised Reduction, submitted for publication.

.... calculus rather than to Category Theory. In fact, we believe that as and ffi are operators of the calculus whose behaviour is well understood, oe, and should also be treated similarly. This approach of treating the calculus via items has proven advantageous in our various extensions as in [6,15,17]. 13] provides an account of explicit substitution which is used to discuss local and global substitution and reduction. No semantics is provided for that account and the precision of this paper is not assumed there. The reduction rules however of the present paper are based on [13] even though ....

C.J. Bloo, F. Kamareddine and R. Nederpelt, The Barendregt Cube with Definitions and Generalised Reduction, Information and Computation 126(2),:123--143, (1996) 25 123--143.

....we explained the third redex of A, we said that two definitions were unfolded in y:ff :fxy. It turns out that this observation is necessary in order to show that the cube extended with term reshuffling and ; fi satisfies Subject Reduction. But then definitions are important on their own (see [BKN 9y] Con 86] Dow 91] KBN 9 ] NGV 94] and [SP 93] We show that the cube extended with TS, fi and definitions, preserves its original properties including Strong Normalisation and Subject Reduction and that term reshuffling preserves typing. 2 The item notation and the formal machinery I ....

....relation generated by (B 1 ffi)s(B 2 x )B 3 , fi s(B 3 [x : B 1 ] for s well balanced, that is, fi reduction contracts an (extended) redex. fi is the reflexive and transitive closure of , fi and fi be the least equivalence relation closed under , fi . fi has been used in [BKN 9y] and [KN 95] We will use [BKN 9y] to obtain Strong Normalisation for the present paper. Example 4.2 Let A j (zffi) wffi) u x ) x y )y. Then [A] fA; wffi) u x ) z ffi) x y )yg. Moreover, A ; fi (wffi) u x )z and A ; fi (z ffi) w y )y. Lemma 4.3 TS(A) fi B iff TS(A) fi B. Proof: This is ....

[Article contains additional citation context not shown here]

Bloo, R., Kamareddine, F., Nederpelt, R., The Barendregt Cube with Definitions and Generalised Reduction, Computing Science Note, University of Glasgow, Computing Science department, 1994. To appear in Information and Computation.

.... in reducing a term [KN95] Bloo, Kamareddine, and Nederpelt show that with generalized reduction, one may indeed avoid size explosion without the cost of a longer reduction path; and, simultaneously, the calculus can be elegantly extended with definitions that result in shorter type derivations [BKN96]. Generalized reduction is strongly normalizing [BKN96] for all systems of the cube [Bar92] and preserves the strong normalization of ordinary fi reduction [Kam96] In particular, generalized reduction allows the postponement of K reductions (which discard their argument) after I reductions ....

.... show that with generalized reduction, one may indeed avoid size explosion without the cost of a longer reduction path; and, simultaneously, the calculus can be elegantly extended with definitions that result in shorter type derivations [BKN96] Generalized reduction is strongly normalizing [BKN96] for all systems of the cube [Bar92] and preserves the strong normalization of ordinary fi reduction [Kam96] In particular, generalized reduction allows the postponement of K reductions (which discard their argument) after I reductions (which use their argument in at least one place) An ....

R. Bloo, F. Kamareddine, and R. Nederpelt. The Barendregt cube with definitions and generalised reduction. Information & Computation, 126:123--143, May 1996.

....let expressions one can avoid such an explosion in complexity. This is, by the way, a very natural thing to do: the apparatus of mathematics, for instance, is unimaginable without a form of let expressions (viz. definitions) There exist already two formal studies of let expressions in the cube [BKN 96, SP 93] where those let expressions are called definitions. In this paper we differ from both accounts and use the simplest way of renaming large expressions and describe such renaming as abbreviations. We differ from [SP 93] in that we do not introduce new terms (let terms) into our syntax and ....

....we differ from both accounts and use the simplest way of renaming large expressions and describe such renaming as abbreviations. We differ from [SP 93] in that we do not introduce new terms (let terms) into our syntax and do not extend fi reduction to deal with those new terms. We differ from [BKN 96] in that we do not use nested definitions, which are needed for generalised reduction in [BKN 96] and not for Pi reduction. We write hx : Aia to describe that x of type A, abbreviates a. We include abbreviations in contexts such that if an abbreviation occurs in a context then it can be used ....

[Article contains additional citation context not shown here]

R. Bloo, F. Kamareddine, and R.P. Nederpelt, The Barendregt Cube with Definitions and Generalised Reduction, Information and Computation 126(2), 123-143, 1996.

