Domain theory is an established part of theoretical computer science, used in giving semantics to programming languages and logics. In mathematics and logic it has also proved to be useful in the study of algorithms. This book is devoted to providing a unified and self-contained treatment of the subject. The theory is presented in a mathematically precise manner which nevertheless is accessible to mathematicians and computer scientists alike. The authors begin with the basic theory including domain equations, various domain representations and universal domains. They then proceed to more specialized topics such as effective and power domains, models of lambda-calculus and so on. In particular, the connections with ultrametric spaces and the Kleene–Kreisel continuous functionals are made precise. Consequently the text will be useful as an introductory textbook (earlier versions have been class-tested in Uppsala, Gothenburg, Passau, Munich and Swansea), or as a general reference for professionals in computer science and logic.
This document has been prepared by the Cyber-Physical Systems Public Working Group (CPS PWG), an open public forum established by the National Institute of Standards and Technology (NIST) to support stakeholder discussions and development of a framework for cyber-physical systems. This document is a freely available contribution of the CPS PWG and is published in the public domain. Certain commercial entities, equipment, or materials may be identified in this document in order to describe a concept adequately. Such identification is not intended to imply recommendation or endorsement by the CPS PWG (or NIST), nor is it intended to imply that these entities, materials, or equipment are necessarily the best available for the purpose. All registered trademarks or trademarks belong to their respective organizations.
Abstract. One objective of this paper is the determination of the proof-theoretic strength of Martin-L6f's type theory with a universe and the type of well-founded trees. It is shown that this type system comprehends the consistency of a rather strong classical subsystem of second order arithmetic, namely the one with A~ comprehension and bar induction. As Martin-L6f intended to formulate a system of constructive (intuitionistic) mathematics that has a sound philosophical basis, this yields a constructive consistency proof of a strong classical theory. Also the proof-theoretic strength of other inductive types like Aczel's type of iterative sets is investigated in various contexts.Further, we study metamathematical relations between type theories and other frameworks for formalizing constructive mathematics, e.g. Aczel's set theories and theories of operations and classes as developed by Feferman. IntroductionMartin-L6f's intuitionistic theory of types was originally introduced as a system for formalizing intuitionistic mathematics. It is a typed theory of constructions containing types which themselves depend on the constructions contained in previously constructed types. These dependent types enable one to express the general Cartesian product of the resulting families of types, as well as the disjoint union of such a famiIy.Using the Brouwer-Heyting semantics of the logical constants one sees how logical notions are obtained in this theory, This is done by interpreting propositions as types and proofs as constructions [ML 84]. In addition to ground types for the finite sets and the set of natural numbers the theory can be taken to contain types which play * The second author would like to thank the National Science Foundation of the USA for support by grant DMS-9203443
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