E. Ritter. Categorical Abstract Machines for Higher-Order Typed Lambda Calculi. PhD thesis, Trinity College, Cambridge, September 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....1 ; x m i ) and h = hx 1 ; x m;Mi ; x : A, then u(g; h) hx 1 ; x m;Mi ; x : A. Alternatively, the de nition of prestructures can be formulated so that the context extension operation, arises from the Grothendieck construction, so that the h;i notation be justi ed as in [13] (cf. The requirement of (4) amounts to the existence of a natural isomorphism, cur W : hom J (W ) D A) p D;A C; B) hom J (W ) D) C; D;A (B) cur 1 W ; 19) with the co unit of the adjunction, the application map, app W : p D;A D;A = 1 J (W ) D A) given by ....

E. Ritter. Categorical Abstract Machines for Higher-Order Typed Lambda Calculi . Ph.D. thesis, University of Cambridge, 1992.

.... ) Why ) Explain why this defines indeed a category and find an extension of the assignment D to a contravariant functor from C to the category of categories. Such a functor is called an indexed category and forms the heart of Curien and Ehrhard s notion of D categories (Curien 1989; Ritter 1992). 3.3 Semantic type formers In order to interpret a type theory in a CwF we must specify how the various type and term formers are to be interpreted. This results in certain requirements on a CwF which follow very closely the syntactic rules. We give a precise definition for closure under Pi; ....

....a type theory with a quotient type former into ordinary type theory and other applications of syntactic models are described. Connections between category theoretic semantics and abstract machines have been noticed in (Curien 1986) and (Ehrhard 1988) and were subsequently exploited and applied in (Ritter 1992) where an evaluator for the Calculus of Constructions is derived from its category theoretic semantics. Last, but not least we mention the use of domain theoretic interpretations of type theory in order to establish the consistency of general recursion and fixpoint combinators with dependent ....

Ritter, E. (1992). Categorical Abstract Machines for Higher-Order Typed Lambda Calculi. Ph. D. thesis, University of Cambridge.

....construction presented in [Wer92] 1.5. The use of categories The machinery of Category Theory has been used and proven useful for the semantic investigation of Type Theories. e.g. this has been extensively studied in the work of Thomas Streicher [Str89] 12 and Bart Jacobs [Jac91] Moreover in [Rit92] the categorical semantics of CC is used directly as a starting point for an implementation. It has been noted 13 that the naive use of categorical notions does not necessarily produce a sound interpretation of the syntax. Often we have to introduce additional assumptions (e.g. split ....

....implicit presentation (Church syntax) We will also show the equivalence of the judgement and the conversion presentation (for pure CC without the j rule) We use a particular notion of reduction tight reduction which is essential to our approach. This presentation should be compared to [Rit92] where essentially the same goals are achieved using categorical combinators. 2.1. The judgement presentation of CC In our presentation we largely follow [Str91] i.e. ffl We use equality as a judgement. ffl We do not confuse types and terms, i.e. we avoid chains of colons. Therefore we have to ....

Eike Ritter. Categorical Abstract Machines for Higher-Order Typed Lambda Calculi. PhD thesis, University of Cambridge, 1992.

....algebraic nature of the category theory corresponding to various kinds of logic and type theory gives rise to var iable free, combinatory presentations of such systems. These have been used as the basis of abstract machines for expression evaluation and type checking: see [ Curien, 1993 ] Ritter, 1992 ] ....

E. Ritter. Categorical Abstract Machines for Higher-Order Typed Lambda Calculi. PhD thesis, Cambridge Univ., 1992. 90 Andrew M. Pitts

.... a 0 : Gamma A Gamma a a 0 2 A : Set There is no set constructor for context equality, since our base category has a set and not a setoid of objects (contexts) We note the similarity to Martin Lof s substitution calculus [17] which (unlike Ehrhard s [10] Curien s [7] and Ritter s [20]) lacks a judgement for context equality. The element constructors in the definition can be divided into three kinds: Those which correspond to operator symbols of the generalized algebraic theory of cwfs, such as the rules for composition, identity, and substitution. General rules for ....

E. Ritter. Categorical Abstract Machines for Higher-Order Typed Lambda Calculi. PhD thesis, Trinity College, Cambridge, September 1992.

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E. Ritter. Categorical Abstract Machines for Higher-Order Typed Lambda Calculi. PhD thesis, Trinity College, Cambridge, September 1992.

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E. Ritter. Categorical Abstract Machines for Higher-Order Typed Lambda Calculi . Ph.D. thesis, University of Cambridge, 1992.

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Eike Ritter. Categorical abstract machine for higher-order typed - calculus. Phd thesis, Univ. of Cambridge, 30 Sept, 1992.

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Eike Ritter. Categorical abstract machine for higher-order typed -calculus. Phd thesis, Univ. of Cambridge, 30 Sept, 1992.

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E. Ritter. Categorical Abstract Machines for Higher-Order Typed Lambda Calculi. PhD thesis, Cambridge U., Trinity College, September 1992.

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