The notion of a universe of types was introduced into constructive type theory by Martin-Löf (1975). According to the propositions-as-types principle inherent in

The significance of type theory to the theory of programming languages has long been recognized. Advances in programming languages have often derived from understanding that stems from type theory. However, these applications of type theory to practical programming languages have been indirect; the differences between practical languages and type theory have prevented direct connections between the two. This dissertation presents systematic techniques directly relating practical programming languages to type theory. These techniques allow programming languages to be interpreted in the rich mathematical domain of type theory. Such interpretations lead to semantics that are at once denotational and operational, combining the advantages of each, and they also lay the foundation for formal verification of computer programs in type theory. Previous type theories either have not provided adequate expressiveness to interpret practical languages, or have provided such expressiveness at the expense of essential features of the type theory. In particular, no previous type theory has supported a notion of partial functions (needed to interpret recursion in practical languages), and a notion of total functions and objects (needed to reason about data values), and an intrinsic notion of equality (needed for most interesting results). This dissertation presents the first type theory incorporating all three, and discusses issues arising in the design of that type theory. This type theory is used as the target of a type­theoretic semantics for a expressive programming calculus. This calculus may serve as an internal language for a variety of functional programming languages. The semantics is stated as a syntax­directed embedding of the programming calculus into type theory. A critical point arising in both the type theory and the type­theoretic semantics is the issue of admissibility. Admissibility governs what types it is legal to form recursive functions over. To build a useful type theory for partial functions it is necessary to have a wide class of admissible types. In particular, it is necessary for all the types arising in the type­theoretic semantics to be admissible. In this dissertation I present a class of admissible types that is considerably wider than any previously known class.

This paper is the first in a series whose objective is to study notions of large sets in the context of formal theories of constructivity. The two theories considered are Aczel's constructive set theory (CZF) and Martin-Lof's intuitionistic theory of types. This paper treats Mahlo's -numbers which give rise classically to the enumerations of inaccessibles of all transfinite orders. We extend the axioms of CZF and show that the resulting theory, when augmented by the tertium non datur, is equivalent to ZF plus the assertion that there are inaccessibles of all transfinite orders. Finally the theorems of that extension of CZF are interpreted in an extension of Martin-Lof's intuitionistic theory of types by a universe. 1 Prefatory and historical remarks The paper is organized as follows: After recalling Mahlo's -numbers and relating the history of universes in Martin-Lof type theory in section 1, we study notions of inaccessibility in the context of Aczel's constructive set theo...

this paper we study some extensions of the Kleene-Kreisel continuous functionals [7, 8] and show that most of the constructions and results, in particular the crucial density theorem, carry over from nite to dependent and transnite types. Following an approach of Ershov we dene the continuous functionals as the total elements in a hierarchy of Ershov-Scott-domains of partial continuous functionals. In this setting the density theorem says that the total functionals are topologically dense in the partial ones, i.e. every nite (compact) functional has a total extension. We will extend this theorem from function spaces to dependent products and sums and universes. The key to the proof is the introduction of a suitable notion of density and associated with it a notion of co-density for dependent domains with totality. We show that the universe obtained by closing a given family of basic domains with totality under some quantiers has a dense and co-dense totality provided the totalities on the basic domains are dense and co-dense and the quantiers preserve density and co-density. In particular we can show that the quantiers and have this preservation property and hence, for example, the closure of the integers and the booleans (which are dense and co-dense) under and has a dense and co-dense totality. We also discuss extensions of the density theorem to iterated universes, i.e. universes closed under universe operators. From our results we derive a dependent continuous choice principle and a simple order-theoretic characterization of extensional equality for total objects. Finally we survey two further applications of density: Waagb's extension of the Kreisel-Lacombe-Shoeneld-Theorem showing the coincidence of the hereditarily eectively continuous hierarchy...

. We study a semantics of dependent types and universe operators based on parametrized domains with totality. The main results are generalizations of the Kleene/Kreisel density theorem for the continuous functionals. This continues work of E. Palmgren and V. Stoltenberg{Hansen on the domain interpretation of dependent types, and of D. Normann on universes of wellfounded types with density. Key words: Continuous functionals, Domains, Totality, Dependent types, Universes 1. Introduction In Mathematical Logic and Computer Science there is growing interest in constructive type theories as developed by Martin{Lof [8]. This paper is concerned with a semantics of such theories within the realm of Ershov{Scott domains [5] with totality [10]. Erik Palmgren and Viggo Stoltenberg{Hansen [15], [17] developed a semantics for a partial type theory (modelling partial functions and functionals) based on the notion of a parametrization, i.e. a domain depending on parameters. Since this semantics wa...

Universes of types were introduced into constructive type theory by Martin-Lof [3]. The idea of forming universes in type theory is to introduce a universe as a set closed under a certain specified ensemble of set constructors, say C. The universe then "reflects" C. This is the second part of a paper which addresses the exact logical strength of a particular such universe construction, the so-called superuniverse due to Palmgren (cf. [4, 5, 6]). It is proved that Martin-Lof type theory with a superuniverse, termed MLS, is a system whose proof-theoretic ordinal resides strictly above the Feferman-Schutte ordinal \Gamma 0 but well below the Bachmann-Howard ordinal. Not many theories of strength between \Gamma 0 and the Bachmann-Howard ordinal have arisen. MLS provides a natural example for such a theory. In this second part of the paper the concern is with the with upper bounds. 1 Introduction One reason for splitting this paper into two parts was its sheer length. But the main reason ...

In this paper a hybrid type theory HTT is defined which combines the programming language notion of partial type with the logical notion of total type into a single theory. A new partial type constructor A is added to the type theory: objects in A may diverge, but if they converge, they must be members of A. A fixed point typing rule is given to allow for typing of fixed points. The underlying theory is based on ideas from Feferman's Class Theory and Martin Lof's Intuitionistic Type Theory. The extraction paradigm of constructive type theory is extended to allow direct extraction of arbitrary fixed points. Important features of general programming logics such as LCF are preserved, including the typing of all partial functions, a partial ordering ! ¸ on computations, and a fixed point induction principle. The resulting theory is thus intended as a general-purpose programming logic. Rules are presented and soundness of the theory established. Keywords: Constructive Type Theory, Logics...

September, 2005 This paper studies categorical models of predicative formal systems, like Aczel’s CZF or Martin-Löf Type Theory. Following the suggestion of Moerdijk and Palmgren that the collection of such categorical models should have the closure properties of toposes, I study the stability under sheaves of five possible axiomatisations. The conclusion will be that all the notions of a “predicative topos ” that I consider, are stable under presheaves, while most are stable under sheaves. This opens up the possibility of using sheaf models for studying predicative theories. 1