S. Berardi, Encoding of data types in Pure Construction Calculus: a semantic justication, in Logical Environments, eds. G. Huet and G. Plotkin, Cambridge University Press, pp 30-60.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....but only for one speci c encoding of the natural numbers, as polymorphic Church numerals. Our proof of non derivability uses a fairly simple model construction which originates from [Geuvers 1996] and [Stefanova and Geuvers 1996] The model we construct has some similarities with the one used in [Berardi 1993] to justify encoding mathematics in the Calculus of Constructions. To establish our main result we construct a model in which the type that represents induction is empty. Apart from the induction principle we also show the non derivability of the Axiom of Choice. 1.1 Small diversion: a possible ....

S. Berardi, Encoding of data types in Pure Construction Calculus: a semantic justication, in Logical Environments, eds. G. Huet and G. Plotkin, Cambridge University Press, pp 30-60.

....in [2, 9] Here that representation is completed, and it is proved that all extra axioms needed are consistent. Among the innovations of this paper is a definition of cdr, whose definition was left for future work in [2, 9] The results are then extended to other abstract data types, those of [1]. The method used to define cdr for lists is extended to obtain the definition of an inverse for each argument of each constructor of an abstract data type. These inverses are used to prove the injective property for the constructors. Also, Dedekind s method of defining the natural numbers is used ....

....of cons with respect to each of its arguments, and the proof uses the car and cdr. In 4, it will be shown that the remaining axiom, which asserts that the empty list is not constructed by cons, is consistent. Finally, in 5, these methods will be extended to all of the abstract data types of [1]. In particular, the methods used to define car in 1 and cdr in 2 will be used to obtain an inverse for each constructor with respect to each argument. This will allow the proof of the injective property of constructors. The use of a predicate associated with each abstract data type that uses ....

Stefano Berardi. Encoding of data types in pure construction calculus: a semantic justification. In Gerard Huet and Gordon Plotkin, editors, Logical Environments, pages 30--60. Cambridge University Press, 1993.

....= take X and Y such that ; X;Y ( A and X Y 6= then ( X ; X = A X and ( Y ; Y = A Y , so [ CL] ae ae (A X) A Y ) We have obtained the following result. Lemma 4.4 In CC there is no term M such that M : CL. Moreover, there is no term M such that M : PI CL. In [4], it is shown that there is a term M such that x:EXT; ff: c; c 0 :ff; h:c 6= ff c 0 M : every f :ff ff has a fixed point : The statement that every f :ff ff has a fixed point is written formally as Pif :oe oe:9x:oe:fx = oe x. In the models we are looking at here, this is even stronger: ....

....fact we have tried to organize these concrete models in a more general scheme to cover the PERs as well, but we have so far not succeeded. However one can use PERs instead of polystructures as interpretations of and redo the rest of the construction. Other (partial) models of the PTS CC (see [8, 4]) In the literature there are models of CC employed for proving strong normalization, in which CC is interpreted via an explicit or implicit syntactical mapping into Girard s system F (see [9, 8] Furthermore, there are models in which type dependencies are not fully disregarded as in [4] ....

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S. Berardi. Encoding of data types in pure construction calculus: a semantic justification. In G. Plotkin and G. Huet, editors, Logical Enviroments, pages 30--60, Edinburgh, 1992.

....see that CC does not prove everything and is a conservative extension of higher order propositional logic. These are more or less standard results by now, but we shall devote some attention to them as this text is meant to be introductory. Further we shall discuss a recent result by Berardi( Berardi 199 ] showing that CC is still an adequate system for higher order reasoning about inductive data types, which is one of the main practical applications of the system. To understand this result, we have to devote some attention to data types and specifications in CC, a subject extensively studied in ....

....distance between CC and HOPL is not so large when it comes to propositions about inductive data types. This follows from a recent result by Berardi, which we shall discuss it here only for what concerns the implications for the formulas as types embedding. For details and proofs we refer to [Berardi 199 ] The point is that for purposes of deriving programs from proofs, it doesn t seem to make sense to declare a theory in the context. Instead one uses the definable impredicative data types and inductive predicates on them, as is done in the examples of 3.4. This is not the place to discuss in ....

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S. Berardi, Encoding of data types in Pure Construction Calculus: a semantic justification. To appear in the Proceedings of the second BRA meeting on Logical Frameworks, Edinburgh, May 1991.

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S. Berardi, Encoding of data types in Pure Construction Calculus: a semantic justification, in Logical Environments, eds. G. Huet and G. Plotkin, Cambridge University Press, pp 30--60.

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