Proofs and Computations.:
This major graduate-level text provides a detailed, self-contained coverage of proof theory.
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Weitere Verfasser: | |
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Cambridge :
Cambridge University Press,
2011.
|
| Schriftenreihe: | Perspectives in logic.
|
| Schlagworte: | |
| Online-Zugang: | DE-862 DE-863 |
| Zusammenfassung: | This major graduate-level text provides a detailed, self-contained coverage of proof theory. |
| Beschreibung: | 1 online resource (482 pages) |
| Bibliographie: | Includes bibliographical references (pages 431-455) and index. |
| ISBN: | 9781139220200 1139220209 9786613579836 6613579831 9781139223638 1139223631 9781139031905 1139031902 9781139217101 1139217100 |
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| 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 | |
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| 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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| contents | 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 |
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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. 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S.</subfield><subfield code="0">http://id.loc.gov/authorities/names/n82129389</subfield></datafield><datafield tag="776" ind1="0" ind2="8"><subfield code="i">Print version:</subfield><subfield code="z">9780521517690</subfield></datafield><datafield tag="830" ind1=" " ind2="0"><subfield code="a">Perspectives in logic.</subfield><subfield code="0">http://id.loc.gov/authorities/names/no2009092095</subfield></datafield><datafield tag="966" ind1="4" ind2="0"><subfield code="l">DE-862</subfield><subfield code="p">ZDB-4-EBA</subfield><subfield code="q">FWS_PDA_EBA</subfield><subfield code="u">https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=416700</subfield><subfield code="3">Volltext</subfield></datafield><datafield tag="966" ind1="4" ind2="0"><subfield code="l">DE-863</subfield><subfield code="p">ZDB-4-EBA</subfield><subfield code="q">FWS_PDA_EBA</subfield><subfield code="u">https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=416700</subfield><subfield code="3">Volltext</subfield></datafield><datafield tag="880" ind1="8" ind2=" "><subfield code="6">505-01/Grek</subfield><subfield code="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.</subfield></datafield><datafield tag="880" ind1="0" ind2=" "><subfield code="6">505-00/(S</subfield><subfield code="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.</subfield></datafield><datafield tag="880" ind1="8" ind2=" "><subfield code="6">505-00/(S</subfield><subfield code="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.</subfield></datafield><datafield tag="880" ind1="8" ind2=" "><subfield code="6">505-00/(S</subfield><subfield code="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. 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| id | ZDB-4-EBA-ocn775869745 |
| illustrated | Not Illustrated |
| indexdate | 2025-04-11T08:37:35Z |
| institution | BVB |
| isbn | 9781139220200 1139220209 9786613579836 6613579831 9781139223638 1139223631 9781139031905 1139031902 9781139217101 1139217100 |
| language | English |
| oclc_num | 775869745 |
| open_access_boolean | |
| owner | MAIN DE-862 DE-BY-FWS DE-863 DE-BY-FWS |
| owner_facet | MAIN DE-862 DE-BY-FWS DE-863 DE-BY-FWS |
| physical | 1 online resource (482 pages) |
| psigel | ZDB-4-EBA FWS_PDA_EBA ZDB-4-EBA |
| publishDate | 2011 |
| publishDateSearch | 2011 |
| publishDateSort | 2011 |
| publisher | Cambridge University Press, |
| record_format | marc |
| series | Perspectives in logic. |
| series2 | Perspectives in Logic |
| spelling | Schwichtenberg, Helmut. Proofs and Computations. Cambridge : Cambridge University Press, 2011. 1 online resource (482 pages) text txt rdacontent computer c rdamedia online resource cr rdacarrier Perspectives in Logic Print version record. This major graduate-level text provides a detailed, self-contained coverage of proof theory. Includes bibliographical references (pages 431-455) and index. 880-01 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 Computable functions. http://id.loc.gov/authorities/subjects/sh85029469 Machine theory. http://id.loc.gov/authorities/subjects/sh85079341 Proof theory. http://id.loc.gov/authorities/subjects/sh85107437 Machine theory. Fonctions calculables. Théorie des automates. Théorie de la preuve. COMPUTERS Machine Theory. bisacsh Demostración, Teoría de la embucm Computable functions fast Machine theory fast Proof theory fast Wainer, S. S. http://id.loc.gov/authorities/names/n82129389 Print version: 9780521517690 Perspectives in logic. http://id.loc.gov/authorities/names/no2009092095 505-01/Grek 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. 505-00/(S 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. 505-00/(S 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. 505-00/(S 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. |
| spellingShingle | Schwichtenberg, Helmut Proofs and Computations. Perspectives in logic. 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 Computable functions. http://id.loc.gov/authorities/subjects/sh85029469 Machine theory. http://id.loc.gov/authorities/subjects/sh85079341 Proof theory. http://id.loc.gov/authorities/subjects/sh85107437 Machine theory. Fonctions calculables. Théorie des automates. Théorie de la preuve. COMPUTERS Machine Theory. bisacsh Demostración, Teoría de la embucm Computable functions fast Machine theory fast Proof theory fast |
| subject_GND | http://id.loc.gov/authorities/subjects/sh85029469 http://id.loc.gov/authorities/subjects/sh85079341 http://id.loc.gov/authorities/subjects/sh85107437 |
| title | Proofs and Computations. |
| title_auth | Proofs and Computations. |
| title_exact_search | Proofs and Computations. |
| title_full | Proofs and Computations. |
| title_fullStr | Proofs and Computations. |
| title_full_unstemmed | Proofs and Computations. |
| title_short | Proofs and Computations. |
| title_sort | proofs and computations |
| topic | Computable functions. http://id.loc.gov/authorities/subjects/sh85029469 Machine theory. http://id.loc.gov/authorities/subjects/sh85079341 Proof theory. http://id.loc.gov/authorities/subjects/sh85107437 Machine theory. Fonctions calculables. Théorie des automates. Théorie de la preuve. COMPUTERS Machine Theory. bisacsh Demostración, Teoría de la embucm Computable functions fast Machine theory fast Proof theory fast |
| topic_facet | Computable functions. Machine theory. Proof theory. Fonctions calculables. Théorie des automates. Théorie de la preuve. COMPUTERS Machine Theory. Demostración, Teoría de la Computable functions Machine theory Proof theory |
| work_keys_str_mv | AT schwichtenberghelmut proofsandcomputations AT wainerss proofsandcomputations |