Jean-Yves Girard, Yves Lafont, and Paul Taylor. Proofs and Types, volume 7 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, Cambridge, 1989. (pp 67, 98)  Home/Search   Document Not in Database   Summary   Related Articles   Check

....j j additive disjunction and false 3.2 A sequent system for LP A sequent style presentation of the logic is given in Table 1. With one exception, the rules in Table 1 are the natural generalizations of the rules given by Girard for the commutative intuitionistic linear logic, cf. [13, 14], to the non commutative case, cf [5] The exceptional axiom is ( Its expected generalization is ; A, as in [5] However, the stronger axiom is not consistent with our earlier discussion. For instance, ldd = yes; which is equivalent to ldd = yes by (L ) would then be equivalent ....

Girard, J.-Y., Lafont, Y. and P. Taylor. Proofs and Types. volume 7 of Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, 1989.

....(R# ) #, #, #, # (R#) L L) L R) #,# # # #, #, # (R# L) R# R) Table 1. A sequent system for LP # . are the natural generalizations of the rules given by Girard for the commutative intuitionistic linear logic, cf. [14, 15], to the non commutative case, cf [6] The exceptional axiom is (#) Its expected generalization is #, A, as in [6] However, the stronger axiom is not consistent with our earlier discussion, and axiom (16) in particular. For instance ldd = yes would then be equivalent to rather than ....

Girard, J.-Y., Lafont, Y. and P. Taylor. Proofs and Types. volume 7 of Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, 1989.

....relatively sophisticated material, because this is more exciting for the reader. Thus, we have assumed that the reader has a certain familiarity with logic and the lambda calculus. If the reader does not feel suciently comfortable with these topics, we suggest consulting Girard, Lafont, Taylor [9] or Gallier [6] for background on logic, and Barendregt [2] Hindley and Seldin [15] or Krivine [19] for background on the lambda calculus. For an in depth study of constructivism in mathematics, we highly recommend Troelstra and van Dalen [32] 2 Natural Deduction, Simply Typed Calculus We ....

.... Actually, in order to salvage some form of subformula property ruined by the introduction of the connectives , 9, and , one can add further conversions known as commuting conversions (or permutative conversions ) A lucid discussion of the necessity for such rules can be found in Girard [9]. Theorem 7.4 and theorem 7.5 can be extended to cover the reduction rules of de nition 7.3 together with the new reductions rules, but at the cost of rather tedious and rather noninstructive technical complications. Due to the lack of space, we will not elaborate any further on this subject and ....

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J.-Y. Girard, Y. Lafont, and P. Taylor. Proofs and Types, volume 7 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989.

....on the length of the list r, analogous to the proof of Lemma 2.1. Induction on rx 2 SN would show the converse direction. 5 Extension to permutative conversions We proceed to a typed calculus with sums and permutative conversions similar to those being studied in [Pra71] Lei75] or [GLT89]. In contrast to their expositions we do not consider product types or other rst order extensions, although the proof method copes with them, too. 5.1. Raw) terms. Let al.ways i 2 f0; 1g. r; s; t : x j yr j rs j inj i r j s(x 0 :t 0 ; x 1 :t 1 ) The variables y; x 0 ; x 1 get bound in r; t ....

Jean-Yves Girard, Yves Lafont, and Paul Taylor. Proofs and Types, volume 7 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989.

....of the di#erent flavours of polymorphism is given by Cardelli and Wegner [CW85] In mathematics (and mathematical branches of computer science) polymorphism is mostly studied in the form of polymorphically typed # calculus. This extension of simply typed # calculus was introduced by Girard [GLT89], who termed it System F. Girard used the system to prove cut elimination for second order Peano arithmetic via a functional interpretation. However, the form of polymorphism used in System F is not without drawbacks. It is impredicative in the sense, that one may define types by referring to the ....

....great times we have together. The systems # T In the following we define three systems of polymorphically typed # calculus. First we define the base system # of predicative polymorphism as given by Mitchell [Mit90, Mit96] is a fragment of System F , as presented for example by Girard [GLT89]. It features a restricted form of polymorphism, which is achieved by splitting the types into two universes: the universe U 1 of small types and the universe U 2 of large types. The crucial feature is, that variables in type expressions are taken to range over the small types only. We then ....

