J. C. Mitchell and P. J. Scott, Typed lambda models and cartesian closed categories, Categories in Computer Science and Logic, Contemp. Math., vol. 92, Amer. Math. Soc., 1989, pp. 301-316.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....interpreted by arbitrary sets equipped with an application operation of the appropriate type. Comprehension is still required; however, the way terms are interpreted is now part of the structure of the model rather than just an existence axiom. Intensional Henkin models are discussed, e.g. in [19, 20]. The notion chosen for HasCasl, for reasons of semantic as well as methodological nature, is that of intensional Henkin models. To begin, moving away from standard models avoids the well known incompleteness problems. Extensionality, moreover, carries the disadvantage of destroying the existence ....

....Similarly, implies precedes formulas that are logical consequences of the previous axioms. It may come as a surprise that the relevant formula shown in Figure 3 expresses a form of extensionality; however, it is well known that all categorical models are internally extensional see [19] for a more detailed discussion of this point. spec InternalLogic = op eq : pred(a a) x : a eq(x ; x) tt(a) x ; y : a x res eq(x ; y) x ; y : a y res eq(x ; y) ops all ; ex : pred(pred(a) or ; impl : pred( unit unit) ff : pred(unit) neg : pred( unit) all(a) ....

J. C. Mitchell and P. J. Scott, Typed lambda models and cartesian closed categories, Categories in Computer Science and Logic, Contemp. Math., vol. 92, Amer. Math. Soc., 1989, pp. 301-316.

....interpreted by arbitrary sets equipped with an application operation of the appropriate type. Comprehension is still required; however, the way terms are interpreted is now part of the structure of the model rather than just an existence axiom. Intensional Henkin models are discussed, e.g. in [24, 22]. The notion chosen for HasCasl, for reasons of semantic as well as methodological nature, is that of intensional Henkin models. To begin, moving away from standard models avoids the well known incompleteness problems. Extensionality, moreover, carries the disadvantage of destroying the existence ....

.... Extensionality of Henkin models becomes, in the categorical setting, wellpointedness of the relevant pccc; in general, a category is called well pointed if 8 1 is a generator (it is easy to see that this implies determination of subobjects by their elements in a pccc) It has been pointed out in [22] that there is no technical reason to restrict attention to categorical models with this property. Finally, extensionality may easily be introduced by an appropriate axiom: spec Extensionality = forall a; b : type; f ; g : a b (8x : a f (x ) g(x ) f = g 2.4 Predicates and total ....

J. C. Mitchell and P. J. Scott, Typed lambda models and cartesian closed categories, Categories in Computer Science and Logic, Contemp. Math., vol. 92, Amer. Math. Soc., 1989, pp. 301-316.

....all standard models in such categories, are (strongly) complete. It has been an open question for some time whether topological semantics are complete in this sense. Results of the kind given here go back to L. Henkin [9] who in effect showed that non standard, set valued semantics are complete ([14] for some fine points) An oft cited result of H. Friedman [8] established the strong completeness of standard, set valued semantics for the theory consisting of a single basic type (no constants or equations) In this same vein, G. Plotkin has extended the result to certain categories of posets ....

J. Mitchell and P. Scott, Typed lambda models and cartesian closed categories, Contemporary Mathematics 92 (1988), 301--316.

....semantics is given, which is intentionally quite weak (it satisfies fi and j conversion for abstraction and a limited form of fi conversion for abstraction) and is tailored to obtaining the results of that paper and no more. See also [1] for a semantics in a similar style; and see [7] for a detailed comparison between the categorical and the environment style models in the case of the simply typed lambda calculus. For both kinds of model, part of the structure is a cartesian closed category K which is used in particular to give denotations to the closed types and terms. ....

J. C. Mitchell and P. J. Scott, Typed Lambda Models and Cartesian Closed Categories. In J. W. Gray (ed.), Categories in Computer Science and Logic (Proc. Boulder, 1987), Contemporary Math. (Amer. Math. Soc., Providence RI), to appear.

....to construct useful models compared with the more traditional sets and elements approach. The work of Reynolds and Oles [ 1985 ] on the semantics of block structure using functor categories is an example. As the example above shows, functor categories are not well pointed in general. See [ Mitchell and Scott, 1989 ] for a comparison of the categorical and set theoretic approaches in the case of the simply typed lambda calculus. 3 Categorical Datatypes This section explores the categorical treatment of various type constructors. We postpone considering types that depend upon variables until section 6, and ....

J. C. Mitchell and P. J. Scott. Typed lambda models and cartesian closed categories (preliminary version). In Gray and Scedrov [ 1989 ] , pages 301--316.

....of extensionality or uniqueness rule. This rule says: any term t satisfying the same defining equations as a given term, must be equal to that term. The uniqueness rule may be expressed in first order logic as a universally quantified conditional equation, possibly between terms of higher type [22] . However, in many interesting cases (e.g. product and coproduct types ) we can do much better: we can present uniqueness equationally and build a terminating ( strongly normalizing) higher typed rewrite system for the theory [20] 1 Here we use a technique of Lambek[18] involving Mal cev ....

