The theory of computability, or basic recursive function theory as it is often called, is usually motivated and developed using Church's Thesis. Here we show that there is an alternative computability theory in which some of the basic results on unsolvability become more absolute, results on completeness become simpler, and many of the central concepts become more abstract. In this approach computations are viewed as mathematical objects, and the major theorems in recursion theory may be classified according to which axioms about computation are needed to prove them. The theory is a typed theory of functions over the natural numbers, and there are unsolvable problems in this setting independent of the existence of indexings. The unsolvability results are interpreted to show that the partial function concept, so important in computer science, serves to distinguish between classical and constructive type theories (in a different way than does the decidability concept as expressed in the ...

In this paper it is shown how operational semantic methods may be naturally extended to encompass many of the concepts of denotational semantics. This work builds on the standard development of an operational semantics as an interpreter and operational equivalence. The key addition is an operational ordering on sets of terms. From properties of this ordering a closure construction directly yields a fully abstract continuous cpo model. Furthermore, it is not necessary to construct the cpo, for principles such as soundness of fixed-point induction may be obtained by direct reasoning from this new ordering. The end result is that traditional denotational techniques may be applied in a purely operational setting in a natural fashion, a matter of practical importance for developing semantics of realistic programming languages. 1 Introduction This paper aims to accomplish a degree of unification between operational and denotational approaches to programming language semantics by recasting d...

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...

It is well-known that the proof theory of many sufficiently powerful logics may be represented internally by Godelization. Here we show that for Constructive Type Theory, it is furthermore possible to define a semantics of the types in the type theory itself, and that this procedure results in new reasoning principles for type theory. Paradoxes are avoided by stratifying the definition in layers. 1 Introduction Given a sufficiently powerful logical theory L such as Peano Arithmethic, it is well-known that the proof theory of L may be expressed internally via Godelization. This is accomplished by defining a metafunction dAe that encodes formulas A as data, and a predicate Provable L (dAe) which is true just when formula A is provable. This gives L knowledge of its own proof theory, but it doesn't know that it knows it: the embedded proof theory could just as well be for some different logic. What is needed then are principles of self-knowledge that connect the provability predicate wit...

Types 83 9.1 Syntax : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 83 9.2 Typing Rules : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 83 9.3 Operational Semantics : : : : : : : : : : : : : : : : : : : : : : : : 84 iv 9.4 Impredicative Existentials : : : : : : : : : : : : : : : : : : : : : : 84 9.5 Representation Independence : : : : : : : : : : : : : : : : : : : : 85 9.6 Projection Notation : : : : : : : : : : : : : : : : : : : : : : : : : 85 9.7 References : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 87 10 Modularity 88 10.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 88 10.2 A Critique of Some Modularity Mechanisms : : : : : : : : : : : : 88 10.3 Basic Modules : : : : : : : : : : : : : : : : : : : : : : : : : : : : 93 10.4 Module Hierarchies : : : : : : : : : : : : : : : : : : : : : : : : : : 97 10.5 Parameterized Modules : : : : : : : : : : : : : : : : : : : : : : : 98 10.6 References : : : : : : : : : : : ...