A Modern Perspektive on Type Theory (gebundenes Buch)

InhaltsangabePreface. Preliminaries. Overview of this book. Acknowledgements. Introduction. I: The Evolution of Type Theory until the 1940s. 1. Prehistory. 1.a. Paradox threats. 1.b. Paradox threats in formal systems. 2. Type theory in Pricipia Mathematica. 2.a. Principia's propositional functions. 2.b. The Ramified Theory of Types RTT. 2.c. Properties of RTT. 2.d. Legal propositional functions. Conclusions. 3. Deramification. 3.a. History of the deramification. 3.b. The Simple Theory of Types STT. 3.c. Are orders to be blamed? Conclusions. II: Propositions as Types, Pure Type Systems, AUTOMATH. 4. Propositions as Types and Pure Type Systems. 4.a. Propositions as types and proofs as terms (PAT). 4.b. Lambda calculus. 4.c. Pure type systems. 5. The prePAT and STT in PATstyle. 5.a. RTT in PATstyle. 5.b. STT in PATstyle. 6. A correspondence between RTT and the system Nuprl. 6.a. On the role of orders. 6.b. The Nuprl type system. 6.c. RTT in Nuprl. Conclusions. 7. Automath. 7.a. Description of AUTOMATH. 7.b. From AUT-68 towards a PTS. 7.c. lambda68. 7.d. More suitable pure type systems for AUTOMATH. Conclusions. III: Extensions of Pure Type Systems. 8. Pure type systems with definitions. 8.a. Definitions in contexts. 8.b. Definitions in the terms and the contexts. 9. The Barendregt cube with parameters. 9.a.On parameters in the Barendregt cube. 9.b. The Barendregt cube refined with parameters. 10. Pure Type Systems with parameters and definitions. 10.a. Parametric constraints and definitions. 10.b. Properties of terms. 10.c. Properties of legal terms. 10.d. Restrictive use of parameters. 10.e. Systems in the redefined Barendregt cube. 10.f. First-order predicate logic. Conclusions: yet another extension of PTSs? Practical motivation. The heart of type theory. Future work. A: Type Systems in this Book. A.a. Pure Type Systems. A.b. The Barendregt cube. A.c. The Ramified Theory of Types. A.d. The Simple Theory of Types. A.e. Church's simply typed lambda-calculus lambda-->Church. A.f. A fragment of Nuprl in PTS-style. A.g. AUTOMATH. A.h. Pure Type Systems with definitions. A.i. Pure Type Systems with parametric constants. A.j. A CD-PTS and its subsystems. Bibliography. Subject Index. Name Index. List of Figures.