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J.-Y. Girard, Y. Lafont, and P. Taylor. Proofs and Types, volume 7 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989.

....produced, it is rather daunting to show that composing them with their inverse yields the identity function. A normalization tool that handles sums is needed. In the presence of sums, however, normalization is known to be a non trivial a#air [1] chiefly because of commuting conversions [27]. Type directed partial evaluation does handle sums, but the redundancy pointed out in Section 1.3 is a major impediment. In ML s type language, the type constructors for products and functions are infix, and the type constructor for sums is postfix. a ( b ( c, d) sum, b ( d ....

Jean-Yves Girard, Yves Lafont, and Paul Taylor. Proofs and Types, volume 7 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989.

....write , the corresponding one step rewrite relations; if is a rewrite relation, we write its reflexive transitive closure, its transitive closure. We write , for the appropriate congruences. The relations , terminate on simply typed terms [14]. Moreover, any ( normal preterm if of the form x 1 ; x n ht 1 : t m , where the head h is a constant, an existential variable or one of x 1 , x n , and t 1 , t m are ( normal. If h is an existential variable, then x 1 ; x n ht 1 : t m is ....

J.-Y. Girard, Y. Lafont, and P. Taylor. Proofs and Types, volume 7 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989.

.... to the syntactic device of cut elimination: let LK S denote Gentzen s sequent calculus for classical logic augmented with non logical axioms taken from S (clauses being read as sequents) then eliminating cuts from any (LK S) proof of the empty sequent yields one where the only rule is (Cut) [6, 7]. That resolution is complete will be assumed in the sequel. To show that some refinement of resolution is complete, we only need to rewrite any given refutation of S into a refutation of S that obeys the constraints of the refinement. We shall do this by using rewrite rules ( 1) 2) below) that ....

J.-Y. Girard, Y. Lafont, and P. Taylor. Proofs and Types, volume 7 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989.

....how categorical models of typed calculus can be used to provide not just an equational theory for typed terms but also rewrite relations which form the basis of a decision procedure for this equality. There are several standard references: for the untyped calculus one may consult [4,38] while [32,55] are good references for typed calculus. 2.1 Untyped Calculi The untyped calculus was first introduced by Church in the 1930s and has been widely studied by logicians and computer scientists as it is a particularly simple yet elegant formalism capable of representing exactly those functions ....

....otherwise stated, henceforth we shall restrict ourselves to the more limited, but better behaved typed calculi. Identity, Structural and Logical Rules The traditional presentation of typed calculi as a unified system masks the fact that several separate processes are involved we follow [32] in identifying three different processes. The first type of inference rules are the identity rules and consist of axiom and cut: x : A x : A (2.2) where Var is the set of variables and cut requires Gamma and Delta to be disjoint to ensure the context in the conclusion is well defined. ....

[Article contains additional citation context not shown here]

J. Y. Girard, P. Taylor, and Y. Lafont. Proofs and Types, volume 7 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989.

....on the length of the list r, analogous to the proof of Lemma 2.1. Induction on rx 2 SN would show the converse direction. 5 Extension to permutative conversions We proceed to a typed calculus with sums and permutative conversions similar to those being studied in [Pra71] Lei75] or [GLT89]. In contrast to their expositions we do not consider product types or other rst order extensions, although the proof method copes with them, too. 5.1. Raw) terms. Let al.ways i 2 f0; 1g. r; s; t : x j yr j rs j inj i r j s(x 0 :t 0 ; x 1 :t 1 ) The variables y; x 0 ; x 1 get bound in r; t ....

Jean-Yves Girard, Yves Lafont, and Paul Taylor. Proofs and Types, volume 7 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989.

....rule: Gamma S U Gamma U T By analogy with proof theory, this rule is sometimes called the cut rule of the subtyping system: the type U appearing in the subderivations is cut out when moving to the conclusion. By analogy with the sequent calculus or the simply typed calculus (c.f. GLT89] this cut rule can be almost completely eliminated by rewriting derivations. But not completely. In one situation, transitivity is actually essential. Statements with variables on the left hand side cannot, in general, be proved without using transitivity. For example, CTop( BC; AB A C ....

