Jean Gallier. On Girard's "candidats de reductibilit'e". In Odifreddi, editor, Logic and Computer Science, pages 123--203. Academic Press, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....naively represent causality incorporating bound names in (4) there is a circular chain a c b a, although this cycle never arises in actual interaction. How can we then prove termination Simple structural inductions would not be usable for the same reason they do not work in typed l calculi [8, 20]. The idea we use is suggested by SN proofs for typed l calculi, due to, among others, Tait [61] His method employs a semantic interpretation of each type [ s] as a collection of strongly normalising l terms, and shows that all typable terms are indeed in these sets. A key step is to prove that ....

.... that MfN=xg 2 [ t] and (lx:M)N MfN=xg imply (lx:M)N 2 [ t] hence we have only to show MfN=xg 2 [ t] To be able to do this we strengthen the induction hypothesis M 2 [ t] to M 2 [ t] r for each environment r, mapping each variable of type s to some term in [ s] Now the result is immediate [8, 20]. While we cannot use an identical framework due to the different nature of reduction in the p calculus, a similar technique works for the induction to go through . A key observation concerns the close correspondence between the substitution MfN=xg and the consumption of a message xhvi by a ....

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Gallier, J. H., On Girard's "Candidats de Reductibilite", 123--203, Logic and Computer Science, Academic Press Limited, 1990.

....naively represent causality incorporating bound names in (1) there is a circular chain a c b a, although this cycle never arises in actual interaction. How can we then prove termination Simple structural inductions would not be usable for the same reason they do not work in typed l calculi [7, 12]. The idea we use is suggested by SN proofs for typed l calculi, due to, among others, Tait [42] His method uses a semantic interpretation of each type [ s] as a collection of strongly normalising l terms, and shows that all typable terms are indeed in these sets. A key step is to prove, for ....

....w.r.t. the observational congruence of l ; justifying all commutative conversions and h equations [15] and automatically leads to SN in the source calculus. Related Work Strong normalisation in typed l calculi has been studied extensively in the past; detailed surveys can be found in [7, 12]. Abramsky extends the CurryHoward correspondence to linear logic [14] using proof expressions and proves SN [1] guiding our present usage of acyclicity in names. This programme is taken further with realisability semantics of linear logic in [5] where CCS processes act as realisers. The ....

Gallier, J. H., On Girard's "Candidats de Reductibilit e", 123--203, Logic and Computer Science, Academic Press Limited, 1990.

....that a terminating CBV evaluation sequence starting from a program terminates with a CBV value of type # CBV evaluation does not get stuck . The restriction of CBV evaluation to pure programs is a particular # reduction strategy. It follows from the strong normalization property of F# [14, 13] that CBV evaluation of pure programs terminates. Termination of CBV evaluation for full F# control will be established in Section 5. The following property of CBV evaluation will be important to that argument. Lemma 3.3 Any infinite CBV evaluation sequence starting from a program contains ....

Jean Gallier. On girard's "candidats de reductibilit e". In P. Odifreddi, editor, Logic and Computation, volume 31 of The APIC Series, pages 123--203. Academic Press, 1990.

....if Gamma S 2 K in F , then F ( Gamma) F (S) 2 K in F . Now, any fi reduction in F from S can be mirrored by a fi reduction from F (S) of the same length in F . The existence of an infinite fi reduction in F would thus contradict the strong normalization of F [Gir72, Gal90] Next, we define the notion of Gamma reduction and establish some of its basic properties. 10 5.3 Definition: Single step Gamma reduction is the least family of relations closed under: Gamma(A) 6= Top(K) Gamma) A Gamma Gamma(A) S ( Gamma; ATop(K) S Fun(A:K)S Gamma ....

Jean H. Gallier. On Girard's "candidats de reductibilit'e". In Piergiorgio Odifreddi, editor, Logic and Computer Science, number 31 in APIC Studies in Data Processing, pages 123--203. Academic Press, 1990.

....relation is applicable to it. Otherwise, by the induction hypothesis at most one of the two rules corresponding to ( Phi Gamma L) is applicable. We now turn to Convergence. Our proof is a simplified and suitably modified version of Girard s original proof of SN for System F [Gir72] see [Gal90] for a good exposition. The idea is to use the evaluation relation to give a realizability semantics for types. We take a semantic type to be a set of closed linear terms, i.e. a subset of T = T . We interpret the Linear connectives over (T ) as follows: 1 = ft 2 T j t g V = ft 2 T j d c ....

J. Gallier. On Girard's "Candidats de Reductibilit'e". In P.-G. Odifreddi, editor, Logic and Computer Science. North Holland, 1990.

