M. Hofmann. Syntax and semantics of dependent types. In A. Pitts and P. Dybjer, editors, Semantics and Logics of Computation, pages 79--130. Cambridge University Press, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... UP , x = 0 (x = 1 P ) By (1) we have 1 ffl U 00 P , 9t 2 Tree(1) 8z 2 Branch(1; t) U P (lf br(1; t; z) 3) Since 1 ffl UP , P we get from (2) that 1 ffl U 00 P , P ; hence : P , 9t 2 Tree(1) 8z 2 Branch(1; t) U(lf br(1; t; z) 4) We claim that (4) is not valid in the model [Hof97]. Indeed, if we take P = Q :Q, 4) implies (9t 2 Tree(1) 8z 2 Branch(1; t) U Q:Q (lf br(1; t; z) because : Q :Q) is logically true. Hence, by the definition of UQ:Q , 9t 2 Tree(1) 8z 2 Branch(1; t) lf br(1; t; z) 0 (lf br(1; t; z) 1 (Q :Q) 5) 34 Uniformity in the model ....

M. Hofmann, Syntax and Semantics of Dependent Types, in "Semantics and Logics of Computation", A. Pitts and P. Dybjer (eds.), Cambridge University Press, 1997, pp. 79-133.

....the last 10 years, and circulating in the form of lecture notes. This material includes among other things a variable free presentation of the logical framework in which Martin Lof style type systems are expressed. In essentials, I have adopted it here. Another source is Hofmann s survey article [48]. What is a logical framework One can try to answer this general question in two ways . by saying what a framework looks like, or its formal representation. by saying what a framework is for, whatever it looks like and however it is represented. In this chapter I am concerned only with a ....

....of the input vector. The extra structure with which the category of contexts is equipped is a contravariant functor to a category which is universal in a certain sense. There are various choices for this universal category. There is a brief survey in section 3. 2 of Hofmann s article [48]) The essential thing is that the objects of this universal category should be worlds rich enough to model both the types available in a context, and the values inhabiting those types. The contexts should be closed under extension of a context # by declaring a new coordinate position, whose ....

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M. Hofmann. Syntax and Semantics of Dependent Types, pages 79--127. Cambridge University Press, 1997.

.... Notes in Typed Lambda Calculus Thierry Coquand Chalmers University November 1998 Introduction Since quite good books [12, 9] or review article [3, 10, 6, 18] are available on typed lambda calculi, these notes limit themselves to some historical remarks and some points that I consider delicate important. In particular, I will emphasize the connections with logic and one goal of this course is that the reader can understand and appreciate the use of ....

M. Hofmann. Syntax and Semantics of Dependent Type. In Semantics of Logics of Computation, P. Dybjer and A. Pitts, eds., Cambridge University Press, 1997.

....etc. Note however, that we here only treat the categorical semantics of induction recursion and not of the logical framework. The reader is referred to the literature on categorical semantics of dependent types for the latter, see for example Cartmell [4] Seely [31] Dybjer [11] or Hofmann [13]. The categorical semantics of universes has previously been investigated by Mendler [20] There he considers various universes which are all inductiverecursive definitions with D = set. Our approach goes further since we consider inductive recursive definitions with arbitrary D and characterize ....

M. Hofmann. Syntax and semantics of dependent types. In A. Pitts and P. Dybjer, editors, Semantics and Logics of Computation, pages 79--130. Cambridge University Press, 1997.

....Category Theory but is generally self contained. Thomas Streicher [Str91] investigates the Calculus of Constructions also using Category Theory but seems to be a bit heavy going for the uninitiated. A good general reference and introduction for categorical models are Martin Hofmann s lecture notes [Hof95b]. On a more practical level it is certainly exciting to use one of the tools mentioned to play with Type Theory. The ALF system on which these notes are based is available by ftp from file:ftp.cs.chalmers.se:pub provers walf. The compiled version there is only for SUNs under Solaris. If you are ....

Martin Hofmann. Syntax and semantics of dependent types, 1995. to appear.

....system preserve the meaning of the forms of judgment. Thus in the presentation of the model we define the interpretation of all the expressions, in particular, of those which are not meaningful. This is in contrast with other presentations of models of type theory like Dybjer [Dyb96] Hofmann [Hof97] and Martin Lof s, in which only meaningful expressions are interpreted. In these presentations a model is required to have the same structure as type theory itself. One way to fulfill this requirement is to look at type theory as an initial algebra in a particular generalized algebraic theory ....