Jean-Yves Girard, Yves Lafont, and Paul Taylor. Proofs and Types, volume 7 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, Cambridge, 1989.

....F and several extensions. They are typed calculi where every term has its type, no type assignment system is set up and the presentation of the term rules strictly follows the idea of natural deduction proof systems. 2. 1 System F The version of system F we will use is in essence the same as in [5]. Types: We have infinitely many type variables (denoted by ff, fi, and with types ae and oe we also have the product type ae Theta oe and the function type ae oe. Moreover, given a variable ff and a type ae we form the universal type 8ffae. The quantifier 8 binds ff in ae. The ....

....In order to better display the construction for MC we define an embedding into the system NPFex which is NPF enriched with existential types. The standard encoding of existential types in system F with essential clause (9ffae) 8fi: 8ff:ae fi) fi; fi = 2 fffg [ FV(ae) see [5] p. 86 for the encoding and [9] for a careful proof in the style of section 3.1 that this indeed gives an embedding) trivially carries over to NPF (but not to NPI or NPC because formation of map ffae and the encoding of the existential do not commute ) Therefore, also NPFex embeds into NPF. We ....

Jean-Yves Girard, Yves Lafont, and Paul Taylor. Proofs and Types, volume 7 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989.

....for the basic case of functional programming languages, span a broad spectrum in terms of expressive power. Thus, ML style types [31] are relatively weak as regards expressing behavioural constraints, but correspondingly tractable as regards efficient algorithms for type checking . System F types [21] are considerably more expressive of polymorphic behaviour, and System F typing guarantees Strong Normalization. However, System F cannot express the fact that a program of type list[nat] list[nat] is actually a sorting function. Martin Lof type theory, with dependent types and equality types, ....

.... Rules of this kind are known to be difficult to obtain [17] ffl The concepts and techniques used in defining this specification structure and verifying that it has the required properties represent a striking transfer of techniques from Proof Theory (Tait Girard proofs of Strong Normalization [21]) to concurrency. This is made possible by our framework of interaction categories and specification structures. We begin with some preliminary definitions. Firstly we define a binary process combinator p u q by the transition rule q p u q u q Note that p u q is the meet of p ....

J.-Y. Girard, Y. Lafont, and P. Taylor. Proofs and Types, volume 7 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989.

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Jean-Yves Girard, Yves Lafont, and Paul Taylor. Proofs and Types, volume 7 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, Cambridge, 1989. (pp 67, 98)

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Jean-Yves Girard, Yves Lafont, and Paul Taylor. Proofs and Types, volume 7 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, Cambridge, 1989.

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J.-Y. Girard, P. Taylor, and Y. Lafont. Proofs and types, volume 7 of Cambridge tracts in theoretical computer science. Cambridge University Press, 1989.

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Jean-Yves Girard, Yves Lafont, and P. Taylor. Proofs and types, volume 7 of Cambridge tracts in theoretical computer science. Cambridge University Press, 1989.

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J.-Y. Girard, Yves Lafont, and Paul Taylor. Proofs and Types, volume 7 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989.

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J. Y. Girard, Y. Lafont, and D. Taylor, Proofs and Types, Volume 7 of Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, 1989.

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Jean-Yves Girard, Yves Lafont, and Paul Taylor. Proofs and Types, volume 7 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989.

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Jean-Yves Girard, Yves Lafont, and Paul Taylor. Proofs and Types, volume 7 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, Cambridge, 1989.

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Jean-Yves Girard, Yves Lafont, and Paul Taylor. Proofs and Types, volume 7 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, Cambridge, 1989. (pp 67, 98)

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J.-Y. Girard, Y. Lafont, and P. Taylor. Proofs and Types, volume 7 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989.

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Jean-Yves Girard. Proofs and types, volume 7 of Cambridge tracts in theoretical computer science. Cambridge University Press, 1989.

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Jean-Yves Girard, Yves Lafont, and P. Taylor. Proofs and types, volume 7 of Cambridge tracts in theoretical computer science. Cambridge University Press, 1989. Cited on pages 8, 31, 32, 33, 34, 37, and 43.

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