....i.e. there is no infinite sequence of rewrites, whenever all rules of R satisfy the General Schema. Due to the formulation of the schema, our proof here is much simpler than the one in [28] although the schema is more general. It is again based on Tait s computability predicate method. See [19] for a comprehensive survey of the method. We first define the interpretation of types and prove important properties about it. In a second part, we prove a computability property for the function symbols: assuming that the rules satisfy the General Schema, a term headed by a function symbol is ....

J. Gallier. On Girard's "Candidats de R'eductibilit'e". In P.-G. Odifreddi, editor, Logic and Computer Science. North-Holland, 1990.

....represent causality 2 incorporating bound names in (1) there is a circular chain a c b a, although this cycle never arises in actual interaction. How can we then prove termination Simple structural inductions would not be usable for the same reason they do not work in typed l calculi [7, 13]. The idea we use is suggested by SN proofs for typed l calculi, due to, among others, Tait [43] His method employs a semantic interpretation of each type [ s] as a collection of strongly normalising l terms, and shows that all typable terms are indeed in these sets. A key step is to prove that ....

....w.r.t. the observational congruence of l ; justifying all commutative conversions and h equations [16] and automatically leads to SN in the source calculus. Related Work Strong normalisation in typed l calculi has been studied extensively in the past; detailed surveys can be found in [7, 13]. Abramsky extends the CurryHoward correspondence to linear logic [15] using proof expressions and proves SN [1] guiding our present usage of acyclicity in names. This programme is taken further with realisability semantics of linear logic in [5] where CCS processes act as realisers. The ....

Gallier, J. H., On Girard's "Candidats de Reductibilit e", 123--203, Logic and Computer Science, Academic Press Limited, 1990. 12

....represent causality 2 incorporating bound names in (1) there is a circular chain a c b a, although this cycle never arises in actual interaction. How can we then prove termination Simple structural inductions would not be usable for the same reason they do not work in typed l calculi [7, 13]. The idea we use is suggested by SN proofs for typed l calculi, due to, among others, Tait [43] His method employs a semantic interpretation of each type [ s] as a collection of strongly normalising l terms, and shows that all typable terms are indeed in these sets. A key step is to prove that ....

....w.r.t. the observational congruence of l ; justifying all commutative conversions and h equations [16] and automatically leads to SN in the source calculus. Related Work Strong normalisation in typed l calculi has been studied extensively in the past; detailed surveys can be found in [7, 13]. Abramsky extends the CurryHoward correspondence to linear logic [15] using proof expressions and proves SN [1] guiding our present usage of acyclicity in names. This programme is taken further with realisability semantics of linear logic in [5] where CCS processes act as realisers. The ....

Gallier, J. H., On Girard's "Candidats de Reductibilite", 123--203, Logic and Computer Science, Academic Press Limited, 1990.

....F plus expansion rules even in the presence of canonical left linear TRS s can be used with no changes at all to show decidability of Girard s F # with expansive #, even with left linear canonical TRS s added. We do not fully introduce here the syntax and typing judgements for System F # (see [16] for a detailed introduction to the topic) but let s recall that this system is basically System F with a simple typed lambda calculus over its types, the types of the types being now called kinds. More formally, kinds, types and terms are defined by the following grammar: Kinds) K : # K # ....

....K : # K # K (Types) T : t A T#T #t : K.T #t : K.T T T (Terms) M : x #x : T.M M M #t : K.M M [T ] and one only works with those types that kind check and terms that type check w.r.t. appropriate kinding and typing rules (here we follow essentially the presentation from [16]) #, t : K 1 # s : K 2 # # (#t : K 1 .s) K 1 # K 2 # # t : K 1 # K 2 # # s : K 1 # # ts : K 2 #, t : # # s : # # # #t : #.s : # # # t : # # # s : # # # t#s : # Table 4. Kinding judgements Over the types, that now form a simple typed lambda calculus, we ....

J. Gallier. On Girard's "Candidats de Reductibilit e", pages 123--203. Logic and Computer Science. Academic Press, 1990.

....terme normalisable a une seule forme normale. Finalement, dans la section 3.6, on d emontre la propri et e de normalisation forte : tout programme qui a un type doit s arreter, quelle que soit la strat egie utilis ee pour r eduire les redexes. La preuve utilise la m ethode des ensembles satur es [29], consistant a d evelopper un mod ele du calcul o u les types sont interpr et es comme des ensembles de lambda termes fortement normalisables satisfaisant certaines conditions de cloture. Les ensembles satur es forment un treillis complet, ce qui nous 7 permet d interpr eter les diff erents ....

....(and used) in the proof of strong normalization. The strong normalization of the terms of CC 1 F directly entails that the same property is enjoyed for the calculus CC 1 , since the former is an extension of the latter. The proof in [15] is based on the Girard Tait s reducibility method [34, 29, 50]. The reducibility method consists in developing an interpretation of the typing calculus in set theory, where types are viewed as set of terms satisfying some closure conditions, called saturated set. Under this interpretation, the judgement a : A is read as the set membership a 2 [ A] where ....

