Thierry Coquand and Peter Dybjer. Intuitionistic model constructions and normalization proofs. Mathematical Structures in Computer Science, 7:75--94, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....(into a non standard model) reduction free normalization is also referred to as normalization by evaluation. The flattening example above is folklore in the normalization by evaluation community. Normalization by evaluation has been variously studied in logic, proof theory, and category theory [2, 3, 8 10, 13] and in partial evaluation [14, 16] Typedirected partial evaluation, which we present next, has been investigated both practically [5, 15, 17, 18, 29, 31, 38] and foundationally [24, 25, 47] 1.2 Type directed partial evaluation Type directed partial evaluation is a practical instance of ....

Thierry Coquand and Peter Dybjer. Intuitionistic model constructions and normalization proofs. Mathematical Structures in Computer Science, 7:75--94, 1997.

....type directed partial evaluation. 1 Introduction The basic idea of normalization by evaluation is to extract the normal form (with respect to some notion of conversion) of a term from its interpretation in a suitably chosen, quasi syntactic denotational model of the conversion relation [5]. For instance, let us consider the interpretation of a pure, simply typed lambdaterm E in a model where all base types are interpreted as the set of well formed lambda terms, and function types are interpreted as full set theoretic function spaces. Then it is fairly simple to (at least ....

....2 Int , we will return to this notion in Section 5. Among the well typed terms E : we distinguish those in normal and atomic (also known as neutral) form: x : c : E1 : 1 2 E1 E2 : 2 A normalization function, in the sense of Coquand and Dybjer [5] (but with a semantic notion of equivalence) then maps any term E to a normal form term norm(E) such that j= norm(E) E, and such that for all E with j= E = E, norm(E ) norm(E) 3 The traditional way of computing norm(E) is by repeated reductions, possibly followed by ....

Thierry Coquand and Peter Dybjer. Intuitionistic model constructions and normalization proofs. Mathematical Structures in Computer Science, 7:75-94, 1997.

....the trivially specialized program #d.p(s, d) contracting # redexes and eliminating static operations as their inputs become known. What makes this approach attractive is the technique of reduction free normalization or normalization by evaluation , already known from logic and category theory [2, 3, 7]. A few challenges arise, however, with extending these results to a programming language setting. Most notably: Interpreted base types and their associated static operations. These need to be properly accounted for, in addition to the # reduction. Unrestricted recursion. This prevents a ....

....constants nor # redexes. Incidentally, this also means that if we had included polymorphic lets in the source language, they would simply get unfolded in the resulting normal forms. We can now define a notion of normalization based on (undirected) equality, rather than on (directed) reduction [3]. Since lambda abstracting a dynamic type term over a dynamic type variable still yields a dynamic term, it su#ces to be able to compute normal forms of closed terms: Definition 3 (static equivalence and normalization) Let s be an interpretation of # s . We say that two terms s ,# d E : # ....

Thierry Coquand and Peter Dybjer. Intuitionistic model constructions and normalization proofs. Mathematical Structures in Computer Science, 7:75--94, 1997.

....Adding types. It is shown that the addition of types leads to strong normalization of the rewrite rules and the computation of long normal forms. In order to relate our approach to customary type directed expositions (see [BS91] for set theoretic, BES98] for domain theoretic, AHS95, CD97, CDS98] for category theoretic and [Dan98, Ves01] for purely syntactic semantics) it is explained how the rewrite rules are emulated by type directed Normalization by Evaluation. This is made possible through a perspicuous characterization of expansive normal forms that also permits as a ....

Thierry Coquand and Peter Dybjer. Intuitionistic model constructions and normalization proofs. Math. Structures in Computer Science, 7(1):75-94, 1997. 19

....such as Martin Lof s type theory. In such a language, one can directly embed simply typed # terms (in normal form or not) express normalization by evaluation, and prove that it preserves types and yields normal forms. 6 Related work: Normalization by evaluation takes its roots in type theory [7, 16], proof theory [4, 5, 6] logic [2] category theory [1, 8, 18] and partial evaluation [9, 12, 19, 21] Long ## normal forms were specified, e.g. in Huet s thesis [14] The particular characterization we use originates in Pfenning s work on Logical Frameworks, and so does higher order abstract ....

Thierry Coquand and Peter Dybjer. Intuitionistic model constructions and normalization proofs. Mathematical Structures in Computer Science, 7:75-- 94, 1997.

