H. Barendregt, calculi with types". In Handbook of Logic in Computer Science, vol.2, Abramsky, Gabbay, and Maibaum, eds., Oxford University Press, pp. 118-309, 1992 18  Home/Search   Document Not in Database   Summary   Related Articles   Check

....and SN are equivalent and that all semi reductionally equivalent terms have the same normalisation behaviour. This counterexample will be better understood if it is translated into the item notation of Section 2. In Section 6 we introduce the famous cube of eight in uential type systems [2] and explain how the concepts and results of the previous sections change in the typed framework. We then extend the cube with equivalence and show that subject reduction fails. In Section 7 we extend the cube with class reduction and show that subject reduction fails for six systems but ....

....We also explain how de nitions solve the problem of subject reduction for the extension of the cube with the reduction based on equivalence given in Section 6. 2 Some formal machinery We assume familiarity with the calculus and its important notions such as compatibility and reduction (see [2]) Bound and free variables and substitution are de ned as usual. We write BV (A) and FV (A) to represent the bound and free variables of A respectively. We write A[x : B] to denote the term where all the free occurrences of x in A have been replaced by B. We take terms to be equivalent up to ....

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H.P. Barendregt. -calculi with types. In S. Abramsky, D. Gabbay, and T. Maibaum, editors, Handbook of Logic in Computer Science, volume II, pages 118-310. Oxford University Press, 1992.

....Laan Weerdstede 45, 3431 LS Nieuwegein, The Netherlands Rob Nederpelt Mathematics and Computing Science, Eindhoven Univ. of Technology, P.O. Box 513, 5600 MB Eindhoven, the Netherlands Abstract We study the position of the Automath systems within the framework of Pure Type Systems (PTSs) In [1,15], a rough relationship has been given between Automath and PTSs. That relationship ignores three of the most important features of Automath: de nitions, parameters and reduction, because at the time, PTSs did not have these features. Since, PTSs have been extended with these features and in ....

....a name to a certain expression making it easy to remember what the use of the de niens is. As Automath was developed independently from other developments in the world of type theory and calculus, and as it invented powerful typing ideas that were later adopted in in uential type systems (cf. [1]) there are many things to be explained in (and learned from) the relation between the various Automath languages and other type theories. Type theory was originally invented by Bertrand Russell to exclude the paradoxes that arose from Frege s Begri schrift [14] It was presented in 1910 in the ....

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H.P. Barendregt. -calculi with types. In Handbook of Logic in Computer Science, pages 117-309. OUP, 1992.

.... Bruijn s Automath and Pure Type Systems Fairouz Kamareddine Twan Laan Rob Nederpelt Abstract We study the position of the Automath systems within the framework of Pure Type Systems (PTSs) In [2, 22], a rough relationship has been given between Automath and PTSs. That relationship ignores three of the most important features of Automath: de nitions, parameters and reduction, because at the time, formulations of PTSs did not have these features. Since, PTSs have been extended with these ....

....a name to a certain expression making it easy to remember what the use of the de niens is. As Automath was developed independently from other developments in the world of type theory and calculus, and as it invented powerful typing ideas that were later adopted in in uential type systems (cf. [2]) there are many things to be explained in (and learned from) the relation between the various Automath languages and other type theories. Type theory was originally invented by Bertrand Russell to exclude the paradoxes that arose from Frege s The rst practical use of the propositions as types ....

[Article contains additional citation context not shown here]

H.P. Barendregt. -calculi with types. In Handbook of Logic in Computer Science, pages 117-309. OUP, 1992. In
, de Bruijn favours trees over character strings and does not make use of AT-couples.

....by B. We take terms to be equivalent up to variable renaming and use to denote syntactical equality of terms. We assume the usual Barendregt variable convention BC (which says that bound variables are always chosen distinct from free variables) and the usual de nition of compatibility (cf. [2]) We say that A is strongly normalizing with respect to a reduction relation (written SN (A) i every reduction path starting at A terminates. 3. TOWARDS CANONICAL FORMS 3.1 Making redexes visible via Transformations like ( are rather powerful in that they can group together terms ....

....THE CUBE WITH CLASS REDUCTION Our study of class reduction has been discussed up to now for the type free calculus. But, for such reduction to be useful in practice, we need to study it within type theory. Alas, when attempting to build class reduction on the systems of the Barendregt cube of [2], we nd that the subject reduction property which states that if A B then B has the same type as A, no longer holds for six of the systems of the cube, although it holds for the systems and . This problem however can be solved by extending the cube not only with class reduction, but also ....

[Article contains additional citation context not shown here]

H.P. Barendregt. -calculi with types. In S. Abramsky, D. Gabbay, and T. Maibaum, editors, Handbook of Logic in Computer Science, volume II, pages 118-310. Oxford University Press, 1992.

