Open AccessBook
Proofs and Computations
Reads0
Chats0
TLDR
This book provides a unique self-contained text for advanced students and researchers in mathematical logic and computer science and develops the theoretical underpinnings of the first author's proof assistant MINLOG.Abstract:
Driven by the question, 'What is the computational content of a (formal) proof?', this book studies fundamental interactions between proof theory and computability. It provides a unique self-contained text for advanced students and researchers in mathematical logic and computer science. Part I covers basic proof theory, computability and Gdel's theorems. Part II studies and classifies provable recursion in classical systems, from fragments of Peano arithmetic up to 11-CA0. Ordinal analysis and the (Schwichtenberg-Wainer) subrecursive hierarchies play a central role and are used in proving the 'modified finite Ramsey' and 'extended Kruskal' independence results for PA and 11-CA0. Part III develops the theoretical underpinnings of the first author's proof assistant MINLOG. Three chapters cover higher-type computability via information systems, a constructive theory TCF of computable functionals, realizability, Dialectica interpretation, computationally significant quantifiers and connectives and polytime complexity in a two-sorted, higher-type arithmetic with linear logic.read more
Citations
More filters
Journal ArticleDOI
Complexity Hierarchies beyond Elementary
TL;DR: A hierarchy of fast-growing complexity classes is introduced and its suitability for completeness statements of many nonelementary problems is shown, with complexities ranging from simple towers of exponentials to Ackermannian and beyond.
Journal ArticleDOI
Mathematical theory of domains
TL;DR: The Hemaspaandra-Ogihara book is clearly intended for people who are already comfortable with the field, and are looking for a new perspective, and this is a book that flows easily and quickly from topic to topic.
Book ChapterDOI
The power of well-structured systems
TL;DR: This paper recalls the basic theory underlying well-structured systems and shows how two classic decision algorithms can be formulated as an exhaustive search for some "bad" sequences and describes new powerful techniques for the complexity analysis of WSTS algorithms.
Proof theoretic analysis by iterated reflection
TL;DR: For a sufficiently broad class of theories T, it was shown in this article that the uniform reflection principle for T is conservative over the corresponding local reflection principle, and that the hierarchy of uniform and local reflection principles can be constructed in a uniform manner.
References
More filters
Journal ArticleDOI
Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I
Journal ArticleDOI
On a Problem of Formal Logic
TL;DR: This paper is primarily concerned with a special case of one of the leading problems of mathematical logic, the problem of finding a regular procedure to determine the truth or falsity of any given logical formula.
Journal ArticleDOI
Recursively enumerable sets and degrees
TL;DR: In this paper, the relation of the structure of an R set to its degree is discussed, and the infinite injury priority method is proposed to solve the problem of scaling and splitting R sets.
Journal ArticleDOI
Untersuchungen über das logische Schließen. II
TL;DR: In this paper, a process for sizing cellulose fibers or cellulose fiber containing materials and a composition for carrying out the process are described, and a method for sizing according to the general formula of R1 is presented.