U. Berger and H. Schwichtenberg, An inverse to the evaluation functional for typed - calculus,6th Annual IEEE LICS Symposium, 1991, 203-211.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... tend to dominate (the internal representation of a term can be 20 times bigger than the polymorphic term displayed on the screen) 6 Related work The present work can be seen as an instance of a certain approach to normalization in logical calculi: so called reduction free normalization [3, 7, 6]. The idea is to construct an appropriate model of the calculus and a function which inverts the interpretation function. Here the appropriate model is the category N and the inversion functor is application to the unit. Another proof of coherence in this style is Lafont s for cccs [16] We ....

U. Berger and H. Schwichtenberg. An inverse to the evaluation functional for typed -calculus. In Proceedings of the 6th Annual IEEE Symposium on Logic in Computer Science, Amsterdam, pages 203--211, July 1991.

....of intuitionistic abstractions on the meta level and the notion of definitional equality [18] He also proved normalization for intuitionistic type theory [19] in this way. While analyzing these ideas, we realized that there was a close connection to the work by Berger and Schwichtenberg [5]. They showed how to get a normalization algorithm (returning long normal forms) for the simply typed fij calculus by inverting an interpretation function. Here we develop this approach for a small functional programming language based on typed combinatory logic. First we study a combinatory ....

....is simple: it is exactly the strategy used at the meta level. The technique in this paper can easily be generalized to typed calculus with weak reduction, where no reduction under is allowed. For details we refer to the preliminary version of the present paper [8] Berger and Schwichtenberg [5] showed how to obtain an algorithm which returns long j normal forms for simply typed calculus by inverting an interpretation function into the standard model. Berger [4] also showed how this function can be obtained from a standard normalization proof by using a modified realizability model for ....

U. Berger and H. Schwichtenberg. An inverse to the evaluation functional for typed -calculus. In Proceedings of the 6th Annual IEEE Symposium on Logic in Computer Science, Amsterdam, pages 203--211, July 1991.

.... would also be interesting to investigate automatic removal of redundant information along the lines of Takayama [26] Berger [4] has provided a related analysis for a strong normalization proof of the typed calculus, and shown that one gets the normalization algorithm of Berger and Schwichtenberg [5]. He uses an alternative framework and explains program extraction in terms of modified realizability. Only the predicate logic part of the proof, and not the parts involving induction, is treated explicitly. 6 This is a model construction in a somewhat different sense than before, since [ ....

....an auxiliary unquote function which is needed for interpreting variables. Both are defined by recursion on the type structure. 19 This algorithm is closely related to the normalization algorithm for full normalization with j expansion in simply typed calculus given by Berger and Schwichtenberg [5]. Their inversion of the evaluation functional corresponds for example to the quote function from the Kripke model. Categorical glueing was used by Lafont [15] for proving a coherence theorem for categorical combinators. However, he argued that the semantic component of his interpretation cannot ....

U. Berger and H. Schwichtenberg. An inverse to the evaluation functional for typed -calculus. In Proceedings of the 6th Annual IEEE Symposium on Logic in Computer Science, Amsterdam, pages 203--211, July 1991.

....TFR, the Swedish Technical Research Council. x Research supported by an operating grant from the Natural Sciences and Engineering Research Council of Canada (NSERC) D. Cubri c, P. Dybjer and P. Scott 2 proving coherence in various structured (bi )categories and is also closely related to the (Berger and Schwichtenberg 1991) method for normalizing terms. We discuss history and related work in more detail in section 5 below. In a certain sense, our program is dual to Lambek s original goal of categorical proof theory (Lambek 1968) in which he used cut elimination to study categorical coherence problems. Here, we use ....

..... s : C) a 0 ) 24) q Gamma1 B;D (w : D . t(s=z)q A;D (a 0 ) B) q Gamma1 (A;B) C (z : C . t : A; s : B) 25) q Gamma1 A;C (z : C . t : A) q Gamma1 B;C (z : C . s : B) A 6= h i This completes the algorithm. Note the similarity between this algorithm and the algorithm by (Berger and Schwichtenberg 1991, p. 203) Our Gamma corresponds to their evaluation functional into Sets. Our q corresponds to their functional procedure expression p e; and our q Gamma1 corresponds to their make self evaluating mse. 3.4. Uniqueness of Normal Forms Following the discussion in the Introduction, ....

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U. Berger and H. Schwichtenberg, An inverse to the evaluation functional for typed -calculus, Proc. of the 6th Annual IEEE Symposium of Logic in Computer Science, 1991, pp. 203-211.

.... terms tend to dominate (the internal representation of a term can be 20 times bigger than the polymorphic term displayed on the screen) 6 Related work The present work can be seen as an instance of a certain approach to normalization in logical calculi: so called reduction free normalization [3, 7, 6]. The idea is to construct an appropriate model of the calculus and a function which inverts the interpretation function. Here the appropriate model is the category N N and the inversion functor is application to the unit. Another proof of coherence in this style is Lafont s for cccs [16] We ....

U. Berger and H. Schwichtenberg. An inverse to the evaluation functional for typed -calculus. In Proceedings of the 6th Annual IEEE Symposium on Logic in Computer Science, Amsterdam, pages 203--211, July 1991.

....of intuitionistic abstractions on the meta level and the notion of definitional equality [18] He also proved normalization for intuitionistic type theory [19] in this way. While analyzing these ideas, we realized that there was a close connection to the work by Berger and Schwichtenberg [5]. They showed how to get a normalization algorithm (returning long normal forms) for the simply typed fij calculus by inverting an interpretation function. Here we develop this approach for a small functional programming language based on typed combinatory logic. First we study a combinatory ....

....is simple: it is exactly the strategy used at the meta level. The technique in this paper can easily be generalized to typed calculus with weak reduction, where no reduction under is allowed. For details we refer to the preliminary version of the present paper [8] Berger and Schwichtenberg [5] showed how to obtain an algorithm which returns long j normal forms for simply typed calculus by inverting an interpretation function into the standard model. Berger [4] also showed how this function can be obtained from a standard normalization proof by using a modified realizability model for ....

U. Berger and H. Schwichtenberg. An inverse to the evaluation functional for typed -calculus. In Proceedings of the 6th Annual IEEE Symposium on Logic in Computer Science, Amsterdam, pages 203--211, July 1991.

....by using the Curry Howard interpretation which makes explicit the connection between the formal proof of normalization and the proof of coherence. The present work can be seen as an application of a certain approach to normalization in logical calculi: so called reduction free normalization [5, 7, 6, 4]. The idea is to construct an appropriate model of the calculus and a function which inverts the interpretation function. Here the appropriate model is the category N N and inversion is application to unit. Another proof of coherence in this style is Lafont s for cccs [17] We would also like to ....

U. Berger and H. Schwichtenberg. An inverse to the evaluation functional for typed -calculus. In Proceedings of the 6th Annual IEEE Symposium on Logic in Computer Science, Amsterdam, pages 203--211, July 1991.

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U. Berger and H. Schwichtenberg, An inverse to the evaluation functional for typed - calculus,6th Annual IEEE LICS Symposium, 1991, 203-211.

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