....and (proper) rewrite rules; NbE seems to be much more efficient for the former than for the latter. In our implementation (in the Minlog system) we therefore use computational rules whenever possible. A related approach (using a glueing construction) is elaborated by T. Coquand and P. Dybjer in [6]. Another related paper is T. Altenkirch, M. Hofmann and T. Streicher [1] there a cartesian closed category is defined which has the property that the interpretation of the simply typed lambda calculus in it yields the reduction free normalization algorithm from [5] as well as its correctness. ....

Thierry Coquand and Peter Dybjer. Intuitionistic model constructions and normalization proofs. Mathematical Structures in Computer Science, 7:73-- 94, 1997.

....a Scheme interpreter when executing the code for NbE. Such an operational explanation of NbE will certainly benefit from mathematical studies like the present one, but the details still have to be done. A related approach (using a glueing construction) is elaborated by T. Coquand and P. Dybjer in [6]. Another related paper is T. Altenkirch, 1 M. Hofmann and T. Streicher [1] there a cartesian closed category is defined which has the property that the interpretation of the simply typed lambda calculus in it yields the reduction free normalization algorithm from [5] as well as its ....

Thierry Coquand and Peter Dybjer. Intuitionistic model constructions and normalization proofs. Mathematical Structures in Computer Science, 7:73-- 94, 1997.

.... for simple types only, as an inverse to the evaluation functional [5] reduction free normalisation [1, 2] type directed partial evaluation [8] as the computational content of Tait s typed method for proving Strong Normalisation [3] as a glueing construction in intuitionistic model theory [6], and as pertaining to the Yoneda embedding in (constructive) category theory [7] Practical Implications The NbE algorithm can be coded in a functional programming language like, e.g. Scheme or ML. In an interactive session you can thus compile your favourite program, apply the compiled NbE ....

....of the language: typically, the relevant syntax is glued in to a standard model. This has been a substantially limiting factor as far as the language constructs for which NbE has been proven correct go and generally only abstraction application # reduction is being addressed; see, however, [4, 6]. # This is a complete first draft where some proofs only are given in sketch form. Aside from more formal proofs, it should also be extended with a more comprehensive exposition of related work as well as a self contained treatment of System F. Visiting LFCS, University of Edinburgh; ....

Thierry Coquand and Peter Dybjer. Intuitionistic model constructions and normalization proofs. Mathematical Structures in Computer Science, 7:73--94, 1997.

....the trivially specialized program d:p(s; d) contracting redexes and eliminating static operations as their inputs become known. What makes this approach attractive is the technique of reduction free normalization or normalization by evaluation , already known from logic and category theory [2, 3, 7]. A few challenges arise, however, with extending these results to a programminglanguage setting. Most notably: Interpreted base types and their associated static operations. These need to be properly accounted for, in addition to the reduction. Unrestricted recursion. This prevents a ....

....constants nor redexes. Incidentally, this also means that if we had included polymorphic lets in the source language, they would simply get unfolded in the resulting normal forms. We can now de ne a notion of normalization based on (undirected) equality, rather than on (directed) reduction [3]. Since lambda abstracting a dynamic type term over a dynamic type variable still yields a dynamic term, it su ces to be able to compute normal forms of closed terms: De nition 3 (static equivalence and normalization) Let I s be an interpretation of s . We say that two terms s ; d E : and ....

Thierry Coquand and Peter Dybjer. Intuitionistic model constructions and normalization proofs. Mathematical Structures in Computer Science, 7:75-94, 1997.

....Introduction In this paper we give a semantical proof of reduction free normalisation for F fij , a version of Girard s system F with full fij equality for both kinds of abstraction. This generalises the semantical normalisation algorithms for simply typed systems (Berger and Schwichtenberg 1991; Coquand and Dybjer 1996; Altenkirch, Hofmann, and Streicher 1995) to polymorphism. As in those approaches we do not prove strong normalisation but construct a function nf sending terms to terms in normal form such that convertible terms are sent to the same normal form and any term t is convertible with nf (t) Such a ....

....been treated by the abovementioned authors. The richer systems like ECC extend the simply typed lambda calculus by two new features: type dependency and polymorphism. We treat the latter in this paper and leave type dependency for future work. The key idea of the present work (and also implicit in (Coquand and Dybjer 1996)) is to construct a model GT (for Girard Tait or Glueing of the Term model) in which types are interpreted as quadruples (A; A pred ; q A ; u A ) where A is a syntactic type, A pred is a family of sets indexed by conversion classes of terms of A, q A ( quote ) is a function mapping an ....