.... types, ii) recursors are necessary for more general type theories (e.g. Girard s system F) as well as for more general categorical frameworks (e.g. Lambek s multicategories, 19] b) We may consider extensions of typed lambda calculus by adding first order logic to the equational theory [23, 22] . This will be discussed in the next section. Finally, we may extend by additional data types. For example, in the last section we consider adding the data type of Brouwer ordinals, among others. Theory and Applications of Categories, Vol. 6, No. 4 50 Notation: As usual, the type expression A 1 ....

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J. C. Mitchell and P. J. Scott,Typed Lambda Models and Cartesian Closed Categories, Contemp. Math. 92, 1989, pp. 301-316.

....between free P ccc s and typed calculi with no free arrows or additional theories. The extension to the more general case is discussed in Remark 3.10. 3.1. Typed Calculus We briefly recall the typed calculus, as presented in (Barendregt 1984; Girard, Lafont, Taylor 1989; Lambek and Scott 1986; Mitchell and Scott 1989). Definition 3.1. Typed calculus) Let Sorts be a set of sorts (or atomic types) The typed calculus generated by Sorts is a formal system consisting of Types, Terms and Equations between terms, as follows: Types This is the set inductively generated from the set of Sorts using the following ....

J. C. Mitchell and P. J. Scott,Typed Lambda Models and Cartesian Closed Categories, in Contemp. Math.: Categories in Computer Science and Logic (J. Gray and A. Scedrov, eds), 92 (1989), pp. 301-316.

....of r, where = is beta eta conversion, and where g o f denotes function composition. The equation can be restated in cartesian closed categories ( ccc s) Recall that simply typed lambda calculi correspond to ccc s and that in any ccc morphisms A B uniquely correspond to morphisms 1 A ) B [23, 19] . We shall blur this latter distinction and abuse the notation accordingly. In any ccc, then, the above equation says that for any morphism f : A B the following diagram commutes. 2 B 2B B 2B r B A2A A2A r A f 2 f f 2 f This means that the instantiations of r at all values of ....

....between cartesian closed categories ( ccc s) and typed lambda calculus and blur, for example, the distinction (in the associated typed lambda calculus) between closed terms m : A ) B, open terms m(x) B containing at most one free variable x : A, and arrows m : A 0 B, 19] pp. 75 78, and [23]) Substitution vs composition will be equally abused. Readers may easily restate results in the notation of their choice. 3 2 The Dinatural Calculus The basic view that led to the work on functorial polymorphism in [2] is that we may interpret polymorphic type expressions oe(ff 1 ; ff ....

J. C. Mitchell and P. J. Scott. Typed Lambda Models and Cartesian Closed Categories, Contemp. Math. 92, pp. 301-316.

....of r, where = is beta eta conversion, and where g o f denotes function composition. The equation can be restated in cartesian closed categories ( ccc s) Recall that simply typed lambda calculi correspond to ccc s and that in any ccc morphisms A B uniquely correspond to morphisms 1 A ) B [23, 19] . We shall blur this latter distinction and abuse the notation accordingly. In any ccc, then, the above equation says that for any morphism f : A B the following diagram commutes. B Theta B B Theta B r B A Theta A A Theta A r A f Theta f f Theta f This means that the ....

....cartesian closed categories ( ccc s) and typed lambda calculus and blur, for example, the distinction (in the associated typed lambda calculus) between closed terms m : A ) B, open terms m(x) B containing at most one free variable x : A, and arrows m : A Gamma B, 19] pp. 75 78, and [23]) Substitution vs composition will be equally abused. Readers may easily restate results in the notation of their choice. 2 The Dinatural Calculus The basic view that led to the work on functorial polymorphism in [2] is that we may interpret polymorphic type expressions oe(ff 1 ; ff n ....

J. C. Mitchell and P. J. Scott. Typed Lambda Models and Cartesian Closed Categories, Contemp. Math. 92, pp. 301-316.

....the three canonical morphisms described above. 3 Logical Relations and Logical Permutations Logical relations play an important role in the recent proof theory and semantics of typed lambda calculi [35, 39, 40, 43] We begin with logical relations on Henkin models; for further developments see [5, 35, 37, 38]. 3.1 Definitions and Examples Consider a simply typed lambda calculus with product types. A Henkin model is a wellpointed cartesian closed category (ccc) Equivalently, a Henkin model is a type indexed family of sets A = fA oe j oe a type g where A 1 = fg; A oe Theta = A oe Theta A ; A ....

J.C. Mitchell, P.J. Scott, Typed Lambda Models and Cartesian Closed Categories, in: Contemporary Mathematics Vol. 92, (1989), pp. 301-316.

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