R. Hindley and G. Longo. Lambda-calculus models and extensionality. Z. 29 Math. Logik Grundlag. Math., 26(4):289--310, 1980.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....) Let lazy be the equivalence induced by v lazy . In order to model the lazy convergence of the calculus, a semantic account of the notion of valuable terms must be given. This can be done by enriching the well known set theoretical definition of calculus model in Hindley Longo style, see [8], by explicitly introducing a subset V of the interpretation domain, where all valuable terms have to be interpreted. Definition 2.2 A model for the lazy calculus is a 4 tuple M = D; ffl; V; Delta] where: D together with ffl is an applicative structure, i.e. it is a set with at least ....

J. R. Hindley and G. Longo. Lambda-calculus models and extensionality. Z. Math. Logik Grundlag. Math., 26:289 -- 310, 1980.

....lambda calculus has been the subject of research by logicians since the early 1930 s, its model theory developed only much later, following the pioneering model construction made by Dana Scott. The notion of an environment model (the name is due to Meyer [11] originated with Hindley and Longo [8]. They are functional domains where terms can be properly interpreted. Meyer describes them as the natural, most general formulation of what might be meant by mathematical models of the 1 untyped lambda calculus . The main result in [11] is a completeness theorem demonstrating that every lambda ....

Hindley, R., and G. Longo, "Lambda-calculus models and extensionality ", Zeit. f. Math. Logik u. Grund. der Math., vol.26 (1980), pp.289310.

....in [16] for the classical calculus and they were based on the intersection type discipline. In that case, however, discovering that filters of types do actually form a structure (a model) was based on the pre existing and independent definition of this kind of mathematical structures (see [37, 48]) Here the problem is the opposite: given the logical interpretation induced by our system, we look for a reasonable definition of what is a model of our calculus. In the extended view of Curry types (see [16, 21] type theories are an instance of information systems (see [63, 23] Taking ....

....analyze compositionally the interpretation of terms defined by [ M ] foe j M : oeg (where M is closed) and devise a category of objects that embodies the minimum needed structure to interpret the calculus. We then get a notion of environment model for the present calculus, in the sense of [37]. The filter model induced by our type assignment turns out to fit into this notion, a fact that will be used to prove completeness of type inference. Our study culminates in the full abstraction theorem, that we will prove by means of characteristic terms extending [19] A preliminary version ....

J.R. Hindley, G. Longo, "Lambda-calculus Models and Extensionality", Zeit. fur Math. Logik 26, 1980, 289-310.

.... calculi 18 Remark 31. As stated in [19] also Gamma pn enjoys the subject reduction property. 4. 2 The Set Semimodel For the classical calculus, a filter model construction with simple types, even considering as a filter any set of types, does not yield a model (see e.g. [25]) Indeed the best one can obtain is a semimodel in the sense of [39] i.e. a model in which interreducible terms are equal, but in general convertible terms are not (M;N are interreducible iff M Gamma N and N Gamma M ) Adapting Plotkin s definition to the present context (see also [1] ....

....this rule properly increases the set of types of the subject. 5.3 The lattices As the set semimodel suggests, when interpreting our calculus we naturally get lattices. We make precise now what is a model of this calculus. We do this by incorporating the notion of lattice into that of model of [25]. Definition51. A lattice is a structure D = hD; v; Delta; u; t; Delta] D i where: i) hD; v; u; ti is a lattice; ii) Delta : D Theta D D is monotonic; iii) 8d; d 0 ; e 2 D: d t d 0 ) Delta e v (d Delta e) t (d 0 Delta e) and (d Delta e) u (d 0 Delta e) v (d u d 0 ....

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R. Hindley, G. Longo, "Lambda Calculus Models and Extensionality", Z. Math. Logik 26, 1980, 289-310.

....finantial support of Ca Foscari University of Venice. Thanks are due to Robert Goldblatt for discussions and for making the visit to Wellington possible. Typeset by AM S T E X 1 2 ANTONINO SALIBRA notion of an environment model (the name is due to Meyer [29] originated with Hindley and Longo [24]. They are functional domains where terms can be properly interpreted. Meyer describes them as the natural, most general formulation of what might be meant by mathematical models of the untyped lambda calculus . The main result in [29] is a completeness theorem demonstrating that every lambda ....

