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Proofs and Computations.
This major graduate-level text provides a detailed, self-contained coverage of proof theory.
| Clasificación: | Libro Electrónico |
|---|---|
| Autor principal: | |
| Otros Autores: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Cambridge :
Cambridge University Press,
2011.
|
| Colección: | Perspectives in logic.
|
| Temas: | |
| Acceso en línea: | Texto completo |
MARC
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| 049 | |a UAMI | ||
| 100 | 1 | |a Schwichtenberg, Helmut. | |
| 245 | 1 | 0 | |a Proofs and Computations. |
| 260 | |a Cambridge : |b Cambridge University Press, |c 2011. | ||
| 300 | |a 1 online resource (482 pages) | ||
| 336 | |a text |b txt |2 rdacontent | ||
| 337 | |a computer |b c |2 rdamedia | ||
| 338 | |a online resource |b cr |2 rdacarrier | ||
| 490 | 1 | |a Perspectives in Logic | |
| 588 | 0 | |a Print version record. | |
| 520 | |a This major graduate-level text provides a detailed, self-contained coverage of proof theory. | ||
| 504 | |a Includes bibliographical references (pages 431-455) and index. | ||
| 505 | 8 | |6 880-01 |a 6.5.3. Dense and separating sets. -- 6.6. Notes -- Chapter 7: EXTRACTING COMPUTATIONAL CONTENT FROM PROOFS -- 7.1. A theory of computable functionals -- 7.1.1. Brouwer-Heyting-Kolmogorov and Gödel. -- 7.1.2. Formulas and predicates. -- 7.1.3. Equalities. -- 7.1.4. Existence, conjunction and disjunction. -- 7.1.5. Further examples. -- 7.1.6. Totality and induction. -- 7.1.7. Coinductive definitions. -- 7.2. Realizability interpretation -- 7.2.1. An informal explanation. -- 7.2.2. Decorating ₂!and | |
| 590 | |a ProQuest Ebook Central |b Ebook Central Academic Complete | ||
| 590 | |a eBooks on EBSCOhost |b EBSCO eBook Subscription Academic Collection - Worldwide | ||
| 650 | 0 | |a Computable functions. | |
| 650 | 0 | |a Machine theory. | |
| 650 | 0 | |a Proof theory. | |
| 650 | 4 | |a Machine theory. | |
| 650 | 6 | |a Fonctions calculables. | |
| 650 | 6 | |a Théorie des automates. | |
| 650 | 6 | |a Théorie de la preuve. | |
| 650 | 7 | |a COMPUTERS |x Machine Theory. |2 bisacsh | |
| 650 | 0 | 7 | |a Demostración, Teoría de la |2 embucm |
| 650 | 7 | |a Computable functions |2 fast | |
| 650 | 7 | |a Machine theory |2 fast | |
| 650 | 7 | |a Proof theory |2 fast | |
| 700 | 1 | |a Wainer, S. S. | |
| 776 | 0 | 8 | |i Print version: |z 9780521517690 |
| 830 | 0 | |a Perspectives in logic. | |
| 856 | 4 | 0 | |u https://ebookcentral.uam.elogim.com/lib/uam-ebooks/detail.action?docID=833389 |z Texto completo |
| 880 | 8 | |6 505-01/Grek |a 5.2.2. Collapsing properties of G. -- 5.2.3. The functors G, B and ϕ. -- 5.2.4. The accessible recursive functions. -- 5.3. Proof-theoretic characterizations of accessibility -- 5.3.1. Finitely iterated inductive definitions. -- 5.3.2. The infinitary system IDk(W)∞. -- 5.3.3. Embedding IDk(W) into IDk(W)∞. -- 5.3.4. Ordinal analysis of IDk. -- 5.3.5. Accessible = provably recursive in ID<∞. -- 5.3.6. Provable ordinals of IDk(W). -- 5.4. ID<∞ and Π11-CA0 -- 5.4.1. Embedding ID<(W) in Π11-CA0. -- 5.4.2. Reduction ofΠ11-forms toWi sets. -- 5.4.3. Conservativity of Π11-CA0 over ID<∞(W). -- 5.5. An independence result: extended Kruskal theorem -- 5.5.1. φ-terms, trees and i-sequences. -- 5.5.2. The computation sequence is bad. -- 5.6. Notes -- Part 3: CONSTRUCTIVE LOGIC AND COMPLEXITY -- Chapter 6: COMPUTABILITY IN HIGHER TYPES -- 6.1. Abstract computability via information systems -- 6.1.1. Information systems. -- 6.1.2. Domains with countable basis. -- 6.1.3. Function spaces. -- 6.1.4. Algebras and types. -- 6.1.5. Partial continuous functionals. -- 6.1.6. Constructors as continuous functions. -- 6.1.7. Total and cototal ideals in a finitary algebra. -- 6.2. Denotational and operational semantics -- 6.2.1. Structural recursion operators and Gödel's T. -- 6.2.2. Conversion. -- 6.2.3. Corecursion. -- 6.2.4. A common extension T+ of Gödel's T and Plotkin's PCF. -- 6.2.5. Confluence. -- 6.2.6. Ideals as denotation of terms. -- 6.2.7. Preservation of values. -- 6.2.8. Operational semantics -- adequacy. -- 6.3. Normalization -- 6.3.1. Strong normalization. -- 6.3.2. Normalization by evaluation. -- 6.4. Computable functionals -- 6.4.1. Fixed point operators. -- 6.4.2. Rules for pcond, ∃ and valmax. -- 6.4.3. Plotkin's definability theorem. -- 6.5. Total functionals -- 6.5.1. Total and structure-total ideals. -- 6.5.2. Equality for total functionals. | |
| 880 | 0 | |6 505-00/(S |a Cover -- Proofs and Computations -- PERSPECTIVES IN LOGIC -- Title -- Copyright -- Dedication -- CONTENTS -- PREFACE -- PRELIMINARIES -- Part 1: BASIC PROOF THEORY AND COMPUTABILITY -- Chapter 1: LOGIC -- 1.1. Natural deduction -- 1.1.1. Terms and formulas. -- 1.1.2. Substitution, free and bound variables. -- 1.1.3. Subformulas. -- 1.1.4. Examples of derivations. -- 1.1.5. Introduction and elimination rules for → and ∀. -- 1.1.6. Properties of negation. -- 1.1.7. Introduction and elimination rules for disjunction ∨, conjunction ∧ and existence ∃. -- 1.1.8. Intuitionistic and classical derivability. -- 1.1.9. Gödel-Gentzen translation. -- 1.2. Normalization -- 1.2.1. The Curry-Howard correspondence. -- 1.2.2. Strong normalization. -- 1.2.3. Uniqueness of normal forms. -- 1.2.4. The structure of normal derivations. -- 1.2.5. Normal vs. non-normal derivations. -- 1.2.6. Conversions for ∨, ∧, ∃. -- 1.2.7. Strong normalization for β-, π- and σ-conversions. -- 1.2.8. The structure of normal derivations, again. -- 1.3. Soundness and completeness for tree models -- 1.3.1. Tree models. -- 1.3.2. Covering lemma. -- 1.3.3. Soundness. -- 1.3.4. Counter models. -- 1.3.5. Completeness. -- 1.4. Soundness and completeness of the classical fragment -- 1.4.1. Models. -- 1.4.2. Soundness of classical logic. -- 1.4.3. Completeness of classical logic. -- 1.4.4. Compactness and Löwenheim-Skolem theorems. -- 1.5. Tait calculus -- 1.6. Notes -- Chapter 2: RECURSION THEORY -- 2.1. Register machines -- 2.1.1. Programs. -- 2.1.2. Program constructs. -- 2.1.3. Register machine computable functions. -- 2.2. Elementary functions -- 2.2.1. Definition and simple properties. -- 2.2.2. Elementary relations. -- 2.2.3. The class ε. -- 2.2.4. Closure properties of ε. -- 2.2.5. Coding finite lists. -- 2.3. Kleene's normal form theorem -- 2.3.1. Program numbers. -- 2.3.2. Normal form. | |