....theorem as: There are no free constants in a book with appropriate preface. 11 A de ning occurrence of a constant c is an introduction of c in a de nition of the form: B; c(x1 ; xn) 10 Next, we reformulate a number of properties which are important in Type Theory (see [1]) for our WTTcase. We write : W for: formula has weak type W. As usual, FV ( stands for the set of free variables in . Proofs of the lemmas and theorems below can be found in the appendix. Lemma 311 (Free Variables lemma) 1) If B : cont, then the declared variables in are distinct. ....

H.P. Barendregt. -calculi with types. In S. Abramsky, D. Gabbay, and T. Maibaum, editors, Handbook of Logic in Computer Science, volume II, pages 118-310. Oxford University Press, 1992.

....[ Fig. 1. Propositions as types as objects The triangle provides a framework within which the proof theory, model theory and computational interpretation of systems can be formulated. For example: Intuitionistic logic: Here the correspondence between propositions and types [12,1] is particularly strong. At the propositional level, one obtains a close proof theoretic analysis via the simply typed calculus which extend to the predicate level via dependently typed calculi. Semantically, Kripke s possible worlds models of propositional consequence can be generalized to ....

Barendregt, H., -calculi with types, in \Handbook of Logic in Computer Science, Volume 2", Abramsky, S., D.M. Gabbay and T.S.E. Maibaum, editors, Oxford University Press, 1992, 118-309.

....the syntactic analysis of programs written as terms. In order to conduct our syntactic study of reductional equivalence of programs written as terms, we need to use a notation that enables us to detect more redexes in a term than can be visible in the known classical notation calculus [3]. For this purpose, we will use the item notation whose advantages are discussed in [7] 6] explains that it is not feasible to syntactically describe generalised reduction in classical notation and therefore item notation is indispensable for our study of reductional equivalence which depends on ....

....write A[x : B] to denote the term where all the free occurrences of x in A have been replaced by B. We take terms to be equivalent up to variable renaming and use to denote syntactical equality of terms. We assume the usual Barendregt variable convention BC and de nition of compatibility (cf. [3]) We say that A is strongly normalizing with respect to a reduction relation (written SN (A) i every reduction path starting at A terminates. 4 Shu e equivalence In this section we follow [8] and rewrite terms so that all the newly visible redexes can be subject to . We shall show ....

H.P. Barendregt. -calculi with types. In S. Abramsky, D. Gabbay, and T. Maibaum, editors, Handbook of Logic in Computer Science, volume II, pages 118-310. Oxford University Press, 1992.

.... ffl terms depend on terms (simple typed calculus, sometimes written ) ffl terms depend on types (F system, polymorphism [GLT89, Kri90] ffl types depend on types (type constructors system, sometimes written ) The different systems are shown on figure 1 (Barendregt s cube [Bar91] Calculus of Constructions being noted C) 2 F s Types and Terms The syntax of F s terms is quite the same as for Calculus of Constructions [CH88] except we distinguish terms from their types. 2.1 Syntax Types : T : V j PiV : T:T j T ) T j V : T:T j TT j C Terms : T : V j V ....

H. P. Barendregt. -calculi with types. Technical Report 91-19, Catholic University Nijmegen, 1991. In Handbook of Logic in Computer Science, Vol II.

....extended to L and R by the addition of a symbol as below. Definition 3 If M 2 L and M j C[ x:N) P ] then M 2 L is created by replacing (x:N) by (x:N) Similarly R 2 R. The idea of using underlining to track redexes was developed from the proof of the Church Rosser Theorem given by Barendregt [Bar92] We assume that the definitions of substitution and reduction are extended in the obvious way to cater for underlined redexes. Underlining will be used to show a correspondence between reductions on L and R. When M N by the reduction of Delta, this can be lifted to M N by the underlining ....

Henk Barendregt. calculi with types. In S. Abramsky, D. M. Gabbay, and T. S. E. Maibaum, editors, Handbook of Logic in Computer Science, volume 2, chapter 1, pages 117--309. Oxford Science Publications, Oxford, UK, 1992.

....The question remains open for the time being. parsing, recognizing capacity. The undecidability result for polymorphic L provides additional motivation for the search for restricted forms of categorial polymorphism with more pleasant computational properties. For a discussion of the options, see [Barendregt 92] Emms [Emms 93a] has explored this issue in a discussion of parsing with L2. It turns out that for the actual linguistic uses of type schemata mentioned above, one can do with a very mild form of polymorphism where the universal quantifier is restricted to outermost position. Such a restricted ....

Barendregt, H. (1992), `-Calculi with Types'. In S. Abramsky, D.M. Gabbay, and T.E. Maibaum (eds.) Handbook of Logic in Computer Science. Vol 2, 117--309. Oxford.

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H. Barendregt, calculi with types". In Handbook of Logic in Computer Science, vol.2, Abramsky, Gabbay, and Maibaum, eds., Oxford University Press, pp. 118-309, 1992 18

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