R. Hindley and G. Longo, Lambda-calculus models and extensionality, Zeit. f. Math. Logik u. Grund. der Math. 26 (1980), 289--310.

....and has excellent properties, and what more could we want This desired isomorphism will be proved in chapter 2. These models are extensional, i.e. every function is uniquely determined by its input output behaviour 1 . There are many subtleties in the notion of a model for the calculus, but in [2], Hindley and Longo, after having analysed several extensionality properties of calculus models, make the following remark. In contrast to the above, with the splendid D1 everything goes perfectly smoothly; Other properties in favour of the reflexive domain in P is that it is a continuous ....

R. Hindley and G. Longo. Lambda Calculus Models and Extensionality. Z. Math. Logik Grundlag. Math., 26, 1980.

....variables in a basis. It turns out that, within this system, types are preserved under fi v conversion. Moreover, the filter structure of the type theory is an instance of call by value syntactical model (a generalisation, 6, 10, 11] of the notion of syntactical model due to Hindley and Longo [7]) 2 The minimal relevant logic B as a type discipline The minimal relevant logic B is a propositional calculus. To see B as a type discipline, we interpret propositions as types: the constant for truth is interpreted as a constant type , whose intended meaning is the type of values ; ....

....of v is denoted by = v . Models of call by value calculus are a generalisation of models, in which a distinguished subset V of an applicative structure D is meant to interpret values: if one takes V = D, the following definition immediately coincides with the notion of syntactical model of [7]. A slightly different definition can be found in [6] the present one is from [10] Definition 4.2 (Models of call by value calculus) A model of call by value calculus is a structure M = D; V; Delta; M ae , such that Delta is a binary operation on D, called application (i.e. D; ....

R. Hindley and G. Longo, "Lambda-calculus models and extensionality", Zeitschrift f ur Mathematische Logik 26 (1980), pp.289-310.

....) Let lazy be the equivalence induced by v lazy . In order to model the lazy convergence of the calculus, a semantic account of the notion of valuable terms must be given. This can be done by enriching the well known set theoretical definition of calculus model in Hindley Longo style, see Hindley Longo (1980), by explicitly introducing a subset V of the interpretation domain, where all valuable terms have to be interpreted. Definition 2 A model for the lazy calculus is a 4 tuple M = D; ffl; V; Delta] where: D together with ffl is an applicative structure, i.e. it is a set with at least two ....

Hindley, J. R. & Longo, G. (1980), `Lambda-calculus models and extensionality.' Z.

....C.so Svizzera 185 10149 TORINO. E mail: fpravato; ronchi; roverg di:unito:it As far as the denotational semantics of the fi v calculus is concerned, a general definition of a model for this language was given in [4] following the HindleyLongo approach for defining a model for fi calculus [6]. i.e. a model is defined as an applicative structure with an interpretation function that maps terms (in a given environment) to elements of the model, and that must satisfy some constraints. The main difference between the original definition of Hindley Longo and this one, is that there must ....

....be contextual closed, namely, if two terms M and N have the same interpretation, then for every context C[ Delta] C[M ] and C[N ] must have the same interpretation. A general definition of a model for the fi v calculus, following the HindleyLongo approach for defining a lambda calculus model [6], has been given in [4] We recall here that definition in a slightly different form, more suitable for our aims. Definition 2.3 Let S and V be two non empty sets such that V ae S, and call V the set of semantic values. Let Env be the set of environments, where an environment is a map : X V , ....

J. R. Hindley and G. Longo. Lambda-calculus models and extensionality. Z. Math. Logik Grundlag. Math., 26:289 -- 310, 1980.