....Formalisms 27 (cffi) P x ) bffi) Q y ) dffi) R z ) affi) z Figure 2: Term reshuffling in item notation [BKN 9x] shows that ; fi (the reflexive transitive closure of ; fi ) is a generalisation of fi . We then show that with ; fi satisfies all the desirable typing properties. BKN 9y] extends the Barendregt cube with this generalised reduction and shows that all the above properties hold for this extension. Moreover, BKN 9x] shows that term reshuffling is correct. In particular, we show that accommodated with term reshuffling TS, satisfies the following: 1. Reshuffling a ....

....definition let x : A be a in b over ( x:A :b)a is that it is convenient to have the freedom of substituting only some of the occurrences of an expression in a given Kamareddine April 7, Foundational Formalisms 28 formula. Another advantage is that defining x to be a in b can be used to type b. BKN 9y] introduces definitions to Barendregt s cube and shows that Church Rosser, Subject Reduction, Unicity of Typing and Strong Normalisation all hold for this extension. ....

Bloo, R., Kamareddine, F. and Nederpelt, R., The Barendregt Cube with definitions and generalised reduction, submitted for publication.

....only separated by the redex ( x :N)P but by many redexes (ordinary and generalised) There are other reasons for using generalised reduction than those mentioned above. KN 95] showed that generalised reduction makes more redexes visible and hence allows for more flexibility in reducing a term. BKN 9y] showed that with generalised reduction one may indeed avoid size explosion without the cost of a longer reduction path and that calculus can be elegantly extended with definitions which result in shorter type derivation. All the research mentioned above is a living proof for the importance and ....

....is a living proof for the importance and usefulness of generalised reduction (from now on, fi e ) For this reason, properties of this reduction must be studied. Confluence of fi e is a direct consequence of the fact that M = fi N , M = fi e N . Subject reduction for fi e has been established in [BKN 9y] with the condition that explicit definitions must be added for some systems of the cube) Strong Normalisation of fi e has been established for the whole Cube (with or without definitions) in [BKN 9y] One important property however, the Preservation of Strong Normalisation (PSN) of fi e has ....

[Article contains additional citation context not shown here]

Bloo, R., Kamareddine, F., Nederpelt, R., The Barendregt Cube with Definitions and Generalised Reduction, Computing Science Note, University of Glasgow, Computing Science department, 1994. To appear in Information and Computation.

....calculus rather than to Category Theory. In fact, we believe that as and ffi are operators of the calculus whose behaviour is well understood, oe, and should also be treated similarly. This approach of treating the calculus via items has proven advantageous in our various extensions as in [BKN 95] KN 95] and [KN 96b] KN 93] provides an account of explicit substitution which is used to discuss local and global substitution and reduction. No semantics is provided for that account and the precision of this paper is not assumed there. The reduction rules however of the present paper are ....

Bloo, R., Kamareddine, F., Nederpelt, R., The Barendregt Cube with Definitions and Generalised Reduction, to appear in Information and Computation.

.... in reducing a term [21] Bloo, Kamareddine, and Nederpelt show that with generalised reduction one may indeed avoid size explosion without the cost of a longer reduction path and that simultaneously the calculus can be elegantly extended with definitions which result in shorter type derivations [7]. Generalised reduction is strongly normalising [7] for all systems of the cube [4] and preserves the strong normalisation of ordinary fi reduction [19] In particular, generalized reduction allows the postponement of K reductions (which discard their argument) after I reductions (which use their ....

.... show that with generalised reduction one may indeed avoid size explosion without the cost of a longer reduction path and that simultaneously the calculus can be elegantly extended with definitions which result in shorter type derivations [7] Generalised reduction is strongly normalising [7] for all systems of the cube [4] and preserves the strong normalisation of ordinary fi reduction [19] In particular, generalized reduction allows the postponement of K reductions (which discard their argument) after I reductions (which use their argument in at least one place) An alternative ....

R. Bloo, F. Kamareddine, and R. Nederpelt. The Barendregt cube with definitions and generalised reduction. Inf. & Comput., 126(2):123--143, May 1996.

....directly corresponds to a formal calculus. 19] uses a more extended version of where Q and N are not only separated by the redex ( x :N)P but by many redexes (ordinary and generalised) 19] shows that generalised reduction makes more redexes visible allowing flexibility in reducing a term. [6] shows that with generalised reduction one may indeed avoid size explosion without the cost of a longer reduction path and that calculus can be elegantly extended with definitions which result in shorter type derivations. Generalised reduction is strongly normalising (cf. 6] for all systems of ....

