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A construction of Type:Type in MartinLöf's partial type theory with one universe
"... ing on w and pairing with oe(p(c); (x)Ap(q(c); x) ! p(c)) in the first coordinate yields hoe(p(c);(x)Ap(q(c); x) ! p(c)); (w)(Ap(q(c); p(w)); (x)Ap(q(c); Ap(q(w); x)))i 2 PAR; i.e. s (c) 2 PAR. We define the operator that builds the universe (U 1 ; T 1 ) by putting f(c) := s (c) +hn 1 ; (x ..."
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ing on w and pairing with oe(p(c); (x)Ap(q(c); x) ! p(c)) in the first coordinate yields hoe(p(c);(x)Ap(q(c); x) ! p(c)); (w)(Ap(q(c); p(w)); (x)Ap(q(c); Ap(q(w); x)))i 2 PAR; i.e. s (c) 2 PAR. We define the operator that builds the universe (U 1 ; T 1 ) by putting f(c) := s (c) +hn 1 ; (x)R 1 (x; p(c))i; for c 2 PAR, and let e := fix((c)f(c)). Hence e 2 PAR is a fixed point of f , e = f(e). The right summand of f corresponds to the rules (2). We now interpret Type:Type. The universe (U 1 ; T 1 ) is defined by letting U 1 := T (p(e)) and T 1 (a) := T (Ap(q(e); a)); for a 2 U 1 . Thus the rules (1) are verified. Using the equality e = f(e) and the commutation of T with \Sigma, \Pi and + we get U 1 = T (p(e)) = T (p(f(e))) (4) = T (oe(p(e); (x)Ap(q(e); x) ! p(e))) + T (n 1 ) = (\Sigmax 2 T (p(e)))[T (Ap(q(e); x)) \Gamma! T (p(e))] +N 1 = (\Sigmax 2 U 1 )[T 1 (x) \Gamma! U 1 ] +N 1 and hence j(0 1 ) 2 U 1 . Furthermore we have T 1 (j(0 1 )) = T (Ap(q(...
LOGIC & COMPUTATION 12 V. BreazuTannen
, 1989
"... Abstract. Inheritance in the form of subtyping is considered in the framework of a polymorphic type discipline with records, variants, and recursive types. We give a denotational semantics based on the paradigm that interprets subtyping as explicit coercion. The main technical result gives a coheren ..."
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Abstract. Inheritance in the form of subtyping is considered in the framework of a polymorphic type discipline with records, variants, and recursive types. We give a denotational semantics based on the paradigm that interprets subtyping as explicit coercion. The main technical result gives a coherent interpretation for a strong rule for deriving inheritances between recursive types. 1
Developing Theories of Types and Computability
, 1999
"... Introduction Domain Theory, type theory (both in the style of MartinLof [40, 41] and in the polymorphic style of Girard/Reynolds [23, 56]), and topos theory (both in the topological/sheaftheoretic treatments and in the realizability approach going back to the early work of Kleene) have attempted ..."
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Introduction Domain Theory, type theory (both in the style of MartinLof [40, 41] and in the polymorphic style of Girard/Reynolds [23, 56]), and topos theory (both in the topological/sheaftheoretic treatments and in the realizability approach going back to the early work of Kleene) have attempted to improve on set theory by providing a large suite of closure conditions on domains/types/objects as well as a farreaching logic of properties emphasizing the computable/constructive aspects of the definitions and qualities of functions. Scott's domain theory, (and the many variations proposed and studied; see [2] and [75] for recent introductions with references) has been especially successful in allowing for recursive definitions of types (i.e., solutions to domain equations) but at the expense of introducing a complex structure of "partial elements" in order to have solutions to fixedpoint equations in the domains. Moreover, the topological and e
1 An expression of closure to efficient causation in terms of lambdacalculus1
"... In this paper, we propose a mathematical expression of closure to efficient causation in terms of λcalculus; we argue that this opens up the perspective of developing principled computer simulations of systems closed to efficient causation in an appropriate programming language. An important implic ..."
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In this paper, we propose a mathematical expression of closure to efficient causation in terms of λcalculus; we argue that this opens up the perspective of developing principled computer simulations of systems closed to efficient causation in an appropriate programming language. An important implication of our formulation is that, by exhibiting an expression in λcalculus, which is a paradigmatic formalism for computability and programming, we show that there are no conceptual or principled problems in realizing a computer simulation or model of closure to efficient causation. We conclude with a brief discussion of the question whether closure to efficient causation captures all relevant properties of living systems. We suggest that it might not be the case, and that more complex definitions could indeed create some obstacles to computability.
Function Spaces of Posets with Projections
, 2002
"... This paper investigates function spaces of structures consisting of a partially ordered set together with some directed family of projections. ..."
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This paper investigates function spaces of structures consisting of a partially ordered set together with some directed family of projections.
A New Model Construction for the Polymorphic Lambda Calculus
"... Various models for the GirardReynolds secondorder lambda calculus have been presented in the literature. Except the term model they are either realizability or domain models. In this paper a further model construction is introduced. Types are interpreted as inverse limits of #cochains of finite s ..."
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Various models for the GirardReynolds secondorder lambda calculus have been presented in the literature. Except the term model they are either realizability or domain models. In this paper a further model construction is introduced. Types are interpreted as inverse limits of #cochains of finite sets. The corresponding morphisms are sequences of maps acting locally on the finte sets in the omegacochains. The model can easily be turned into an effectively given one. Moreover, it can be arranged in such a way that the universal type (ForAll t.t) representing absurdity in the higherorder logic defined by the type structure is interpreted by the empty set, which means that it is also a model of this logic.
Finitely Generated RankOrdered Sets as a Model for Type: Type
"... The collection of isomorphism classes of finitely generated rankordered sets is shown to be a finitely generated rankordered set again. This is used to construct a model of the simply typed lambda calculus extended by the assumption Type: Type. Beside this, the structure of rankordered sets is st ..."
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The collection of isomorphism classes of finitely generated rankordered sets is shown to be a finitely generated rankordered set again. This is used to construct a model of the simply typed lambda calculus extended by the assumption Type: Type. Beside this, the structure of rankordered sets is studied. They can be represented as inverse limits of !cochains of substructures, each being a retract of the following. The category of such limits is equivalent to the category of rankordered sets. Introduction There has been a controversial discussion among type theoreticians whether, or not, one should accept that the collection of all types is itself a type (cf. e. g. [3]). In the recent systems by MartinLof, in Girard's systems F and F ! , and CoquandHuet's Calculus of Construction this assumption is carefully avoided, whereas in programming languages like those studied by Burstall and Lampson [5], or by Cardelli [6], it is explicitly used. Models of such calculi with the axiom Type...
This is a preprint of a paper that has been submitted to Information and Computation. On Functors Expressible in the Polymorphic Typed Lambda Calculus
, 1991
"... Given a model of the polymorphic typed lambda calculus based upon a Cartesian closed category K, there will be functors from K to K whose action on objects can be expressed by type expressions and whose action on morphisms can be expressed by ordinary expressions. We show that if T is such a functor ..."
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Given a model of the polymorphic typed lambda calculus based upon a Cartesian closed category K, there will be functors from K to K whose action on objects can be expressed by type expressions and whose action on morphisms can be expressed by ordinary expressions. We show that if T is such a functor then there is a weak initial Talgebra and if, in addition, K possesses equalizers of all subsets of its morphism sets, then there is an initial Talgebra. These results are used to establish the impossibility of certain models, including those in which types denote sets and S → S ′ denotes the set of all functions from S to S ′.