In 1933 G odel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that G odel's provability calculus is nothing but the forgetful projection of LP. This also achieves G odel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a Brouwer-Heyting-Kolmogorov style provability semantics for Int which resisted formalization since the early 1930s. LP may be regarded as a unified underlying structure for intuitionistic, modal logics, typed combinatory logic and #-calculus.

We present a possible computational content of the negative translation of classical analysis with the Axiom of Choice. Our interpretation seems computationally more direct than the one based on Godel's Dialectica interpretation [10, 18]. Interestingly, thisinterpretation uses a re nement of the realizibility semantics of the absurdity proposition, which is not interpreted as the empty type here. We alsoshowhow to compute witnesses from proofs in classical analysis, and how to interpret the axiom of Dependent Choice and Spector's Double Negation Shift.

This paper gives an introduction to type theory, focusing on its recent use as a logical framework for proofs and programs. The first two sections give a background to type theory intended for the reader who is new to the subject. The following presents Martin-Lof's monomorphic type theory and an implementation, ALF, of this theory. Finally, a few small tutorial examples in ALF are given.

We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a first step in this programme, we give an extended survey of the di#erent strands of research on higher type computability to date, bringing together material from recursion theory, constructive logic and computer science. The paper thus serves as a reasonably complete overview of the literature on higher type computability. Two sequel papers will be devoted to developing a more systematic account of the material reviewed here.

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We give a proof for the strong normalization result in the intersection type discipline, which we obtain by putting together some well-known results and proof techniques. Our proof uses a variant of Klop's extended -calculus, for which it is shown that strong normalization is equivalent to weak normalization. This is

In this paper we present a simple argument for normalization of the fragment of Martin-Löf's type theory that contains the natural numbers, dependent function types and the first universe. We do this by building a realizability model of this theory which directly reflects that terms and types are generated simultaneously.

This paper aims to give a readable and reasonably accessible account of some ideas linking the currently still largely separate worlds of concurrency theory and process algebra, on the one hand, and type theory, categorical models and realizability on the other. Background in process algebra may be found in standard texts such as [Hoa85, Hen88, Mil89, Ros97]; while background in realizability, categorical models etc. is provided by texts such as [GLT89, AL91, Cro93, AC98, BW99]. A modest background in either or both of these fields should suffice to understand the main ideas. Most of the detailed verification of properties of the formal definitions we will present is left as a series of exercises. The diligent reader who attempts a number of these should get some feeling for the interplay between concrete process-theoretic notions, and more abstract logical and categorical ideas, which is characteristic of this topic. It is this interplay which makes the topic a fascinating one for the author; I hope this brief introduction, to a field which is still wide open for further development, succeeds in conveying something of this fascination to the reader.

Constructive Mathematics might be regarded as a fragment of classical mathematics in which any proof of an existence theorem is equipped with a computable function giving the solution of the theorem. Limit Computable Mathematics (LCM) considered in this note is a fragment of classical mathematics in which any proof of an existence theorem is equipped with a function computing the solution of the theorem in the limit.

Constructive Zermelo-Fraenkel Set Theory, CZF, has emerged as a standard reference theory that relates to constructive predicative mathematics as ZFC relates to classical Cantorian mathematics. A hallmark of this theory is that it possesses a type-theoretic model. Aczel showed that it has a formulae-as-types interpretation in Martin-Löf’s intuitionist theory of types [14, 15]. This paper, though, is concerned with a rather different interpretation. It is shown that Kleene realizability provides a self-validating semantics for CZF, viz. this notion of realizability can be formalized in CZF and demonstrably in CZF it can be verified that every theorem of CZF is realized. This semantics, then, is put to use in establishing several equiconsistency results. Specifically, augmenting CZF by well-known principles germane to Russian constructivism and Brouwer’s intuitionism turns out to engender theories of equal proof-theoretic strength with the same stock of provably recursive functions.