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Coquand, T. and P. Dybjer (1996). Intuitionistic model constructions and normalization proofs.

.... Introduction and Summary In this paper we give a semantical proof of reductionfree normalisation for F SK , a variant of Girard s system F based on combinators rather than abstraction (for object variables) This generalises the semantical normalisation algorithms for a simply typed systems [2, 5, 1] to polymorphism. As in loc.cit. we do not prove strong normalisation but construct a function nf sending terms to terms in normal form such that convertible terms are sent to the same normal form and any term t is convertible with nf (t) Such a function is sufficient for practical purposes as ....

....algorithms can employ the interpreter of the underlying functional programming language as e.g. Standard ML. This gain of efficiency was the initial motivation of Berger and Schwichtenberg for studying reduction free normalisation. The case of simply typed lambda calculus has been treated in [2, 5, 1]. The richer systems like ECC extend the simply typed lambda calculus by two new features: type dependency and polymorphism. We make a first step towards the latter in this paper and leave type dependency for future work. The key idea of the present work (and also implicit in [5] is to construct ....

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T. Coquand and P. Dybjer. Intuitionistic model constructions and normalization proofs. Mathematical Structures in Computer Science, 1993. to appear, previous version has appeared in the informal proceedings of the BRA-Types workshop, 1993 held in Turin.

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T. Coquand and P. Dybjer, Intuitionistic Model Constructions and Normalization Proofs, Math. Structures in Compter Science 7, 1997, 75-94.

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T. Coquand and P. Dybjer. Intuitionistic model constructions and normalization proofs. Mathematical Structures in Computer Science, 1996. To appear.

.... 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 ....

T. Coquand and P. Dybjer. Intuitionistic model constructions and normalization proofs. Preliminary Proceedings of the 1993.

.... it is also perfectly possible to consider convertibility in a combinatory setting, especially for a di erent basis, such as categorical combinators [AHS95] The following (obviously incomplete) table summarizes the situation: notion of conversion syntax weak strong combinators [CD97] AHS95] lambda terms [ML75b,CD93b] Mog92] BS91] We should also mention that a main advantage with the glueing technique is that it extends smoothly to datatypes. We showed only how to treat natural numbers in the previous section, but the approach extends smoothly to arbitrary 17 strictly ....

....also mention that a main advantage with the glueing technique is that it extends smoothly to datatypes. We showed only how to treat natural numbers in the previous section, but the approach extends smoothly to arbitrary 17 strictly positive datatypes such as the datatype of Brouwer ordinals [CD97] For TDPE it is of course essential to deal with functions on datatypes, and in Section 4 we show how to deal with this problem by combining the idea of binding time separation with normalization by evaluation. 3.1 The setting: simply typed lambda calculus For the purpose of presenting the ....

Thierry Coquand and Peter Dybjer. Intuitionistic model constructions and normalization proofs. Mathematical Structures in Computer Science, 7:75{ 94, 1997.

....that normalization is achieved by interpreting an object language term as a meta language term and then evaluating the latter. For this purpose one can use di erent meta languages, and we make use of several such in these notes. In Section 2 we follow Martin L of [ML75b] and Coquand and Dybjer [CD93b] who used an intuitionistic meta language, which is both a mathematical language and a programming language. Martin L of worked in an informal intuitionistic meta language, where one uses that a function is a function in the intuitionistic sense, that is, an algorithm. Coquand and Dybjer [CD93b] ....

....[CD93b] who used an intuitionistic meta language, which is both a mathematical language and a programming language. Martin L of worked in an informal intuitionistic meta language, where one uses that a function is a function in the intuitionistic sense, that is, an algorithm. Coquand and Dybjer [CD93b] implemented a formal construction similar to Martin L of s in the meta language of Martin L of s 4 intuitionistic type theory using the proof assistant ALF [MN93] This NBEalgorithm makes essential use of the dependent type structure of that language. It is important to point out that the ....

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Thierry Coquand and Peter Dybjer. Intuitionistic model constructions and normalization proofs. In H. Barendregt and T. Nipkow, editors, Types for Proofs and Programs, International Workshop TYPES'93, number 806 in Lecture Notes in Computer Science, Nijmegen, The Netherlands, May 1993.