J. Gallier. On girard's "candidats de reductibilit'e". In Logic and Computer Science, number 31 in The APIC series, pages 123--203. Academic Press, 1990.

....our system G types strictly more terms than all the above systems altogether, it is a nontrivial problem whether our system is also SN. SN for the higher order systems is a difficult enterprise, requiring strong induction principles non formalizable in the second order arithmetic, cf. GLT89, Gal90] Recently McAllester, Kucan, and Otth [MKO95] suggested a nice, simple, and modular approach to the SN proofs based on evaluation semantics. The ingeniously simple idea of their method is to define a semantics of types, terms, subtyping and typing judgments in such a way that: 1) the validity ....

J. H. Gallier. Logic and Computer Scuence, volume 31 of APIC Studies in Data Processing, chapter On Girard's "candidats de reductibilit'e", pages 123--201. Academic Press, 1990.

....R # and the unique R normal form of t is denoted R(t) In this section, two versions of F will be defined. Extensional F uses fij equality for type conversion while non extensional F has only fi equality for type conversion. Our presentation of these calculi is based on Gallier s [24] and we only give here the typing rules, which are necessary for a self contained introduction of expansions, while referring the reader to that paper for the formal definition of the usual fi rewriting rules. Formally, let be a distinguished symbol and let TVar and Var be disjoint sets of type ....

J. Gallier. On Girard's "Candidats de Reductibilite", pages 123--203. Logic and Computer Science. Academic Press, 1990. Odifreddi, editor.

....a set theoretic interpretation of types. There are many possible reducibility candidates such as those of Girard [GLT88] the saturated sets of Tait [Tai67] or those of Parigot [Par93] to which we can add the condition of stability by reduction without loosing those of admissibility. A survey is [Gal90] In the following, RC will refer to any admissible set of reducibility candidates. Definition 5.5 (Interpretation of the kinds) Geu95] The interpretation of the kinds is inductively defined as follows, where F(X;Y ) denotes the set theoretic function space from X to Y : Delta [ RC ....

J. Gallier. On Girard's "Candidats de Reductibilit'e". In P.-G. Odifreddi, editor, Logic and Computer Science. North Holland, 1990.

....can be encoded into non mutually recursive ones, therefore the General Schema applies in this case too. Due to the formulation of the schema, our proof here is much simpler than the one in [27] although the schema is more general. It is again based on Tait s computability predicate method, see [19] for a comprehensive survey of the method. In this section, we first recall the definition of a computability predicate, define an interpretation of types and prove that the interpretation of each type is a computability predicate. In a second 10 part, we prove a computability property for the ....

J. Gallier. On Girard's "Candidats de R'eductibilit'e". In P.-G. Odifreddi, editor, Logic and Computer Science. North Holland, 1990.

....rewrite rules. The case of the polymorphic calculus is indeed straightforward: it suffices to generalize the main lemma for strong normalization, by using a richer notion of computability predicate due to Girard [18] called reducibility candidates . For a comprehensive survey of the method, see [17]. The case of the Calculus of Constructions [12] needs more work, because dependent types require to relativize every single concept used here to a typing environment. In the second, the General Schema should be generalized again, so as to capture the rules for recursors of (non strictly) positive ....

J. Gallier. On Girard's "Candidats de R'eductibilit'e". In P.-G. Odifreddi, editor, Logic and Computer Science. North Holland, 1990.

....The original proof of SN for terms typable in F 2 is due to Girard and was given simultaneously with his introduction of the system itself. Since then the proof has been simplified by a variety of authors, e.g. Tai75, Mit86] An extensive overview of the SN proofs for F 2 can be found in [Gal90]. The inference rules of F 2 (and other type systems) allow one to derive sequents of the form Sigma . M : where Sigma is a set of variable declarations, M is a term, and is a type expression. Intuitively, a type denotes a set of values and a formula of the form M : expresses the statement ....

....[Sce87] We emphasize that, like in some earlier SN proofs [Mit86, Sce87] we work with the implicit type systems in the style of Curry (i.e. the types do not occur in terms) Additional work is needed to prove the SN property of the explicit versions of these systems. The reader is referred to [Gal90] for a full review of earlier proofs of SN for F 2 . Although the term model semantics presented here is designed to make SN a corollary of soundness, it is also a natural semantics in its own right. The evaluation process which defines 1 the semantics is similar to standard implementations of a ....

Jean Gallier. On Girard's "Candidats de Reductibilit'e". In P. Odifreddi, editor, Logic and Computer Science, volume 31 of APIC Studies in Data Processing, pages 123--203. Academic Press, 1990.