....as inclusion of PERs, which is simply subset inclusion on D Theta D. The construction is mostly standard (see e.g. CL91, BL90] but incorporates type term dependency. We make use of a model of the untyped calculus to interpret terms and to build PERs over. Definition 5. 1 (Lambda Model [HL80]) A lambda model is a triple, D = hD; Delta; i, where D is a set, Delta is a binary operation on D and for untyped lambda terms M , the interpretation of M in an environment ae: Var D is [ M ] ae 2 D, such that: x] ae = ae(x) var) MN ] ae = M ] ae Delta [ N ] ae ....

R. Hindley and G. Longo. Lambda calculus models and extensionality. Z. Math. Logik Grundlag. Math., 26:289--310, 1980.

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Sci. 22 (1-18). Hindley R., Longo G. (1980) "Lambda-calculus models and extensionality," Zeit. Math.

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Hindley R., Longo G. [1980] "Lambda-calculus models and extensionality," Zeit. Math. Logik Grund. Math. n. 2, vol. 26 ( 289-310).

....of CL by Curry s five axioms, then one can prove CLb (a = b) from a derivation CLb (a(x) b(x) Thus (z b ) is admissible for CLb. However, it is not derivable in CLb, since there exist models of CLb which may realize the assumption a(x) b(x) but not the consequence a = b (see [HL80] where these distinctions were first made, from the point of view of models: the experienced reader may take as a model the interpretations of closed terms in Scott s Pw model) Finally, as said above, lbh is an extension of CL where (z b ) is just an instance of the rule (z) of ....

R. Hindley, G. Longo, Lambda-calculus models and extensionality, Zeit. Math. Logik Grund. Math. n. 2, vol. 26 ( 289-310), 1980.

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R. Hindley and G. Longo. Lambda-calculus models and extensionality. Z. 29 Math. Logik Grundlag. Math., 26(4):289--310, 1980.

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R. Hindley and G. Longo. Lambda-calculus models and extensionality. Z. Math. Logik Grundlag. Math., 26(4):289-310, 1980.

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R. Hindley and G. Longo. Lambda-calculus models and extensionality. Z. Math. Logik Grundlag. Math., 26(4):289-310, 1980.

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Roger Hindley and Giuseppe Longo. Lambda-calculus models and extensionality. Z. Math. Logik Grundlag. Math., 26(4):289--310, 1980.

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Roger Hindley and Giuseppe Longo. Lambda-calculus models and extensionality. Z. Math. Logik Grundlag. Math., 26(4):289--310, 1980.

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R. Hindley and G. Longo. Lambda-calculus models and extensionality. Z. Math. Logik Grundlag. Math., 26(4):289--310, 1980.

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J.R. Hindley, G. Longo, "Lambda Calculus Models and Extensionality ", Z. Math. Logik Grundlag. Math. 26, 1980.

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J.R. Hindley, G. Longo, "Lambda Calculus Models and Extensionality", Z. Math. Logik Grundlag. Math. 26, 1980.

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R. Hindley and G. Longo, "Lambda-calculus models and extensionality", Zeitschrift fur Mathematische Logik 26 (1980), pp.289-310.

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J.R. Hindley, G. Longo, "Lambda-calculus Models and Extensionality", Zeit. fur Math. Logik 26, 1980, 289-310.

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J.R. Hindley, G. Longo, "Lambda Calculus Models and Extensionality", Z. Math. Logik Grundlag. Math. 26, 1980.

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R. Hindley and G. Longo. Lambda-calculus models and extensionality. Z. Math. Logik Grundlag. Math., 26(4):289-310, 1980.

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R. Hindley and G. Longo. Lambda-calculus models and extensionality. Z. Math. Logik Grundlag. Math., 26(4):289-310, 1980.

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Hindley R., Longo G., "Lambda Calculus Models and Extensionality", Z. Math. Logik Grundlag. Math., 26, 1980, pp. 289-310.

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R. Hindley and G. Longo, Lambda-calculus models and extensionality, Z. Math. Logik Grundlag. Math.,26 (1980), 289--310.

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Hindley, J. R., Longo, G.: Lambda Calculus Models and Extensionality. Z. Math. Logik Grundlag. Math. 26 (1980) 289--310

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