| 880 | 8 | |6 505-00/(S |a 2.3.3. Στ̔̈ΒΑ·1-definable relations and μ-recursive functions. -- 2.3.4. Computable functions. -- 2.3.5. Undecidability of the halting problem. -- 2.4. Recursive definitions -- 2.4.1. Least fixed points of recursive definitions. -- 2.4.2. The principles of finite support and monotonicity, and the effective index property. -- 2.4.3. Recursion theorem. -- 2.4.4. Recursive programs and partial recursive functions. -- 2.4.5. Relativized recursion. -- 2.5. Primitive recursion and for-loops -- 2.5.1. Primitive recursive functions. -- 2.5.2. Loop-programs. -- 2.5.3. Reduction to primitive recursion. -- 2.5.4. A complexity hierarchy for Prim. -- 2.6. The arithmetical hierarchy -- 2.6.1. Kleene's second recursion theorem. -- 2.6.2. Characterization of Σ01-definable and recursive relations. -- 2.6.3. Arithmetical relations. -- 2.6.4. Closure properties. -- 2.6.6. Σ0r-complete relations. -- 2.7. The analytical hierarchy -- 2.7.1. Analytical relations. -- 2.7.2. Closure properties. -- 2.7.3. Universal Σ1r+1-definable relations. -- 2.7.4. Σ1r-complete relations. -- 2.8. Recursive type-2 functionals and well-foundedness -- 2.8.1. Computation trees. -- 2.8.2. Ordinal assignments -- recursive ordinals. -- 2.8.3. A hierarchy of total recursive functionals. -- 2.9. Inductive definitions -- 2.9.1. Monotone operators. -- 2.9.2. Induction and coinduction principles. -- 2.9.3. Approximation of the least and greatest fixed point. -- 2.9.4. Continuous operators. -- 2.9.5. The accessible part of a relation. -- 2.9.6. Inductive definitions over N. -- 2.9.7. Definability of least fixed points for monotone operators. -- 2.9.8. Some counter examples. -- 2.10. Notes -- Chapter 3: GÖDEL'S THEOREMS -- 3.1. IΔ0(exp) -- 3.1.1. Basic arithmetic in IΔ0(exp). -- 3.1.2. Provable recursion in IΔ0(exp). -- 3.1.3. Proof-theoretic characterization. -- 3.2. Gödel numbers. | |
| 880 | 8 | |6 505-00/(S |a 3.2.1. Gödel numbers of terms, formulas and derivations. -- 3.2.2. Elementary functions on Gödel numbers. -- 3.2.3. Axiomatized theories. -- 3.2.4. Undefinability of the notion of truth. -- 3.3. The notion of truth in formal theories -- 3.3.1. Representable relations and functions. -- 3.3.2. Undefinability of the notion of truth in formal theories. -- 3.4. Undecidability and incompleteness -- 3.4.1. Undecidability. -- 3.4.2. Incompleteness. -- 3.5. Representability -- 3.5.1. Weak arithmetical theories. -- 3.5.2. Robinson's theory Q. -- 3.5.3. Σ1ı-formulas. -- 3.6. Unprovability of consistency -- 3.6.1. Σ1ı-completeness. -- 3.6.2. Derivability conditions. -- 3.7. Notes -- Part 2: PROVABLE RECURSION IN CLASSICAL SYSTEMS -- Chapter 4: THE PROVABLY RECURSIVE FUNCTIONS OF ARITHMETIC -- 4.1. Primitive recursion and IΣı -- 4.1.1. Primitive recursive functions are provable in IΣı. -- 4.1.2. IΣı-provable functions are primitive recursive. -- 4.2. εο-recursion in Peano arithmetic -- 4.2.1. Ordinals below εο. -- 4.2.2. Introducing the fast-growing hierarchy. -- 4.2.3. α-recursion and εο-recursion. -- 4.2.4. Provable recursiveness of Hα and Fα. -- 4.2.5. Gentzen's theorem on transfinite induction in PA. -- 4.3. Ordinal bounds for provable recursion in PA -- 4.3.1. The infinitary system n : N α Γ. -- 4.3.2. Embedding of PA. -- 4.3.3. Cut elimination. -- 4.3.4. The classification theorem. -- 4.4. Independence results for PA -- 4.4.1. Goodstein sequences. -- 4.4.2. The modified finite Ramsey theorem. -- 4.5. Notes -- Chapter 5: ACCESSIBLE RECURSIVE FUNCTIONS, ID<∞ AND Π11-CA0 -- 5.1. The subrecursive stumblingblock -- 5.1.1. An old result of Myhill, Routledge and Liu. -- 5.1.2. Subrecursive hierarchies and constructive ordinals. -- 5.1.3. Incompleteness along Π11-paths through W. -- 5.2. Accessible recursive functions -- 5.2.1. Structured tree ordinals. | |
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