M. Hofmann. Syntax and semantics of dependent types. In A Pitts and P. Dybjer, editors, Semantics and Logics of Computation, pages 79--130. Cambridge University Press, 1997.

.... paper is a revised version of a paper that appears in the proceedings of the Joint CLICS TYPES Workshop on Categories and Type Theory, Goteborg, January 1995 [9] Much useful information on cwfs can also be found in the lecture notes on Syntax and Semantics of Dependent Types by Martin Hofmann [14]. He uses ordinary set theoretic cwfs as the central semantic notion and gives several examples. He also discusses the relationship with other categorical notions of model for dependent types and gives a detailed proof of the equivalence of cwfs and categories with attributes. Cwfs have also been ....

M. Hofmann. Syntax and semantics of dependent types. In A. Pitts and P. Dybjer, editors, Semantics and Logics of Computation. Cambridge University Press, 1996. To appear.

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Hofmann, M. (199?). Syntax and semantics of dependent types. In P. Dybjer and A. M. Pitts (Eds.), Semantics and Logics of Computation. Cambridge University Press. Hofmann, M. (1993, July). A model of intensional Martin-Lof type theory in which unicity of identity proofs does not hold. unpublished note, available on email request.

....by parametrisation from the obvious algorithms computing them. 2 5 Realisability sets In this section we define and explore an analogue of the category of realisability sets introduced by Moggi and others based on a BCK algebra supporting truth values and natural numbers. We refer to, e.g. [9] for an introduction to modest sets and realisability. We shall see that due to the absence of an S combinator hence of diagonalisation, the thus obtained category of modest sets is not cartesian closed. It is, however, an affine linear category w.r.t. to a natural tensor product based on the ....

Martin Hofmann. Syntax and semantics of dependent types. In A. M. Pitts and P. Dybjer, editors, Semantics and Logics of Computation, Publications of the Newton Institute, pages 79--130. Cambridge University Press, 1997.

....seems to be in the air . In a semantical context it has been around for a while, notably in the theory of idealised Algol [16, 15] and in semantic models of the calculus [18, 6] The only place in the literature where functor categories have been used explicitly to justify higher order syntax is [10]. It is, however, fair to say that the possibility of using functor categories for HOAS is part of the folklore. Indeed, I believe that most if not all of the results to be elaborated in this paper are known to one or the other person who has used functor categories in a semantic context. The ....

....Then use Equation 17 to analyse the types of constants. If our metalanguage is merely simply typed lambda calculus then the structure of functor categories exhibited so far suffices to interpret it. Dependent types in the metalanguage can also be accommodated in any presheaf category, see [10] for details. Also universes can be easily modelled. If impredicativity is desired like for COQ s Set one needs to consider presheaves relative to some constructive set theory which supports such universes in the first place. See [1] for details. Being a topos every presheaf category also ....

Martin Hofmann. Syntax and semantics of dependent types. In A. M. Pitts and P. Dybjer, editors, Semantics and Logics of Computation, Publications of the Newton Institute, pages 79--130. Cambridge University Press, 1997.

....obtained by parametrisation from the obvious algorithms computing them. 2 4.2 Realisability sets In this section we define and explore an analogue of the category of modest sets introduced by Moggi and others based on a BCK algebra supporting truth values and natural numbers. We refer to, e.g. [16] for an introduction to modest sets and realisability. We shall see that due to the absence of an S combinator hence of diagonalisation, the thus obtained category of modest sets is not cartesian closed. It is, however, an affine linear category w.r.t. to a natural tensor product based on the ....

Martin Hofmann. Syntax and semantics of dependent types. In A. M. Pitts and P. Dybjer, editors, Semantics and Logics of Computation, Publications of the Newton Institute, pages 79--130. Cambridge University Press, 1997.

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M. Hofmann. Syntax and semantics of dependent types. In A. Pitts and P. Dybjer, editors, Semantics and Logics of Computation, pages 79--130. Cambridge University Press, 1997.

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Martin Hofmann. Syntax and semantics of dependent types. In Semantics and logics of computation (Cambridge, 1995.

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M. Hofmann. Syntax and Semantics of Dependent Types, pages 79--127. Cambridge University Press, 1997.

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M. Hofmann. Syntax and semantics of dependent types. In A. Pitts and P. Dybjer, editors, Semantics and Logics of Computation. Cambridge University Press, 1996. To appear.

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