....arguments follow The answer 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 ....

T. Coquand and P. Dybjer. Intuitionistic model constructions and normalization proofs. Preliminary Proceedings of the 1993.

....by first interpreting a term in a suitable model and then map this interpretation back to the normal form of the term. By working in an intuitionistic framework one ensures that the normalization function thus obtained is an algorithm. Martin Lof s approach was investigated further by (T. Coquand and P. Dybjer 1997) They formulated the abstract conditions NF1 and NF2 (for the special case that j is syntactic equality of terms, so that NF3 and NF4 are trivially satisfied) Moreover, they focussed on algebraic aspects and used the fact that syntax modulo conversion is a free model and hence has a unique ....

T. Coquand and P. Dybjer, Intuitionistic Model Constructions and Normalization Proofs , Math. Structures in Computer Science, Vol.7, No.1 (1997), pp. 75-94.

....and Bjorn von Sydow describes how to use a proof editor for type theory. ALF supports pattern matching with dependent types, a feature that has proved very useful in the context of formal proofs. 5. Intuitionistic Model Construction and Normalization Proof by Thierry Coquand and Peter Dybjer [4] describes an example of mechanical proof in ALF using type theory with inductive definitions and pattern matching. The example is normalization in typed combinatory logic and lambda calculus and shows the power and elegance of the machinery of dependent types in the context of metamathematics. ....

T. Coquand and P. Dybjer. Intuitionistic model constructions and normalization proofs. Preliminary Proceedings of the 1993 TYPES Workshop, Nijmegen, 1993.

....with their function and ours is that we extend this technique to calculus with explicit contexts and substitutions and that we give the the complete set of conversion rules for this calculus. The treatment of variables is also different in our approach. The inversion function is also presented in [6] where it is discussed for combinatory logic. In [3] a more detailed study of how to extract the normalization function from a normalization proof using Tait s method is presented. One important difference with other works on explicit substitution is that we address the problem of using named ....

....for comp(ffi; fl) Derivations of terms and substitutions will be named M;N and fl; ffi; respectively. The substitution c is not a standard primitive for explicit substitutions. Often one rather has an identity substitution (in Gamma Gamma) 1, 12] or the empty substitution (in Gamma [ [6] in either case c would be derivable. Instead we have taken c as primitive and the identity and the empty substitution is then derivable. In [1] they also have a substitution which corresponds to a shift on substitutions; this substitution is here defined as c where c 2 [ Gamma; x : A] ....

Th. Coquand and P. Dybjer. Intuitionistic Model Constructions and Normalization Proofs. Math. Structures Comput. Sci. 7 (1997), no. 1, p. 75--94.

....their function and ours is that we extend this technique to calculus with explicit contexts and substitutions and that we give the the complete set of conversion rules for this calculus. The treatment of variables is also different in our approach. The inversion function is also presented in [6] where it is discussed for combinatory logic. In [3] a more detailed study of how to extract the normalization function from a normalization proof using Tait s method is presented. One important difference with other works on explicit substitution is that we address the problem of using named ....

....comp(ffi; fl) Derivations of terms and substitutions will be named M;N and fl; ffi; respectively. The substitution c is not a standard primitive for explicit substitutions. Often one rather has an identity substitution (in Gamma Gamma ) 1, 12] or the empty substitution (in Gamma [ [6] in either case c would be derivable. Instead we have taken c as primitive and the identity and the empty substitution is then derivable. In [1] they also have a substitution which corresponds to a shift on substitutions; this substitution is here defined as c where c 2 [ Gamma; x : A] ....

Th. Coquand and P. Dybjer. Intuitionistic Model Constructions and Normalization Proofs. Paper in preparation.

.... 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 ....

T. Coquand and P. Dybjer. Intuitionistic model constructions and normalization proofs. Preliminary Proceedings of the 1993 TYPES Workshop, Nijmegen, 1993.

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Thierry Coquand and Peter Dybjer. Intuitionistic model constructions and normalization proofs. Mathematical Structures in Computer Science, 7:75--94, 1997.

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Thierry Coquand and Peter Dybjer. Intuitionistic model constructions and normalization proofs. Mathematical Structures in Computer Science, 7:75--94, 1997.

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Thierry Coquand and Peter Dybjer. Intuitionistic model constructions and normalization proofs. Mathematical Structures in Computer Science, 7:75--94, 1997.

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