First-Order Programming Theories

This work presents a purely classical first-order logical approach to the field of study in theoretical computer science sometimes referred to as the theory of programs, or programming theory. This field essentially attempts to provide a precise mathematical basis for the common activities involved in reasoning about computer programs and programming languages, and it also attempts to find practical applications in the areas of program specification, verification and programming language design. Many different approaches with different mathematical frameworks have been proposed as a basis for programming theory. They differ in the mathe matical machinery they use to define and investigate programs and program properties and they also differ in the concepts they deal with to understand the programming paradigm. Different approaches use different tools and viewpoints to characterize the data environment of programs. Most of the approaches are related to mathe matical logic and they provide their own logic. These logics, however, are very eclectic since they use special entities to reflect a special world of programs, and also, they are usually incomparable with each other. This Babel's mess irritated us and we decided to peel off the eclectic com ponents and try to answer all the questions by using classical first-order logic.

This book proposes a purely classical first-order logical approach to the theory of programming. It provides a unified framework to present new programming theories and logics systematically and to investigate and compare existing ones with respect to their expressive and proof-theoretic powers.

Mathematical Background.- 1. Logic and Model Theory.- 2. Inductive Definability.- I Computability.- 3. Introduction to Part I.- 4. Main Properties of Program Schemas.- 5. Extension of Program Schemas.- 6. Program Schemas with Stacks.- 7. Computability.- 8. On Inductive Definability of 1- and 2-Computable Relations.- II Extended Dynamic Logics.- 9. Introduction to Part II.- 10. Description of Program Properties.- 11. Den-based Descriptive Languages.- 12. The Problem of Completeness.- 13. Dynamic Logic Generated by Extension.- 14. Continuous Denotational Semantics.- 15. Definable Denotational Semantics.- III Temporal Characterization of Programs.- 16. Introduction to Part III.- 17. Temporal Logic.- 18. Temporal Logical Description of Program Properties.- 19. Is Temporal Logic Expressible in Dynamic Logic?.- 20. Is Dynamic Logic Expressible in Temporal Logic?.- 21. The Case of Enumerable Models.- 22. Temporal Axiomatization of Program Verification Methods.- IV Programming Logic with Explicit Time.- 23. Introduction to Part IV.- 24. Time Logic.- 25. Definability in Regular Time Theories.- 26. Expressive Power of Time.- Epilogue.- References.- Notations.