....There exists proofs with lighter notations based on untyped candidates [16] but which do not allow one to reason about the type of the elements of a reducibility candidate, as it will be necessary to do with our extension of the General Schema. For a comprehensive survey of the method, see [15]. The strong normalization proof of Coquand and Gallier can easily be tailored to our need. It suffices to define an adequate interpretation for the inductive types, and to prove that, if the arguments of a function call belong to the interpretation of their type, then the function call itself ....

J. Gallier. On Girard's "Candidats de R'eductibilit'e". In P.-G. Odifreddi, editor, Logic and Computer Science. North Holland, 1990.

.... ] Y : t] Then, since [ 8Y: oe 1 ] X : 8(t 2 T : oe 1 ] X : Y : t] we have [ 8Y: oe 1 ) X ] 8Y: oe 1 ] X : The next definition can be viewed as a semantic version of Girard s candidats de r eductibilit e (see Girard [7] Gallier [4]) Definition 16.4 Given a type interpretation T and a pre applicative structure A, a sheaf valuation is a pair = h ; ji, where : V T is a type valuation, and j: V S Sheaf(A; P) is a function called a candidate assignment , such that: j(X) S (X) where S (X) 2 Sheaf(A; P) X) ....

....] and for every s 2 T ; every S 2 Sheaf(A; P) s ; Ms 2 r[ oe 1 ] X : h[ r[ i; Y : hs; Si]g; and so, we have r[ 8Y: oe 1 ) X ] r[ 8Y: oe 1 ] X : h[ r[ i] The following lemma shows that the notion of a P sheaf is an inductive invariant. In Gallier [4], this is the lemma we call Girard s trick . 4 Lemma 16.8 Given a scenic P site hA; P ; Covi, for every sheaf valuation , if P satisfies conditions (P1) P3) then the family (r[ oe] oe2T is a P sheaf, and if A [ oe] 6= then each r[ oe] contains all stubborn elements in P [ oe] ....

Jean H. Gallier. On Girard's "candidats de reductibilit'e". In P. Odifreddi, editor, Logic And Computer Science, pages 123--203. Academic Press, London, New York, May 1990.

....as well. The method was pioneered by Tait [22] for the simply typed calculus, and brilliantly extended to various higher order typed calculi by Girard [9, 10] see also Tait [23] Various expositions and analyses of such proofs are given in Mitchell [15] Krivine [14] Huet [11] and Gallier [5, 6, 7, 8], among others. Another crucial concept is that of a partial equivalence relation, or PER. PER s were introduced by Hyland [12] and Mulry [17] PERs are a major tool in defining categories of domains in an effective setting (see Freyd, Mulry, Rosolini, and Scott [3] PERs also often show up as ....

Jean H. Gallier. On Girard's "candidats de reductibilit'e". In P. Odifreddi, editor, Logic And Computer Science, pages 123--203. Academic Press, London, New York, May 1990.

....strongly normalizing. The main technique involved is a kind of realizability argument known as reducibility . The crux of the reducibility method is to interpret every type oe as a set [ oe] of terms having certain closure properties (see Tait [18, 19] Girard [8, 9] Krivine [13] and Gallier [5, 6]) One of the crucial properties is that for a nice type oe, the terms in [ oe] satisfy the predicate P (but this does not have to be the case for ugly types ) If the sets [ oe] are defined right, then the following realizability property holds (for example, see lemma 3.8) If P is a ....

Jean H. Gallier. On Girard's "candidats de reductibilit'e". In P. Odifreddi, editor, Logic And Computer Science, pages 123--203. Academic Press, London, New York, May 1990.

....strongly normalizing. The main technique involved is a kind of realizability argument known as reducibility . The crux of the reducibility method is to interpret every type oe as a set [ oe] of terms having certain closure properties (see Tait [18, 19] Girard [8, 9] Krivine [13] and Gallier [5, 6]) One of the crucial properties is that for a nice type oe, the terms in [ oe] satisfy the predicate P (but this does not have to be the case for ugly types ) If the sets [ oe] are defined right, then the following realizability property holds (for example, see lemma 3.8) If P is a ....

Jean H. Gallier. On Girard's "candidats de reductibilit'e". In P. Odifreddi, editor, Logic And Computer Science, pages 123--203. Academic Press, London, New York, May 1990.

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Jean Gallier. On Girard's "candidats de reductibilit'e". In Odifreddi, editor, Logic and Computer Science, pages 123--203. Academic Press, 1990.

No context found.

Jean H. Gallier. On Girard's "Candidats de R'eductibilit'e". In P. Odifreddi, editor, Logic and Computer Science. Academic Press, 1990.

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J. H. Gallier. On Girard's "candidats de reductibilit 'e". In Odifreddi [Odi90], pp. 123--203.

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Jean Gallier. On girard's "candidats de reductibilit'e". In P. Odifreddi, editor, Logic and Computation, volume 31 of The APIC Series, pages 123--203. Academic Press, 1990.

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