....in reducing a term. 6] shows that with generalised reduction one may indeed avoid size explosion without the cost of a longer reduction path and that calculus can be elegantly extended with definitions which result in shorter type derivations. Generalised reduction is strongly normalising (cf. [6]) for all systems of the cube (cf. 3] and preserves strong normalisation of classical reduction (cf. 16] 1.2 The calculus with explicit substitution Functional programming and in particular partial evaluation may benefit from explicit substitution. For example, given xx[x : y] we may not ....

R. Bloo, F. Kamareddine, and R. Nederpelt. The Barendregt Cube with Definitions and Generalised Reduction. To appear in Information and Computation, 1996.

....one can avoid such an explosion in complexity. This is, by the way, a very natural thing to do: the apparatus of mathematics, for instance, is unimaginable without definitions. Introducing definitions in Pure Type Systems is an interesting subject of research at the moment (see [SP 93] and [BKN 9y] Furthermore, BKN 9y] has shown that the generated type derivations for terms in the Cube with definitions become much shorter than those in the absence of definitions. Our approach in this paper is to introduce definitions as redexes where the body is not written. For example, x:A : Gamma)a ....

....an explosion in complexity. This is, by the way, a very natural thing to do: the apparatus of mathematics, for instance, is unimaginable without definitions. Introducing definitions in Pure Type Systems is an interesting subject of research at the moment (see [SP 93] and [BKN 9y] Furthermore, BKN 9y] has shown that the generated type derivations for terms in the Cube with definitions become much shorter than those in the absence of definitions. Our approach in this paper is to introduce definitions as redexes where the body is not written. For example, x:A : Gamma)a defines x to be a. ....

[Article contains additional citation context not shown here]

Bloo, R., Kamareddine, F., Nederpelt, R., The Barendregt Cube with Definitions and Generalised Reduction, Computing Science Note 94/08, University of Glasgow, Computing Science department, 1994.

....we explained the third redex of A, we said that two definitions were unfolded in y:ff :fxy. It turns out that this observation is necessary in order to show that the cube extended with term reshuffling and ; fi satisfies Subject Reduction. But then definitions are important on their own (see [BKN 9y] Con 86] Dow 91] KBN 9 ] NGV 94] and [SP 93] We show that the cube extended with TS, fi and definitions, preserves its original properties including Strong Normalisation and Subject Reduction and that term reshuffling preserves typing. 2 The item notation and the formal machinery I ....

....relation generated by (B 1 ffi)s(B 2 x )B 3 , fi s(B 3 [x : B 1 ] for s well balanced, that is, fi reduction contracts an (extended) redex. fi is the reflexive and transitive closure of , fi and fi be the least equivalence relation closed under , fi . fi has been used in [BKN 9y] and [KN 95] We will use [BKN 9y] to obtain Strong Normalisation for the present paper. Example 4.2 Let A j (zffi) wffi) u x ) x y )y. Then [A] fA; wffi) u x ) z ffi) x y )yg. Moreover, A ; fi (wffi) u x )z and A ; fi (z ffi) w y )y. Lemma 4.3 TS(A) fi B iff TS(A) fi B. Proof: This is ....

[Article contains additional citation context not shown here]

Bloo, R., Kamareddine, F., Nederpelt, R., The Barendregt Cube with Definitions and Generalised Reduction, Computing Science Note, University of Glasgow, Computing Science department, 1994. To appear in Information and Computation.

....directly corresponds to a formal calculus. 16] uses a more extended version of where Q and N are not only separated by the redex ( x :N)P but by many redexes (ordinary and generalised) 16] shows that generalised reduction makes more redexes visible allowing flexibility in reducing a term. [6] shows that with generalised reduction one may indeed avoid size explosion without the cost of a longer reduction path and that calculus can be elegantly extended with definitions which result in shorter type derivations. Generalised reduction is strongly normalising (cf. 6] for all systems of ....

....in reducing a term. 6] shows that with generalised reduction one may indeed avoid size explosion without the cost of a longer reduction path and that calculus can be elegantly extended with definitions which result in shorter type derivations. Generalised reduction is strongly normalising (cf. [6]) for all systems of the cube (cf. 3] and preserves strong normalisation of classical reduction (cf. 13] 1.2 The calculus with explicit substitution Functional programming and in particular partial evaluation may benefit from explicit substitution. For example, given xx[x : y] we may not ....

R. Bloo, F. Kamareddine, and R. Nederpelt. The Barendregt Cube with Definitions and Generalised Reduction. Information and Computation, 126 (2):123--143, 1996.

No context found.

Roel Bloo, Fairouz Kamareddine, and Rob Nederpelt. The Barendregt cube with definitions and generalised reduction. Information and Computation, 126(2):123--143, 1996.

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